check_quotas: avoid re-fetching object from object table

Reviewed-on: #646

.cargo/config.toml

@@ -0,0 +1,3 @@
+[target.x86_64-unknown-linux-gnu]
+linker = "clang"
+rustflags = ["-C", "link-arg=-fuse-ld=mold"]

.drone.yml

@@ -20,6 +20,7 @@ steps:
     image: nixpkgs/nix:nixos-22.05
     environment:
       GARAGE_TEST_INTEGRATION_EXE: result-bin/bin/garage
+      GARAGE_TEST_INTEGRATION_PATH: tmp-garage-integration
     commands:
       - nix-build --no-build-output --attr clippy.amd64 --argstr git_version ${DRONE_TAG:-$DRONE_COMMIT}
       - nix-build --no-build-output --attr test.amd64
@@ -31,8 +32,9 @@ steps:
       - ./result/bin/garage_util-*
       - ./result/bin/garage_web-*
       - ./result/bin/garage-*
-      - ./result/bin/integration-*
+      - ./result/bin/integration-* || (cat tmp-garage-integration/stderr.log; false)
       - rm result
+      - rm -rv tmp-garage-integration
 
   - name: integration tests
     image: nixpkgs/nix:nixos-22.05
@@ -63,11 +65,16 @@ steps:
       - nix-build --no-build-output --attr pkgs.amd64.release --argstr git_version ${DRONE_TAG:-$DRONE_COMMIT}
       - nix-shell --attr rust --run "./script/not-dynamic.sh result-bin/bin/garage"
 
-  - name: integration
+  - name: integration tests
     image: nixpkgs/nix:nixos-22.05
     commands:
       - nix-shell --attr integration --run ./script/test-smoke.sh || (cat /tmp/garage.log; false)
 
+  - name: upgrade tests
+    image: nixpkgs/nix:nixos-22.05
+    commands:
+      - nix-shell --attr integration --run "./script/test-upgrade.sh v0.8.4 x86_64-unknown-linux-musl" || (cat /tmp/garage.log; false)
+
   - name: push static binary
     image: nixpkgs/nix:nixos-22.05
     environment:
@@ -114,11 +121,16 @@ steps:
       - nix-build --no-build-output --attr pkgs.i386.release --argstr git_version ${DRONE_TAG:-$DRONE_COMMIT}
       - nix-shell --attr rust --run "./script/not-dynamic.sh result-bin/bin/garage"
 
-  - name: integration
+  - name: integration tests
     image: nixpkgs/nix:nixos-22.05
     commands:
       - nix-shell --attr integration --run ./script/test-smoke.sh || (cat /tmp/garage.log; false)
 
+  - name: upgrade tests
+    image: nixpkgs/nix:nixos-22.05
+    commands:
+      - nix-shell --attr integration --run "./script/test-upgrade.sh v0.8.4 i686-unknown-linux-musl" || (cat /tmp/garage.log; false)
+
   - name: push static binary
     image: nixpkgs/nix:nixos-22.05
     environment:
@@ -283,6 +295,6 @@ trigger:
 
 ---
 kind: signature
-hmac: ac09a5a8c82502f67271f93afa1e1e21ce66383b8e24a6deb26b285cc1c378ba
+hmac: 0c4b57eb4b27b7c6a6ff21ab87f0767fe3eb90f5d95d5cbcdccf794e9d2a5d86
 
 ...

Cargo.toml

@@ -18,14 +18,14 @@ default-members = ["src/garage"]
 
 [workspace.dependencies]
 format_table = { version = "0.1.1", path = "src/format-table" }
-garage_api = { version = "0.8.2", path = "src/api" }
-garage_block = { version = "0.8.2", path = "src/block" }
-garage_db = { version = "0.8.2", path = "src/db", default-features = false }
-garage_model = { version = "0.8.2", path = "src/model", default-features = false }
-garage_rpc = { version = "0.8.2", path = "src/rpc" }
-garage_table = { version = "0.8.2", path = "src/table" }
-garage_util = { version = "0.8.2", path = "src/util" }
-garage_web = { version = "0.8.2", path = "src/web" }
+garage_api = { version = "0.9.0", path = "src/api" }
+garage_block = { version = "0.9.0", path = "src/block" }
+garage_db = { version = "0.9.0", path = "src/db", default-features = false }
+garage_model = { version = "0.9.0", path = "src/model", default-features = false }
+garage_rpc = { version = "0.9.0", path = "src/rpc" }
+garage_table = { version = "0.9.0", path = "src/table" }
+garage_util = { version = "0.9.0", path = "src/util" }
+garage_web = { version = "0.9.0", path = "src/web" }
 k2v-client = { version = "0.0.4", path = "src/k2v-client" }
 
 [profile.dev]

doc/api/garage-admin-v0.yml

@@ -632,7 +632,7 @@ paths:
       operationId: "UpdateBucket"
       summary: "Update a bucket"
       description: |
-        All fields (`websiteAccess` and `quotas`) are optionnal.
+        All fields (`websiteAccess` and `quotas`) are optional.
         If they are present, the corresponding modifications are applied to the bucket, otherwise nothing is changed.
 
         In `websiteAccess`: if `enabled` is `true`, `indexDocument` must be specified.

doc/book/build/_index.md

@@ -1,6 +1,6 @@
 +++
 title = "Build your own app"
-weight = 4
+weight = 40
 sort_by = "weight"
 template = "documentation.html"
 +++

doc/book/connect/_index.md

@@ -1,6 +1,6 @@
 +++
 title = "Existing integrations"
-weight = 3
+weight = 30
 sort_by = "weight"
 template = "documentation.html"
 +++

doc/book/connect/apps/index.md

@@ -11,6 +11,7 @@ In this section, we cover the following web applications:
 | [Peertube](#peertube)     | ✅       | Supported with the website endpoint, proxifying private videos unsupported     |
 | [Mastodon](#mastodon)     | ✅       | Natively supported    |
 | [Matrix](#matrix)     | ✅       |  Tested with `synapse-s3-storage-provider`    |
+| [ejabberd](#ejabberd)     | ✅       |  `mod_s3_upload`    |
 | [Pixelfed](#pixelfed)     | ❓       |  Not yet tested    |
 | [Pleroma](#pleroma)     | ❓       |  Not yet tested    |
 | [Lemmy](#lemmy)     | ✅        |  Supported with pict-rs    |
@@ -36,7 +37,7 @@ Second, we suppose you have created a key and a bucket.
 As a reminder, you can create a key for your nextcloud instance as follow:
 
 ```bash
-garage key new --name nextcloud-key
+garage key create nextcloud-key
 ```
 
 Keep the Key ID and the Secret key in a pad, they will be needed later.  
@@ -138,7 +139,7 @@ a reasonable trade-off for some instances.
 Create a key for Peertube:
 
 ```bash
-garage key new --name peertube-key
+garage key create peertube-key
 ```
 
 Keep the Key ID and the Secret key in a pad, they will be needed later.  
@@ -252,7 +253,7 @@ As such, your Garage cluster should be configured appropriately for good perform
 This is the usual Garage setup:
 
 ```bash
-garage key new --name mastodon-key
+garage key create mastodon-key
 garage bucket create mastodon-data
 garage bucket allow mastodon-data --read --write --key mastodon-key
 ```
@@ -378,7 +379,7 @@ Supposing you have a working synapse installation, you can add the module with p
 Now create a bucket and a key for your matrix instance (note your Key ID and Secret Key somewhere, they will be needed later):
 
 ```bash
-garage key new --name matrix-key
+garage key create matrix-key
 garage bucket create matrix
 garage bucket allow matrix --read --write --key matrix-key
 ```
@@ -420,7 +421,7 @@ Now we can write a simple script (eg `~/.local/bin/matrix-cache-gc`):
 ## CONFIGURATION ##
 AWS_ACCESS_KEY_ID=GKxxx
 AWS_SECRET_ACCESS_KEY=xxxx
-S3_ENDPOINT=http://localhost:3900
+AWS_ENDPOINT_URL=http://localhost:3900
 S3_BUCKET=matrix
 MEDIA_STORE=/var/lib/matrix-synapse/media
 PG_USER=matrix
@@ -441,7 +442,7 @@ EOF
 
 s3_media_upload update-db 1d
 s3_media_upload --no-progress check-deleted $MEDIA_STORE
-s3_media_upload --no-progress upload $MEDIA_STORE $S3_BUCKET --delete --endpoint-url $S3_ENDPOINT
+s3_media_upload --no-progress upload $MEDIA_STORE $S3_BUCKET --delete --endpoint-url $AWS_ENDPOINT_URL
 ```
 
 This script will list all the medias that were not accessed in the 24 hours according to your database.
@@ -474,6 +475,52 @@ And add a new line. For example, to run it every 10 minutes:
 
 *External link:* [matrix-media-repo Documentation > S3](https://docs.t2bot.io/matrix-media-repo/configuration/s3-datastore.html)
 
+## ejabberd
+
+ejabberd is an XMPP server implementation which, with the `mod_s3_upload`
+module in the [ejabberd-contrib](https://github.com/processone/ejabberd-contrib)
+repository, can be integrated to store chat media files in Garage.
+
+For uploads, this module leverages presigned URLs - this allows XMPP clients to
+directly send media to Garage. Receiving clients then retrieve this media
+through the [static website](@/documentation/cookbook/exposing-websites.md)
+functionality.
+
+As the data itself is publicly accessible to someone with knowledge of the
+object URL - users are recommended to use
+[E2EE](@/documentation/cookbook/encryption.md) to protect this data-at-rest
+from unauthorized access.
+
+Install the module with:
+
+```bash
+ejabberdctl module_install mod_s3_upload
+```
+
+Create the required key and bucket with:
+
+```bash
+garage key new --name ejabberd
+garage bucket create objects.xmpp-server.fr
+garage bucket allow objects.xmpp-server.fr --read --write --key ejabberd
+garage bucket website --allow objects.xmpp-server.fr
+```
+
+The module can then be configured with:
+
+```
+  mod_s3_upload:
+    #bucket_url: https://objects.xmpp-server.fr.my-garage-instance.mydomain.tld
+    bucket_url: https://my-garage-instance.mydomain.tld/objects.xmpp-server.fr
+    access_key_id: GK...
+    access_key_secret: ...
+    region: garage
+    download_url: https://objects.xmpp-server.fr
+```
+
+Other configuration options can be found in the
+[configuration YAML file](https://github.com/processone/ejabberd-contrib/blob/master/mod_s3_upload/conf/mod_s3_upload.yml).
+
 ## Pixelfed
 
 [Pixelfed Technical Documentation > Configuration](https://docs.pixelfed.org/technical-documentation/env.html#filesystem)
@@ -539,7 +586,7 @@ secret_key = 'abcdef0123456789...'
 
 ```
 PICTRS__STORE__TYPE=object_storage
-PICTRS__STORE__ENDPOINT=http:/my-garage-instance.mydomain.tld:3900
+PICTRS__STORE__ENDPOINT=http://my-garage-instance.mydomain.tld:3900
 PICTRS__STORE__BUCKET_NAME=pictrs-data
 PICTRS__STORE__REGION=garage
 PICTRS__STORE__ACCESS_KEY=GK...

doc/book/connect/backup.md

@@ -54,7 +54,7 @@ how to configure this.
 Create your key and bucket:
 
 ```bash
-garage key new my-key
+garage key create my-key
 garage bucket create backup
 garage bucket allow backup --read --write --key my-key
 ```
@@ -105,6 +105,7 @@ restic restore 79766175 --target /var/lib/postgresql
 Restic has way more features than the ones presented here.
 You can discover all of them by accessing its documentation from the link below.
 
+Files on Android devices can also be backed up with [restic-android](https://github.com/lhns/restic-android).
 
 *External links:* [Restic Documentation > Amazon S3](https://restic.readthedocs.io/en/stable/030_preparing_a_new_repo.html#amazon-s3)
 

doc/book/connect/cli.md

@@ -70,16 +70,17 @@ Then a file named `~/.aws/config` and put:
 ```toml
 [default]
 region=garage
+endpoint_url=http://127.0.0.1:3900
 ```
 
 Now, supposing Garage is listening on `http://127.0.0.1:3900`, you can list your buckets with:
 
 ```bash
-aws --endpoint-url http://127.0.0.1:3900 s3 ls
+aws s3 ls
 ```
 
-Passing the `--endpoint-url` parameter to each command is annoying but AWS developers do not provide a corresponding configuration entry.
-As a workaround, you can redefine the aws command by editing the file `~/.bashrc`:
+If you're using awscli `<1.29.0` or `<2.13.0`, you need to pass `--endpoint-url` to each CLI invocation explicitly.
+As a workaround, you can redefine the aws command by editing the file `~/.bashrc` in this case:
 
 ```
 function aws { command aws --endpoint-url http://127.0.0.1:3900 $@ ; }

doc/book/connect/repositories.md

@@ -23,7 +23,7 @@ You can configure a different target for each data type (check `[lfs]` and `[att
 Let's start by creating a key and a bucket (your key id and secret will be needed later, keep them somewhere):
 
 ```bash
-garage key new --name gitea-key
+garage key create gitea-key
 garage bucket create gitea
 garage bucket allow gitea --read --write --key gitea-key
 ```
@@ -118,7 +118,7 @@ through another support, like a git repository.
 As a first step, we will need to create a bucket on Garage and enabling website access on it:
 
 ```bash
-garage key new --name nix-key
+garage key create nix-key
 garage bucket create nix.example.com
 garage bucket allow nix.example.com --read --write --key nix-key
 garage bucket website nix.example.com --allow

doc/book/cookbook/_index.md

@@ -1,7 +1,7 @@
 +++
 title="Cookbook"
 template = "documentation.html"
-weight = 2
+weight = 20
 sort_by = "weight"
 +++
 
@@ -37,7 +37,3 @@ This chapter could also be referred as "Tutorials" or "Best practices".
 - **[Monitoring Garage](@/documentation/cookbook/monitoring.md)** This page
   explains the Prometheus metrics available for monitoring the Garage
   cluster/nodes.
-
-- **[Recovering from failures](@/documentation/cookbook/recovering.md):** Garage's first selling point is resilience
-  to hardware failures. This section explains how to recover from such a failure in the
-  best possible way.

doc/book/cookbook/binary-packages.md

@@ -7,10 +7,23 @@ Garage is also available in binary packages on:
 
 ## Alpine Linux
 
+If you use Alpine Linux, you can simply install the
+[garage](https://pkgs.alpinelinux.org/packages?name=garage) package from the
+Alpine Linux repositories (available since v3.17):
+
 ```bash
-apk install garage
+apk add garage
 ```
 
+The default configuration file is installed to `/etc/garage.toml`. You can run
+Garage using: `rc-service garage start`. If you don't specify `rpc_secret`, it
+will be automatically replaced with a random string on the first start.
+
+Please note that this package is built without Consul discovery, Kubernetes
+discovery, OpenTelemetry exporter, and K2V features (K2V will be enabled once
+it's stable).
+
+
 ## Arch Linux
 
 Garage is available in the [AUR](https://aur.archlinux.org/packages/garage).

doc/book/cookbook/encryption.md

@@ -0,0 +1,116 @@
++++
+title = "Encryption"
+weight = 50
++++
+
+Encryption is a recurring subject when discussing Garage. 
+Garage does not handle data encryption by itself, but many things can
+already be done with Garage's current feature set and the existing ecosystem.
+
+This page takes a high level approach to security in general and data encryption
+in particular.
+
+
+# Examining your need for encryption
+
+- Why do you want encryption in Garage?
+
+- What is your threat model? What are you fearing?
+  - A stolen HDD?
+  - A curious administrator?
+  - A malicious administrator?
+  - A remote attacker?
+  - etc.
+
+- What services do you want to protect with encryption?
+  - An existing application? Which one? (eg. Nextcloud)
+  - An application that you are writing
+
+- Any expertise you may have on the subject
+
+This page explains what Garage provides, and how you can improve the situation by yourself
+by adding encryption at different levels.
+
+We would be very curious to know your needs and thougs about ideas such as
+encryption practices and things like key management, as we want Garage to be a
+serious base platform for the developpment of secure, encrypted applications.
+Do not hesitate to come talk to us if you have any thoughts or questions on the
+subject.
+
+
+# Capabilities provided by Garage
+
+## Traffic is encrypted between Garage nodes
+
+RPCs between Garage nodes are encrypted. More specifically, contrary to many
+distributed software, it is impossible in Garage to have clear-text RPC.  We
+use the [kuska handshake](https://github.com/Kuska-ssb/handshake) library which
+implements a protocol that has been clearly reviewed, Secure ScuttleButt's
+Secret Handshake protocol.  This is why setting a `rpc_secret` is mandatory,
+and that's also why your nodes have super long identifiers.
+
+## HTTP API endpoints provided by Garage are in clear text
+
+Adding TLS support built into Garage is not currently planned.
+
+## Garage stores data in plain text on the filesystem
+
+Garage does not handle data encryption at rest by itself, and instead delegates
+to the user to add encryption, either at the storage layer (LUKS, etc) or on
+the client side (or both). There are no current plans to add data encryption
+directly in Garage.
+
+Implementing data encryption directly in Garage might make things simpler for
+end users, but also raises many more questions, especially around key
+management: for encryption of data, where could Garage get the encryption keys
+from ? If we encrypt data but keep the keys in a plaintext file next to them,
+it's useless. We probably don't want to have to manage secrets in garage as it
+would be very hard to do in a secure way. Maybe integrate with an external
+system such as Hashicorp Vault?
+
+
+# Adding data encryption using external tools
+
+## Encrypting traffic between a Garage node and your client
+
+You have multiple options to have encryption between your client and a node:
+
+  - Setup a reverse proxy with TLS / ACME / Let's encrypt
+  - Setup a Garage gateway locally, and only contact the garage daemon on `localhost`
+  - Only contact your Garage daemon over a secure, encrypted overlay network such as Wireguard
+
+## Encrypting data at rest
+
+Protects against the following threats:
+
+- Stolen HDD
+
+Crucially, does not protect againt malicious sysadmins or remote attackers that
+might gain access to your servers.
+
+Methods include full-disk encryption with tools such as LUKS.
+
+## Encrypting data on the client side
+
+Protects againt the following threats:
+
+- A honest-but-curious administrator
+- A malicious administrator that tries to corrupt your data
+- A remote attacker that can read your server's data
+
+Implementations are very specific to the various applications. Examples:
+
+- Matrix: uses the OLM protocol for E2EE of user messages. Media files stored
+  in Matrix are probably encrypted using symmetric encryption, with a key that is
+  distributed in the end-to-end encrypted message that contains the link to the object.
+
+- XMPP: clients normally support either OMEMO / OpenPGP for the E2EE of user
+  messages. Media files are encrypted per
+  [XEP-0454](https://xmpp.org/extensions/xep-0454.html).
+
+- Aerogramme: use the user's password as a key to decrypt data in the user's bucket
+
+- Cyberduck: comes with support for
+  [Cryptomator](https://docs.cyberduck.io/cryptomator/) which allows users to
+  create client-side vaults to encrypt files in before they are uploaded to a
+  cloud storage endpoint.

doc/book/cookbook/monitoring.md

@@ -49,9 +49,5 @@ add the following lines in your Prometheus scrape config:
 To visualize the scraped data in Grafana,
 you can either import our [Grafana dashboard for Garage](https://git.deuxfleurs.fr/Deuxfleurs/garage/raw/branch/main/script/telemetry/grafana-garage-dashboard-prometheus.json)
 or make your own.
-We detail below the list of exposed metrics and their meaning.
 
-
-## List of exported metrics
-
-See our [dedicated page](@/documentation/reference-manual/monitoring.md) in the Reference manual section.
+The list of exported metrics is available on our [dedicated page](@/documentation/reference-manual/monitoring.md) in the Reference manual section.

doc/book/cookbook/real-world.md

@@ -19,9 +19,10 @@ To run a real-world deployment, make sure the following conditions are met:
 
 - You have at least three machines with sufficient storage space available.
 
-- Each machine has a public IP address which is reachable by other machines. It
-  is highly recommended that you use IPv6 for this end-to-end connectivity. If
-  IPv6 is not available, then using a mesh VPN such as
+- Each machine has an IP address which makes it directly reachable by all other machines.
+  In many cases, nodes will be behind a NAT and will not each have a public
+  IPv4 addresses. In this case, is recommended that you use IPv6 for this
+  end-to-end connectivity if it is available. Otherwise, using a mesh VPN such as
   [Nebula](https://github.com/slackhq/nebula) or
   [Yggdrasil](https://yggdrasil-network.github.io/) are approaches to consider
   in addition to building out your own VPN tunneling.
@@ -42,7 +43,7 @@ For our example, we will suppose the following infrastructure with IPv6 connecti
 | Brussels | Mars    | fc00:F::1  | 1.5 TB     |
 
 Note that Garage will **always** store the three copies of your data on nodes at different
-locations. This means that in the case of this small example, the available capacity
+locations. This means that in the case of this small example, the usable capacity
 of the cluster is in fact only 1.5 TB, because nodes in Brussels can't store more than that.
 This also means that nodes in Paris and London will be under-utilized.
 To make better use of the available hardware, you should ensure that the capacity
@@ -75,28 +76,23 @@ to store 2 TB of data in total.
 
 - For the metadata storage, Garage does not do checksumming and integrity
   verification on its own. If you are afraid of bitrot/data corruption,
-  put your metadata directory on a BTRFS partition. Otherwise, just use regular
+  put your metadata directory on a ZFS or BTRFS partition. Otherwise, just use regular
   EXT4 or XFS.
 
-- Having a single server with several storage drives is currently not very well
-  supported in Garage ([#218](https://git.deuxfleurs.fr/Deuxfleurs/garage/issues/218)).
-  For an easy setup, just put all your drives in a RAID0 or a ZFS RAIDZ array.
-  If you're adventurous, you can try to format each of your disk as
-  a separate XFS partition, and then run one `garage` daemon per disk drive,
-  or use something like [`mergerfs`](https://github.com/trapexit/mergerfs) to merge
-  all your disks in a single union filesystem that spreads load over them.
+- Servers with multiple HDDs are supported natively by Garage without resorting
+  to RAID, see [our dedicated documentation page](@/documentation/operations/multi-hdd.md).
 
 ## Get a Docker image
 
 Our docker image is currently named `dxflrs/garage` and is stored on the [Docker Hub](https://hub.docker.com/r/dxflrs/garage/tags?page=1&ordering=last_updated).
-We encourage you to use a fixed tag (eg. `v0.8.0`) and not the `latest` tag.
-For this example, we will use the latest published version at the time of the writing which is `v0.8.0` but it's up to you
+We encourage you to use a fixed tag (eg. `v0.9.0`) and not the `latest` tag.
+For this example, we will use the latest published version at the time of the writing which is `v0.9.0` but it's up to you
 to check [the most recent versions on the Docker Hub](https://hub.docker.com/r/dxflrs/garage/tags?page=1&ordering=last_updated).
 
 For example:
 
 ```
-sudo docker pull dxflrs/garage:v0.8.0
+sudo docker pull dxflrs/garage:v0.9.0
 ```
 
 ## Deploying and configuring Garage
@@ -161,12 +157,13 @@ docker run \
   -v /etc/garage.toml:/etc/garage.toml \
   -v /var/lib/garage/meta:/var/lib/garage/meta \
   -v /var/lib/garage/data:/var/lib/garage/data \
-  dxflrs/garage:v0.8.0
+  dxflrs/garage:v0.9.0
 ```
 
-It should be restarted automatically at each reboot.
-Please note that we use host networking as otherwise Docker containers
-can not communicate with IPv6.
+With this command line, Garage should be started automatically at each boot.
+Please note that we use host networking as otherwise the network indirection
+added by Docker would prevent Garage nodes from communicating with one another
+(especially if using IPv6).
 
 If you want to use `docker-compose`, you may use the following `docker-compose.yml` file as a reference:
 
@@ -174,7 +171,7 @@ If you want to use `docker-compose`, you may use the following `docker-compose.y
 version: "3"
 services:
   garage:
-    image: dxflrs/garage:v0.8.0
+    image: dxflrs/garage:v0.9.0
     network_mode: "host"
     restart: unless-stopped
     volumes:
@@ -183,10 +180,12 @@ services:
       - /var/lib/garage/data:/var/lib/garage/data
 ```
 
-Upgrading between Garage versions should be supported transparently,
-but please check the relase notes before doing so!
-To upgrade, simply stop and remove this container and
-start again the command with a new version of Garage.
+If you wish to upgrade your cluster, make sure to read the corresponding
+[documentation page](@/documentation/operations/upgrading.md) first, as well as
+the documentation relevant to your version of Garage in the case of major
+upgrades.  With the containerized setup proposed here, the upgrade process
+will require stopping and removing the existing container, and re-creating it
+with the upgraded version.
 
 ## Controling the daemon
 
@@ -197,6 +196,12 @@ The `garage` binary has two purposes:
 Ensure an appropriate `garage` binary (the same version as your Docker image) is available in your path.
 If your configuration file is at `/etc/garage.toml`, the `garage` binary should work with no further change.
 
+You can also use an alias as follows to use the Garage binary inside your docker container:
+
+```bash
+alias garage="docker exec -ti <container name> /garage"
+```
+
 You can test your `garage` CLI utility by running a simple command such as:
 
 ```bash
@@ -264,12 +269,12 @@ of a role that is assigned to each active cluster node.
 For our example, we will suppose we have the following infrastructure
 (Capacity, Identifier and Zone are specific values to Garage described in the following):
 
-| Location | Name    | Disk Space | `Capacity` | `Identifier` | `Zone` |
-|----------|---------|------------|------------|--------------|--------------|
-| Paris    | Mercury | 1 TB       | `10`       | `563e`     | `par1`       |
-| Paris    | Venus   | 2 TB       | `20`       | `86f0`     | `par1`       |
-| London   | Earth   | 2 TB       | `20`       | `6814`     | `lon1`       |
-| Brussels | Mars    | 1.5 TB     | `15`       | `212f`     | `bru1`       |
+| Location | Name    | Disk Space | Identifier | Zone (`-z`) | Capacity (`-c`) |
+|----------|---------|------------|------------|-------------|-----------------|
+| Paris    | Mercury | 1 TB       | `563e`     | `par1`      | `1T`            |
+| Paris    | Venus   | 2 TB       | `86f0`     | `par1`      | `2T`            |
+| London   | Earth   | 2 TB       | `6814`     | `lon1`      | `2T`            |
+| Brussels | Mars    | 1.5 TB     | `212f`     | `bru1`      | `1.5T`          |
 
 #### Node identifiers
 
@@ -291,6 +296,8 @@ garage status
 It will display the IP address associated with each node;
 from the IP address you will be able to recognize the node.
 
+We will now use the `garage layout assign` command to configure the correct parameters for each node.
+
 #### Zones
 
 Zones are simply a user-chosen identifier that identify a group of server that are grouped together logically.
@@ -300,29 +307,29 @@ In most cases, a zone will correspond to a geographical location (i.e. a datacen
 Behind the scene, Garage will use zone definition to try to store the same data on different zones,
 in order to provide high availability despite failure of a zone.
 
+Zones are passed to Garage using the `-z` flag of `garage layout assign` (see below).
+
 #### Capacity
 
-Garage reasons on an abstract metric about disk storage that is named the *capacity* of a node.
-The capacity configured in Garage must be proportional to the disk space dedicated to the node.
+Garage needs to know the storage capacity (disk space) it can/should use on
+each node, to be able to correctly balance data.
 
-Capacity values must be **integers** but can be given any signification.
-Here we chose that 1 unit of capacity = 100 GB.
+Capacity values are expressed in bytes and are passed to Garage using the `-c` flag of `garage layout assign` (see below).
 
-Note that the amount of data stored by Garage on each server may not be strictly proportional to
-its capacity value, as Garage will priorize having 3 copies of data in different zones,
-even if this means that capacities will not be strictly respected. For example in our above examples,
-nodes Earth and Mars will always store a copy of everything each, and the third copy will
-have 66% chance of being stored by Venus and 33% chance of being stored by Mercury.
+#### Tags
+
+You can add additional tags to nodes using the `-t` flag of `garage layout assign` (see below).
+Tags have no specific meaning for Garage and can be used at your convenience.
 
 #### Injecting the topology
 
 Given the information above, we will configure our cluster as follow:
 
 ```bash
-garage layout assign 563e -z par1 -c 10 -t mercury
-garage layout assign 86f0 -z par1 -c 20 -t venus
-garage layout assign 6814 -z lon1 -c 20 -t earth 
-garage layout assign 212f -z bru1 -c 15 -t mars 
+garage layout assign 563e -z par1 -c 1T -t mercury
+garage layout assign 86f0 -z par1 -c 2T -t venus
+garage layout assign 6814 -z lon1 -c 2T -t earth 
+garage layout assign 212f -z bru1 -c 1.5T -t mars 
 ```
 
 At this point, the changes in the cluster layout have not yet been applied.
@@ -332,6 +339,7 @@ To show the new layout that will be applied, call:
 garage layout show
 ```
 
+Make sure to read carefully the output of `garage layout show`.
 Once you are satisfied with your new layout, apply it with:
 
 ```bash
@@ -339,7 +347,7 @@ garage layout apply
 ```
 
 **WARNING:** if you want to use the layout modification commands in a script,
-make sure to read [this page](@/documentation/reference-manual/layout.md) first.
+make sure to read [this page](@/documentation/operations/layout.md) first.
 
 
 ## Using your Garage cluster

doc/book/cookbook/reverse-proxy.md

@@ -378,6 +378,47 @@ admin.garage.tld {
 But at the same time, the `reverse_proxy` is very flexible.
 For a production deployment, you should [read its documentation](https://caddyserver.com/docs/caddyfile/directives/reverse_proxy) as it supports features like DNS discovery of upstreams, load balancing with checks, streaming parameters, etc.
 
+### Caching
+
+Caddy can compiled with a
+[cache plugin](https://github.com/caddyserver/cache-handler) which can be used
+to provide a hot-cache at the webserver-level for static websites hosted by
+Garage.
+
+This can be configured as follows:
+
+```caddy
+# Caddy global configuration section
+{
+	# Bare minimum configuration to enable cache.
+	order cache before rewrite
+
+	cache
+
+	#cache
+	#	allowed_http_verbs GET
+	#	default_cache_control public
+	#	ttl 8h
+	#}
+}
+
+# Site specific section
+https:// {
+	cache
+
+	#cache {
+	#	timeout {
+	#		backend 30s
+	#	}
+	#}
+
+	reverse_proxy ...
+}
+```
+
+Caching is a complicated subject, and the reader is encouraged to study the
+available options provided by the plugin.
+
 ### On-demand TLS
 
 Caddy supports a technique called
@@ -428,3 +469,6 @@ https:// {
 	reverse_proxy localhost:3902 192.168.1.2:3902 example.tld:3902
 }
 ```
+
+More information on how this endpoint is implemented in Garage is available
+in the [Admin API Reference](@/documentation/reference-manual/admin-api.md) page.

doc/book/cookbook/systemd.md

@@ -33,7 +33,20 @@ NoNewPrivileges=true
 WantedBy=multi-user.target
 ```
 
-*A note on hardening: garage will be run as a non privileged user, its user id is dynamically allocated by systemd. It cannot access (read or write) home folders (/home, /root and /run/user), the rest of the filesystem can only be read but not written, only the path seen as /var/lib/garage is writable as seen by the service (mapped to /var/lib/private/garage on your host). Additionnaly, the process can not gain new privileges over time.*
+**A note on hardening:** Garage will be run as a non privileged user, its user
+id is dynamically allocated by systemd (set with `DynamicUser=true`). It cannot
+access (read or write) home folders (`/home`, `/root` and `/run/user`), the
+rest of the filesystem can only be read but not written, only the path seen as
+`/var/lib/garage` is writable as seen by the service. Additionnaly, the process
+can not gain new privileges over time.
+
+For this to work correctly, your `garage.toml` must be set with
+`metadata_dir=/var/lib/garage/meta` and `data_dir=/var/lib/garage/data`. This
+is mandatory to use the DynamicUser hardening feature of systemd, which
+autocreates these directories as virtual mapping. If the directory
+`/var/lib/garage` already exists before starting the server for the first time,
+the systemd service might not start correctly.  Note that in your host
+filesystem, Garage data will be held in `/var/lib/private/garage`.
 
 To start the service then automatically enable it at boot:
 

doc/book/design/_index.md

@@ -1,6 +1,6 @@
 +++
 title = "Design"
-weight = 6
+weight = 70
 sort_by = "weight"
 template = "documentation.html"
 +++

doc/book/design/goals.md

@@ -42,15 +42,13 @@ locations. They use Garage themselves for the following tasks:
 
 - As a [Matrix media backend](https://github.com/matrix-org/synapse-s3-storage-provider)
 
-- To store personal data and shared documents through [Bagage](https://git.deuxfleurs.fr/Deuxfleurs/bagage), a homegrown WebDav-to-S3 proxy
+- As a Nix binary cache
+
+- To store personal data and shared documents through [Bagage](https://git.deuxfleurs.fr/Deuxfleurs/bagage), a homegrown WebDav-to-S3 and SFTP-to-S3 proxy
+
+- As a backup target using `rclone` and `restic`
 
 - In the Drone continuous integration platform to store task logs
 
-- As a Nix binary cache
-
-- As a backup target using `rclone`
-
 The Deuxfleurs Garage cluster is a multi-site cluster currently composed of
-4 nodes in 2 physical locations. In the future it will be expanded to at
-least 3 physical locations to fully exploit Garage's potential for high
-availability.
+9 nodes in 3 physical locations.

doc/book/design/internals.md

@@ -61,7 +61,7 @@ Garage prioritizes which nodes to query according to a few criteria:
 
 
 For further reading on the cluster structure look at the [gateway](@/documentation/cookbook/gateways.md) 
-and [cluster layout management](@/documentation/reference-manual/layout.md) pages.
+and [cluster layout management](@/documentation/operations/layout.md) pages.
 
 ## Garbage collection
 

doc/book/development/_index.md

@@ -1,6 +1,6 @@
 +++
 title = "Development"
-weight = 7
+weight = 80
 sort_by = "weight"
 template = "documentation.html"
 +++

doc/book/operations/_index.md

@@ -0,0 +1,23 @@
++++
+title = "Operations & Maintenance"
+weight = 50
+sort_by = "weight"
+template = "documentation.html"
++++
+
+This section contains a number of important information on how to best operate a Garage cluster,
+to ensure integrity and availability of your data:
+
+- **[Upgrading Garage](@/documentation/operations/upgrading.md):** General instructions on how to
+  upgrade your cluster from one version to the next. Instructions specific for each version upgrade
+  can bef ound in the [working documents](@/documentation/working-documents/_index.md) section.
+
+- **[Layout management](@/documentation/operations/layout.md):** Best practices for using the `garage layout`
+  commands when adding or removing nodes from your cluster.
+
+- **[Durability and repairs](@/documentation/operations/durability-repairs.md):** How to check for small things
+  that might be going wrong, and how to recover from such failures.
+
+- **[Recovering from failures](@/documentation/operations/recovering.md):** Garage's first selling point is resilience
+  to hardware failures. This section explains how to recover from such a failure in the
+  best possible way.

doc/book/operations/durability-repairs.md

@@ -0,0 +1,126 @@
++++
+title = "Durability & Repairs"
+weight = 30
++++
+
+To ensure the best durability of your data and to fix any inconsistencies that may
+pop up in a distributed system, Garage provides a series of repair operations.
+This guide will explain the meaning of each of them and when they should be applied.
+
+
+# General syntax of repair operations
+
+Repair operations described below are of the form `garage repair <repair_name>`.
+These repairs will not launch without the `--yes` flag, which should
+be added as follows: `garage repair --yes <repair_name>`.
+By default these repair procedures will only run on the Garage node your CLI is
+connecting to. To run on all nodes, add the `-a` flag as follows:
+`garage repair -a --yes <repair_name>`.
+
+# Data block operations
+
+## Data store scrub
+
+Scrubbing the data store means examining each individual data block to check that
+their content is correct, by verifying their hash. Any block found to be corrupted
+(e.g. by bitrot or by an accidental manipulation of the datastore) will be
+restored from another node that holds a valid copy.
+
+Scrubs are automatically scheduled by Garage to run every 25-35 days (the
+actual time is randomized to spread load across nodes). The next scheduled run
+can be viewed with `garage worker get`.
+
+A scrub can also be launched manually using `garage repair scrub start`.
+
+To view the status of an ongoing scrub, first find the task ID of the scrub worker
+using `garage worker list`. Then, run `garage worker info <scrub_task_id>` to
+view detailed runtime statistics of the scrub. To gather cluster-wide information,
+this command has to be run on each individual node.
+
+A scrub is a very disk-intensive operation that might slow down your cluster.
+You may pause an ongoing scrub using `garage repair scrub pause`, but note that
+the scrub will resume automatically 24 hours later as Garage will not let your
+cluster run without a regular scrub. If the scrub procedure is too intensive
+for your servers and is slowing down your workload, the recommended solution
+is to increase the "scrub tranquility" using `garage repair scrub set-tranquility`.
+A higher tranquility value will make Garage take longer pauses between two block
+verifications. Of course, scrubbing the entire data store will also take longer.
+
+## Block check and resync
+
+In some cases, nodes hold a reference to a block but do not actually have the block
+stored on disk. Conversely, they may also have on disk blocks that are not referenced
+any more. To fix both cases, a block repair may be run with `garage repair blocks`.
+This will scan the entire block reference counter table to check that the blocks
+exist on disk, and will scan the entire disk store to check that stored blocks
+are referenced.
+
+It is recommended to run this procedure when changing your cluster layout,
+after the metadata tables have finished synchronizing between nodes
+(usually a few hours after `garage layout apply`).
+
+## Inspecting lost blocks
+
+In extremely rare situations, data blocks may be unavailable from the entire cluster.
+This means that even using `garage repair blocks`, some nodes may be unable
+to fetch data blocks for which they hold a reference.
+
+These errors are stored on each node in a list of "block resync errors", i.e.
+blocks for which the last resync operation failed.
+This list can be inspected using `garage block list-errors`.
+These errors usually fall into one of the following categories:
+
+1. a block is still referenced but the object was deleted, this is a case
+   of metadata reference inconsistency (see below for the fix)
+2. a block is referenced by a non-deleted object, but could not be fetched due
+   to a transient error such as a network failure
+3. a block is referenced by a non-deleted object, but could not be fetched due
+   to a permanent error such as there not being any valid copy of the block on the
+   entire cluster
+
+To help make the difference between cases 1 and cases 2 and 3, you may use the
+`garage block info` command to see which objects hold a reference to each block.
+
+In the second case (transient errors), Garage will try to fetch the block again
+after a certain time, so the error should disappear naturally. You can also
+request Garage to try to fetch the block immediately using `garage block retry-now`
+if you have fixed the transient issue.
+
+If you are confident that you are in the third scenario and that your data block
+is definitely lost, then there is no other choice than to declare your S3 objects
+as unrecoverable, and to delete them properly from the data store. This can be done
+using the `garage block purge` command.
+
+## Rebalancing data directories
+
+In [multi-HDD setups](@/documentation/operations/multi-hdd.md), to ensure that
+data blocks are well balanced between storage locations, you may run a
+rebalance operation using `garage repair rebalance`. This is usefull when
+adding storage locations or when capacities of the storage locations have been
+changed.  Once this is finished, Garage will know for each block of a single
+possible location where it can be, which can increase access speed.  This
+operation will also move out all data from locations marked as read-only.
+
+
+# Metadata operations
+
+## Metadata table resync
+
+Garage automatically resyncs all entries stored in the metadata tables every hour,
+to ensure that all nodes have the most up-to-date version of all the information
+they should be holding.
+The resync procedure is based on a Merkle tree that allows to efficiently find
+differences between nodes.
+
+In some special cases, e.g. before an upgrade, you might want to run a table
+resync manually. This can be done using `garage repair tables`.
+
+## Metadata table reference fixes
+
+In some very rare cases where nodes are unavailable, some references between objects
+are broken. For instance, if an object is deleted, the underlying versions or data
+blocks may still be held by Garage. If you suspect that such corruption has occurred
+in your cluster, you can run one of the following repair procedures:
+
+- `garage repair versions`: checks that all versions belong to a non-deleted object, and purges any orphan version
+- `garage repair block_refs`: checks that all block references belong to a non-deleted object version, and purges any orphan block reference (this will then allow the blocks to be garbage-collected)

doc/book/operations/layout.md

@@ -0,0 +1,274 @@
++++
+title = "Cluster layout management"
+weight = 20
++++
+
+The cluster layout in Garage is a table that assigns to each node a role in
+the cluster. The role of a node in Garage can either be a storage node with
+a certain capacity, or a gateway node that does not store data and is only
+used as an API entry point for faster cluster access.
+An introduction to building cluster layouts can be found in the [production deployment](@/documentation/cookbook/real-world.md) page.
+
+In Garage, all of the data that can be stored in a given cluster is divided
+into slices which we call *partitions*. Each partition is stored by
+one or several nodes in the cluster
+(see [`replication_mode`](@/documentation/reference-manual/configuration.md#replication-mode)).
+The layout determines the correspondence between these partition,
+which exist on a logical level, and actual storage nodes.
+
+## How cluster layouts work in Garage
+
+A cluster layout is composed of the following components:
+
+- a table of roles assigned to nodes, defined by the user
+- an optimal assignation of partitions to nodes, computed by an algorithm that is ran once when calling `garage layout apply` or the ApplyClusterLayout API endpoint
+- a version number
+
+Garage nodes will always use the cluster layout with the highest version number.
+
+Garage nodes also maintain and synchronize between them a set of proposed role
+changes that haven't yet been applied. These changes will be applied (or
+canceled) in the next version of the layout.
+
+All operations on the layout can be realized using the `garage` CLI or using the
+[administration API endpoint](@/documentation/reference-manual/admin-api.md).
+We give here a description of CLI commands, the admin API semantics are very similar.
+
+The following commands insert modifications to the set of proposed role changes
+for the next layout version (but they do not create the new layout immediately):
+
+```bash
+garage layout assign [...]
+garage layout remove [...]
+```
+
+The following command can be used to inspect the layout that is currently set in the cluster
+and the changes proposed for the next layout version, if any:
+
+```bash
+garage layout show
+```
+
+The following commands create a new layout with the specified version number,
+that either takes into account the proposed changes or cancels them:
+
+```bash
+garage layout apply --version <new_version_number>
+garage layout revert --version <new_version_number>
+```
+
+The version number of the new layout to create must be 1 + the version number
+of the previous layout that existed in the cluster.  The `apply` and `revert`
+commands will fail otherwise.
+
+## Warnings about Garage cluster layout management
+
+**⚠️ Never make several calls to `garage layout apply` or `garage layout
+revert` with the same value of the `--version` flag. Doing so can lead to the
+creation of several different layouts with the same version number, in which
+case your Garage cluster will become inconsistent until fixed.** If a call to
+`garage layout apply` or `garage layout revert` has failed and `garage layout
+show` indicates that a new layout with the given version number has not been
+set in the cluster, then it is fine to call the command again with the same
+version number.
+
+If you are using the `garage` CLI by typing individual commands in your
+shell, you shouldn't have much issues as long as you run commands one after
+the other and take care of checking the output of `garage layout show`
+before applying any changes.
+
+If you are using the `garage` CLI or the admin API to script layout changes,
+follow the following recommendations:
+
+- If using the CLI, make all of your `garage` CLI calls to the same RPC host.
+  If using the admin API, make all of your API calls to the same Garage node. Do
+  not connect to individual nodes to send them each a piece of the layout changes
+  you are making, as the changes propagate asynchronously between nodes and might
+  not all be taken into account at the time when the new layout is applied.
+
+- **Only call `garage layout apply`/ApplyClusterLayout once**, and call it
+  **strictly after** all of the `layout assign` and `layout remove`
+  commands/UpdateClusterLayout API calls have returned.
+
+
+## Understanding unexpected layout calculations
+
+When adding, removing or modifying nodes in a cluster layout, sometimes
+unexpected assigntations of partitions to node can occur. These assignations
+are in fact normal and logical, given the objectives of the algorihtm.  Indeed,
+**the layout algorithm prioritizes moving less data between nodes over the fact
+of achieving equal distribution of load. It also tries to use all links between
+pairs of nodes in equal proportions when moving data.**  This section presents
+two examples and illustrates how one can control Garage's behavior to obtain
+the desired results.
+
+### Example 1
+
+In this example, a cluster is originally composed of 3 nodes in 3 different
+zones (data centers).  The three nodes are of equal capacity, therefore they
+are all fully exploited and all store a copy of all of the data in the cluster.
+
+Then, a fourth node of the same size is added in the datacenter `dc1`.
+As illustrated by the following, **Garage will by default not store any data on the new node**:
+
+```
+$ garage layout show
+==== CURRENT CLUSTER LAYOUT ====
+ID                Tags   Zone  Capacity   Usable capacity
+b10c110e4e854e5a  node1  dc1   1000.0 MB  1000.0 MB (100.0%)
+a235ac7695e0c54d  node2  dc2   1000.0 MB  1000.0 MB (100.0%)
+62b218d848e86a64  node3  dc3   1000.0 MB  1000.0 MB (100.0%)
+
+Zone redundancy: maximum
+
+Current cluster layout version: 6
+
+==== STAGED ROLE CHANGES ====
+ID                Tags   Zone  Capacity
+a11c7cf18af29737  node4  dc1   1000.0 MB
+
+
+==== NEW CLUSTER LAYOUT AFTER APPLYING CHANGES ====
+ID                Tags   Zone  Capacity   Usable capacity
+b10c110e4e854e5a  node1  dc1   1000.0 MB  1000.0 MB (100.0%)
+a11c7cf18af29737  node4  dc1   1000.0 MB  0 B (0.0%)
+a235ac7695e0c54d  node2  dc2   1000.0 MB  1000.0 MB (100.0%)
+62b218d848e86a64  node3  dc3   1000.0 MB  1000.0 MB (100.0%)
+
+Zone redundancy: maximum
+
+==== COMPUTATION OF A NEW PARTITION ASSIGNATION ====
+
+Partitions are replicated 3 times on at least 3 distinct zones.
+
+Optimal partition size:                     3.9 MB (3.9 MB in previous layout)
+Usable capacity / total cluster capacity:   3.0 GB / 4.0 GB (75.0 %)
+Effective capacity (replication factor 3):  1000.0 MB
+
+A total of 0 new copies of partitions need to be transferred.
+
+dc1                 Tags   Partitions        Capacity   Usable capacity
+  b10c110e4e854e5a  node1  256 (0 new)       1000.0 MB  1000.0 MB (100.0%)
+  a11c7cf18af29737  node4  0 (0 new)         1000.0 MB  0 B (0.0%)
+  TOTAL                    256 (256 unique)  2.0 GB     1000.0 MB (50.0%)
+
+dc2                 Tags   Partitions        Capacity   Usable capacity
+  a235ac7695e0c54d  node2  256 (0 new)       1000.0 MB  1000.0 MB (100.0%)
+  TOTAL                    256 (256 unique)  1000.0 MB  1000.0 MB (100.0%)
+
+dc3                 Tags   Partitions        Capacity   Usable capacity
+  62b218d848e86a64  node3  256 (0 new)       1000.0 MB  1000.0 MB (100.0%)
+  TOTAL                    256 (256 unique)  1000.0 MB  1000.0 MB (100.0%)
+```
+
+While unexpected, this is logical because of the following facts:
+
+- storing some data on the new node does not help increase the total quantity
+  of data that can be stored on the cluster, as the two other zones (`dc2` and
+  `dc3`) still need to store a full copy of everything, and their capacity is
+  still the same;
+
+- there is therefore no need to move any data on the new node as this would be pointless;
+
+- moving data to the new node has a cost which the algorithm decides to not pay if not necessary.
+
+This distribution of data can however not be what the administrator wanted: if
+they added a new node to `dc1`, it might be because the existing node is too
+slow, and they wish to divide its load by half. In that case, what they need to
+do to force Garage to distribute the data between the two nodes is to attribute
+only half of the capacity to each node in `dc1` (in our example, 500M instead of 1G).
+In that case, Garage would determine that to be able to store 1G in total, it
+would need to store 500M on the old node and 500M on the added one.
+
+
+### Example 2
+
+The following example is a slightly different scenario, where `dc1` had two
+nodes that were used at 50%, and `dc2` and `dc3` each have one node that is
+100% used. All node capacities are the same.
+
+Then, a node from `dc1` is moved into `dc3`. One could expect that the roles of
+`dc1` and `dc3` would simply be swapped: the remaining node in `dc1` would be
+used at 100%, and the two nodes now in `dc3` would be used at 50%. Instead,
+this happens:
+
+```
+==== CURRENT CLUSTER LAYOUT ====
+ID                Tags   Zone  Capacity   Usable capacity
+b10c110e4e854e5a  node1  dc1   1000.0 MB  500.0 MB (50.0%)
+a11c7cf18af29737  node4  dc1   1000.0 MB  500.0 MB (50.0%)
+a235ac7695e0c54d  node2  dc2   1000.0 MB  1000.0 MB (100.0%)
+62b218d848e86a64  node3  dc3   1000.0 MB  1000.0 MB (100.0%)
+
+Zone redundancy: maximum
+
+Current cluster layout version: 8
+
+==== STAGED ROLE CHANGES ====
+ID                Tags   Zone  Capacity
+a11c7cf18af29737  node4  dc3   1000.0 MB
+
+
+==== NEW CLUSTER LAYOUT AFTER APPLYING CHANGES ====
+ID                Tags   Zone  Capacity   Usable capacity
+b10c110e4e854e5a  node1  dc1   1000.0 MB  1000.0 MB (100.0%)
+a235ac7695e0c54d  node2  dc2   1000.0 MB  1000.0 MB (100.0%)
+62b218d848e86a64  node3  dc3   1000.0 MB  753.9 MB (75.4%)
+a11c7cf18af29737  node4  dc3   1000.0 MB  246.1 MB (24.6%)
+
+Zone redundancy: maximum
+
+==== COMPUTATION OF A NEW PARTITION ASSIGNATION ====
+
+Partitions are replicated 3 times on at least 3 distinct zones.
+
+Optimal partition size:                     3.9 MB (3.9 MB in previous layout)
+Usable capacity / total cluster capacity:   3.0 GB / 4.0 GB (75.0 %)
+Effective capacity (replication factor 3):  1000.0 MB
+
+A total of 128 new copies of partitions need to be transferred.
+
+dc1                 Tags   Partitions        Capacity   Usable capacity
+  b10c110e4e854e5a  node1  256 (128 new)     1000.0 MB  1000.0 MB (100.0%)
+  TOTAL                    256 (256 unique)  1000.0 MB  1000.0 MB (100.0%)
+
+dc2                 Tags   Partitions        Capacity   Usable capacity
+  a235ac7695e0c54d  node2  256 (0 new)       1000.0 MB  1000.0 MB (100.0%)
+  TOTAL                    256 (256 unique)  1000.0 MB  1000.0 MB (100.0%)
+
+dc3                 Tags   Partitions        Capacity   Usable capacity
+  62b218d848e86a64  node3  193 (0 new)       1000.0 MB  753.9 MB (75.4%)
+  a11c7cf18af29737  node4  63 (0 new)        1000.0 MB  246.1 MB (24.6%)
+  TOTAL                    256 (256 unique)  2.0 GB     1000.0 MB (50.0%)
+```
+
+As we can see, the node that was moved to `dc3` (node4) is only used at 25% (approximatively),
+whereas the node that was already in `dc3` (node3) is used at 75%.
+
+This can be explained by the following:
+
+- node1 will now be the only node remaining in `dc1`, thus it has to store all
+  of the data in the cluster. Since it was storing only half of it before, it has
+  to retrieve the other half from other nodes in the cluster.
+
+- The data which it does not have is entirely stored by the other node that was
+  in `dc1` and that is now in `dc3` (node4). There is also a copy of it on node2
+  and node3 since both these nodes have a copy of everything.
+
+- node3 and node4 are the two nodes that will now be in a datacenter that is
+ under-utilized (`dc3`), this means that those are the two candidates from which
+ data can be removed to be moved to node1.
+
+- Garage will move data in equal proportions from all possible sources, in this
+  case it means that it will tranfer 25% of the entire data set from node3 to
+  node1 and another 25% from node4 to node1.
+
+This explains why node3 ends with 75% utilization (100% from before minus 25%
+that is moved to node1), and node4 ends with 25% (50% from before minus 25%
+that is moved to node1).
+
+This illustrates the second principle of the layout computation: **if there is
+a choice in moving data out of some nodes, then all links between pairs of
+nodes are used in equal proportions** (this is approximately true, there is
+randomness in the algorihtm to achieve this so there might be some small
+fluctuations, as we see above).

doc/book/operations/multi-hdd.md

@@ -0,0 +1,101 @@
++++
+title = "Multi-HDD support"
+weight = 15
++++
+
+
+Since v0.9, Garage natively supports nodes that have several storage drives
+for storing data blocks (not for metadata storage).
+
+## Initial setup
+
+To set up a new Garage storage node with multiple HDDs,
+format and mount all your drives in different directories,
+and use a Garage configuration as follows:
+
+```toml
+data_dir = [
+    { path = "/path/to/hdd1", capacity = "2T" },
+    { path = "/path/to/hdd2", capacity = "4T" },
+]
+```
+
+Garage will automatically balance all blocks stored by the node
+among the different specified directories, proportionnally to the
+specified capacities.
+
+## Updating the list of storage locations
+
+If you add new storage locations to your `data_dir`,
+Garage will not rebalance existing data between storage locations.
+Newly written blocks will be balanced proportionnally to the specified capacities,
+and existing data may be moved between drives to improve balancing,
+but only opportunistically when a data block is re-written (e.g. an object
+is re-uploaded, or an object with a duplicate block is uploaded).
+
+To understand precisely what is happening, we need to dive in to how Garage
+splits data among the different storage locations.
+
+First of all, Garage divides the set of all possible block hashes
+in a fixed number of slices (currently 1024), and assigns
+to each slice a primary storage location among the specified data directories.
+The number of slices having their primary location in each data directory
+is proportionnal to the capacity specified in the config file.
+
+When Garage receives a block to write, it will always write it in the primary
+directory of the slice that contains its hash.
+
+Now, to be able to not lose existing data blocks when storage locations
+are added, Garage also keeps a list of secondary data directories
+for all of the hash slices. Secondary data directories for a slice indicates
+storage locations that once were primary directories for that slice, i.e. where
+Garage knows that data blocks of that slice might be stored.
+When Garage is requested to read a certain data block,
+it will first look in the primary storage directory of its slice,
+and if it doesn't find it there it goes through all of the secondary storage
+locations until it finds it. This allows Garage to continue operating
+normally when storage locations are added, without having to shuffle
+files between drives to place them in the correct location.
+
+This relatively simple strategy works well but does not ensure that data
+is correctly balanced among drives according to their capacity.
+To rebalance data, two strategies can be used:
+
+- Lazy rebalancing: when a block is re-written (e.g. the object is re-uploaded),
+  Garage checks whether the existing copy is in the primary directory of the slice
+  or in a secondary directory. If the current copy is in a secondary directory,
+  Garage re-writes a copy in the primary directory and deletes the one from the
+  secondary directory. This might never end up rebalancing everything if there
+  are data blocks that are only read and never written.
+
+- Active rebalancing: an operator of a Garage node can explicitly launch a repair
+  procedure that rebalances the data directories, moving all blocks to their
+  primary location. Once done, all secondary locations for all hash slices are
+  removed so that they won't be checked anymore when looking for a data block.
+
+## Read-only storage locations
+
+If you would like to move all data blocks from an existing data directory to one
+or several new data directories, mark the old directory as read-only:
+
+```toml
+data_dir = [
+    { path = "/path/to/old_data", read_only = true },
+    { path = "/path/to/new_hdd1", capacity = "2T" },
+    { path = "/path/to/new_hdd2", capacity = "4T" },
+]
+```
+
+Garage will be able to read requested blocks from the read-only directory.
+Garage will also move data out of the read-only directory either progressively
+(lazy rebalancing) or if requested explicitly (active rebalancing).
+
+Once an active rebalancing has finished, your read-only directory should be empty:
+it might still contain subdirectories, but no data files. You can check that
+it contains no files using:
+
+```bash
+find -type f /path/to/old_data      # should not print anything
+```
+
+at which point it can be removed from the `data_dir` list in your config file.

doc/book/operations/recovering.md

@@ -1,6 +1,6 @@
 +++
 title = "Recovering from failures"
-weight = 50
+weight = 40
 +++
 
 Garage is meant to work on old, second-hand hardware.

doc/book/operations/upgrading.md

@@ -1,6 +1,6 @@
 +++
 title = "Upgrading Garage"
-weight = 60
+weight = 10
 +++
 
 Garage is a stateful clustered application, where all nodes are communicating together and share data structures.
@@ -58,7 +58,7 @@ From a high level perspective, a major upgrade looks like this:
 
 ### Major upgarades with minimal downtime
 
-There is only one operation that has to be coordinated cluster-wide: the passage of one version of the internal RPC protocol to the next.
+There is only one operation that has to be coordinated cluster-wide: the switch of one version of the internal RPC protocol to the next.
 This means that an upgrade with very limited downtime can simply be performed from one major version to the next by restarting all nodes
 simultaneously in the new version.
 The downtime will simply be the time required for all nodes to stop and start again, which should be less than a minute.
@@ -80,6 +80,6 @@ The entire procedure would look something like this:
 5. If any specific migration procedure is required, it is usually in one of the two cases:
 
   - It can be run on online nodes after the new version has started, during regular cluster operation.
-  - it has to be run offline
+  - it has to be run offline, in which case you will have to again take all nodes offline one after the other to run the repair
 
    For this last step, please refer to the specific documentation pertaining to the version upgrade you are doing.

doc/book/quick-start/_index.md

@@ -1,6 +1,6 @@
 +++
 title = "Quick Start"
-weight = 0
+weight = 10
 sort_by = "weight"
 template = "documentation.html"
 +++
@@ -35,6 +35,9 @@ Place this binary somewhere in your `$PATH` so that you can invoke the `garage`
 command directly (for instance you can copy the binary in `/usr/local/bin`
 or in `~/.local/bin`).
 
+You may also check whether your distribution already includes a
+[binary package for Garage](@/documentation/cookbook/binary-packages.md).
+
 If a binary of the last version is not available for your architecture,
 or if you want a build customized for your system,
 you can [build Garage from source](@/documentation/cookbook/from-source.md).
@@ -81,9 +84,8 @@ admin_token = "$(openssl rand -base64 32)"
 EOF
 ```
 
-Now that your configuration file has been created, you can put
-it in the right place. By default, garage looks at **`/etc/garage.toml`.**
-
+Now that your configuration file has been created, you may save it to the directory of your choice.
+By default, Garage looks for **`/etc/garage.toml`.**
 You can also store it somewhere else, but you will have to specify `-c path/to/garage.toml`
 at each invocation of the `garage` binary (for example: `garage -c ./garage.toml server`, `garage -c ./garage.toml status`).
 
@@ -100,12 +102,14 @@ your data to be persisted properly.
 
 ### Launching the Garage server
 
-Use the following command to launch the Garage server with our configuration file:
+Use the following command to launch the Garage server:
 
 ```
-garage server
+garage -c path/to/garage.toml server
 ```
 
+If you have placed the `garage.toml` file in `/etc` (its default location), you can simply run `garage server`.
+
 You can tune Garage's verbosity as follows (from less verbose to more verbose):
 
 ```
@@ -123,7 +127,7 @@ Log level `debug` can help you check why your S3 API calls are not working.
 The `garage` utility is also used as a CLI tool to configure your Garage deployment.
 It uses values from the TOML configuration file to find the Garage daemon running on the
 local node, therefore if your configuration file is not at `/etc/garage.toml` you will
-again have to specify `-c path/to/garage.toml`.
+again have to specify `-c path/to/garage.toml` at each invocation.
 
 If the `garage` CLI is able to correctly detect the parameters of your local Garage node,
 the following command should be enough to show the status of your cluster:
@@ -137,7 +141,7 @@ This should show something like this:
 ```
 ==== HEALTHY NODES ====
 ID                 Hostname  Address         Tag                   Zone  Capacity
-563e1ac825ee3323…  linuxbox  127.0.0.1:3901  NO ROLE ASSIGNED
+563e1ac825ee3323   linuxbox  127.0.0.1:3901  NO ROLE ASSIGNED
 ```
 
 ## Creating a cluster layout
@@ -150,12 +154,12 @@ For our test deployment, we are using only one node. The way in which we configu
 it does not matter, you can simply write:
 
 ```bash
-garage layout assign -z dc1 -c 1 <node_id>
+garage layout assign -z dc1 -c 1G <node_id>
 ```
 
 where `<node_id>` corresponds to the identifier of the node shown by `garage status` (first column).
 You can enter simply a prefix of that identifier.
-For instance here you could write just `garage layout assign -z dc1 -c 1 563e`.
+For instance here you could write just `garage layout assign -z dc1 -c 1G 563e`.
 
 The layout then has to be applied to the cluster, using:
 
@@ -206,7 +210,7 @@ one key can access multiple buckets, multiple keys can access one bucket.
 Create an API key using the following command:
 
 ```
-garage key new --name nextcloud-app-key
+garage key create nextcloud-app-key
 ```
 
 The output should look as follows:
@@ -266,12 +270,14 @@ named `~/.awsrc` with this content:
 export AWS_ACCESS_KEY_ID=xxxx      # put your Key ID here
 export AWS_SECRET_ACCESS_KEY=xxxx  # put your Secret key here
 export AWS_DEFAULT_REGION='garage'
-export AWS_ENDPOINT='http://localhost:3900'
+export AWS_ENDPOINT_URL='http://localhost:3900'
 
-function aws { command aws --endpoint-url $AWS_ENDPOINT $@ ; }
 aws --version
 ```
 
+Note you need to have at least `awscli` `>=1.29.0` or `>=2.13.0`, otherwise you
+need to specify `--endpoint-url` explicitly on each `awscli` invocation.
+
 Now, each time you want to use `awscli` on this target, run:
 
 ```bash

doc/book/reference-manual/_index.md

@@ -1,6 +1,6 @@
 +++
 title = "Reference Manual"
-weight = 5
+weight = 60
 sort_by = "weight"
 template = "documentation.html"
 +++

doc/book/reference-manual/admin-api.md

@@ -39,11 +39,95 @@ Authorization: Bearer <token>
 
 ## Administration API endpoints
 
-### Metrics-related endpoints
-
-#### Metrics `GET /metrics`
+### Metrics `GET /metrics`
 
 Returns internal Garage metrics in Prometheus format.
+The metrics are directly documented when returned by the API.
+
+**Example:**
+
+```
+$ curl -i http://localhost:3903/metrics
+HTTP/1.1 200 OK
+content-type: text/plain; version=0.0.4
+content-length: 12145
+date: Tue, 08 Aug 2023 07:25:05 GMT
+
+# HELP api_admin_error_counter Number of API calls to the various Admin API endpoints that resulted in errors
+# TYPE api_admin_error_counter counter
+api_admin_error_counter{api_endpoint="CheckWebsiteEnabled",status_code="400"} 1
+api_admin_error_counter{api_endpoint="CheckWebsiteEnabled",status_code="404"} 3
+# HELP api_admin_request_counter Number of API calls to the various Admin API endpoints
+# TYPE api_admin_request_counter counter
+api_admin_request_counter{api_endpoint="CheckWebsiteEnabled"} 7
+api_admin_request_counter{api_endpoint="Health"} 3
+# HELP api_admin_request_duration Duration of API calls to the various Admin API endpoints
+...
+```
+
+### Health `GET /health`
+
+Returns `200 OK` if enough nodes are up to have a quorum (ie. serve requests),
+otherwise returns `503 Service Unavailable`.
+
+**Example:**
+
+```
+$ curl -i http://localhost:3903/health
+HTTP/1.1 200 OK
+content-type: text/plain
+content-length: 102
+date: Tue, 08 Aug 2023 07:22:38 GMT
+
+Garage is fully operational
+Consult the full health check API endpoint at /v0/health for more details
+```
+
+### On-demand TLS `GET /check`
+
+To prevent abuses for on-demand TLS, Caddy developpers have specified an endpoint that can be queried by the reverse proxy
+to know if a given domain is allowed to get a certificate. Garage implements this endpoints to tell if a given domain is handled by Garage or is garbage.
+
+Garage responds with the following logic:
+ - If the domain matches the pattern `<bucket-name>.<s3_api.root_domain>`, returns 200 OK
+ - If the domain matches the pattern `<bucket-name>.<s3_web.root_domain>` and website is configured for `<bucket>`, returns 200 OK
+ - If the domain matches the pattern `<bucket-name>` and website is configured for `<bucket>`, returns 200 OK
+ - Otherwise, returns 404 Not Found, 400 Bad Request or 5xx requests.
+
+*Note 1: because in the path-style URL mode, there is only one domain that is not known by Garage, hence it is not supported by this API endpoint.
+You must manually declare the domain in your reverse-proxy. Idem for K2V.*
+
+*Note 2: buckets in a user's namespace are not supported yet by this endpoint. This is a limitation of this endpoint currently.*
+
+**Example:** Suppose a Garage instance configured with `s3_api.root_domain = .s3.garage.localhost` and `s3_web.root_domain = .web.garage.localhost`.
+
+With a private `media` bucket (name in the global namespace, website is disabled), the endpoint will feature the following behavior:
+
+```
+$ curl -so /dev/null -w "%{http_code}" http://localhost:3903/check?domain=media.s3.garage.localhost
+200
+$ curl -so /dev/null -w "%{http_code}" http://localhost:3903/check?domain=media
+400
+$ curl -so /dev/null -w "%{http_code}" http://localhost:3903/check?domain=media.web.garage.localhost
+400
+```
+
+With a public `example.com` bucket (name in the global namespace, website is activated), the endpoint will feature the following behavior:
+
+```
+$ curl -so /dev/null -w "%{http_code}" http://localhost:3903/check?domain=example.com.s3.garage.localhost
+200
+$ curl -so /dev/null -w "%{http_code}" http://localhost:3903/check?domain=example.com
+200
+$ curl -so /dev/null -w "%{http_code}" http://localhost:3903/check?domain=example.com.web.garage.localhost
+200
+```
+
+
+**References:**
+ - [Using On-Demand TLS](https://caddyserver.com/docs/automatic-https#using-on-demand-tls)
+ - [Add option for a backend check to approve use of on-demand TLS](https://github.com/caddyserver/caddy/pull/1939)
+ - [Serving tens of thousands of domains over HTTPS with Caddy](https://caddy.community/t/serving-tens-of-thousands-of-domains-over-https-with-caddy/11179)
 
 ### Cluster operations
 

doc/book/reference-manual/configuration.md

@@ -10,13 +10,16 @@ Here is an example `garage.toml` configuration file that illustrates all of the
 ```toml
 metadata_dir = "/var/lib/garage/meta"
 data_dir = "/var/lib/garage/data"
+metadata_fsync = true
+data_fsync = false
 
 db_engine = "lmdb"
 
 block_size = 1048576
 
-sled_cache_capacity = 134217728
+sled_cache_capacity = "128MiB"
 sled_flush_every_ms = 2000
+lmdb_map_size = "1T"
 
 replication_mode = "3"
 
@@ -89,6 +92,19 @@ This folder can be placed on an HDD. The space available for `data_dir`
 should be counted to determine a node's capacity
 when [adding it to the cluster layout](@/documentation/cookbook/real-world.md).
 
+Since `v0.9.0`, Garage supports multiple data directories with the following syntax:
+
+```toml
+data_dir = [
+    { path = "/path/to/old_data", read_only = true },
+    { path = "/path/to/new_hdd1", capacity = "2T" },
+    { path = "/path/to/new_hdd2", capacity = "4T" },
+]
+```
+
+See [the dedicated documentation page](@/documentation/operations/multi-hdd.md)
+on how to operate Garage in such a setup.
+
 ### `db_engine` (since `v0.8.0`)
 
 By default, Garage uses the Sled embedded database library
@@ -130,11 +146,54 @@ convert-db -a <input db engine> -i <input db path> \
 Make sure to specify the full database path as presented in the table above,
 and not just the path to the metadata directory.
 
+### `metadata_fsync`
+
+Whether to enable synchronous mode for the database engine or not.
+This is disabled (`false`) by default.
+
+This reduces the risk of metadata corruption in case of power failures,
+at the cost of a significant drop in write performance,
+as Garage will have to pause to sync data to disk much more often
+(several times for API calls such as PutObject).
+
+Using this option reduces the risk of simultaneous metadata corruption on several
+cluster nodes, which could lead to data loss.
+
+If multi-site replication is used, this option is most likely not necessary, as
+it is extremely unlikely that two nodes in different locations will have a 
+power failure at the exact same time.
+
+(Metadata corruption on a single node is not an issue, the corrupted data file
+can always be deleted and reconstructed from the other nodes in the cluster.)
+
+Here is how this option impacts the different database engines:
+
+| Database | `metadata_fsync = false` (default) | `metadata_fsync = true`       |
+|----------|------------------------------------|-------------------------------|
+| Sled     | default options                    | *unsupported*                 |
+| Sqlite   | `PRAGMA synchronous = OFF`         | `PRAGMA synchronous = NORMAL` |
+| LMDB     | `MDB_NOMETASYNC` + `MDB_NOSYNC`    | `MDB_NOMETASYNC`              |
+
+Note that the Sqlite database is always ran in `WAL` mode (`PRAGMA journal_mode = WAL`).
+
+### `data_fsync`
+
+Whether to `fsync` data blocks and their containing directory after they are
+saved to disk.
+This is disabled (`false`) by default.
+
+This might reduce the risk that a data block is lost in rare
+situations such as simultaneous node losing power,
+at the cost of a moderate drop in write performance.
+
+Similarly to `metatada_fsync`, this is likely not necessary
+if geographical replication is used.
+
 ### `block_size`
 
 Garage splits stored objects in consecutive chunks of size `block_size`
-(except the last one which might be smaller). The default size is 1MB and
-should work in most cases. We recommend increasing it to e.g. 10MB if
+(except the last one which might be smaller). The default size is 1MiB and
+should work in most cases. We recommend increasing it to e.g. 10MiB if
 you are using Garage to store large files and have fast network connections
 between all nodes (e.g. 1gbps).
 
@@ -160,6 +219,14 @@ Increase this if sled is thrashing your SSD, at the risk of losing more data in
 of a power outage (though this should not matter much as data is replicated on other
 nodes). The default value, 2000ms, should be appropriate for most use cases.
 
+### `lmdb_map_size`
+
+This parameters can be used to set the map size used by LMDB,
+which is the size of the virtual memory region used for mapping the database file.
+The value of this parameter is the maximum size the metadata database can take.
+This value is not bound by the physical RAM size of the machine running Garage.
+If not specified, it defaults to 1GiB on 32-bit machines and 1TiB on 64-bit machines.
+
 ### `replication_mode`
 
 Garage supports the following replication modes:
@@ -401,6 +468,8 @@ manually.
 The IP and port on which to bind for accepting S3 API calls.
 This endpoint does not suport TLS: a reverse proxy should be used to provide it.
 
+Alternatively, since `v0.8.5`, a path can be used to create a unix socket with 0222 mode.
+
 ### `s3_region`
 
 Garage will accept S3 API calls that are targetted to the S3 region defined here.
@@ -409,7 +478,7 @@ message that redirects the client to the correct region.
 
 ### `root_domain` {#root_domain}
 
-The optionnal suffix to access bucket using vhost-style in addition to path-style request.
+The optional suffix to access bucket using vhost-style in addition to path-style request.
 Note path-style requests are always enabled, whether or not vhost-style is configured.
 Configuring vhost-style S3 required a wildcard DNS entry, and possibly a wildcard TLS certificate,
 but might be required by softwares not supporting path-style requests.
@@ -430,9 +499,11 @@ The IP and port on which to bind for accepting HTTP requests to buckets configur
 for website access.
 This endpoint does not suport TLS: a reverse proxy should be used to provide it.
 
+Alternatively, since `v0.8.5`, a path can be used to create a unix socket with 0222 mode.
+
 ### `root_domain`
 
-The optionnal suffix appended to bucket names for the corresponding HTTP Host.
+The optional suffix appended to bucket names for the corresponding HTTP Host.
 
 For instance, if `root_domain` is `web.garage.eu`, a bucket called `deuxfleurs.fr`
 will be accessible either with hostname `deuxfleurs.fr.web.garage.eu`
@@ -449,6 +520,9 @@ If specified, Garage will bind an HTTP server to this port and address, on
 which it will listen to requests for administration features.
 See [administration API reference](@/documentation/reference-manual/admin-api.md) to learn more about these features.
 
+Alternatively, since `v0.8.5`, a path can be used to create a unix socket. Note that for security reasons,
+the socket will have 0220 mode. Make sure to set user and group permissions accordingly.
+
 ### `metrics_token`, `metrics_token_file` or `GARAGE_METRICS_TOKEN` (env)
 
 The token for accessing the Metrics endpoint. If this token is not set, the

doc/book/reference-manual/features.md

@@ -35,7 +35,7 @@ This makes setting up and administering storage clusters, we hope, as easy as it
 
 A Garage cluster can very easily evolve over time, as storage nodes are added or removed.
 Garage will automatically rebalance data between nodes as needed to ensure the desired number of copies.
-Read about cluster layout management [here](@/documentation/reference-manual/layout.md).
+Read about cluster layout management [here](@/documentation/operations/layout.md).
 
 ### No RAFT slowing you down
 

doc/book/reference-manual/k2v.md

@@ -3,7 +3,7 @@ title = "K2V"
 weight = 100
 +++
 
-Starting with version 0.7.2, Garage introduces an optionnal feature, K2V,
+Starting with version 0.7.2, Garage introduces an optional feature, K2V,
 which is an alternative storage API designed to help efficiently store
 many small values in buckets (in opposition to S3 which is more designed
 to store large blobs).

doc/book/reference-manual/layout.md

@@ -1,77 +0,0 @@
-+++
-title = "Cluster layout management"
-weight = 50
-+++
-
-The cluster layout in Garage is a table that assigns to each node a role in
-the cluster. The role of a node in Garage can either be a storage node with
-a certain capacity, or a gateway node that does not store data and is only
-used as an API entry point for faster cluster access.
-An introduction to building cluster layouts can be found in the [production deployment](@/documentation/cookbook/real-world.md) page.
-
-## How cluster layouts work in Garage
-
-In Garage, a cluster layout is composed of the following components:
-
-- a table of roles assigned to nodes
-- a version number
-
-Garage nodes will always use the cluster layout with the highest version number.
-
-Garage nodes also maintain and synchronize between them a set of proposed role
-changes that haven't yet been applied. These changes will be applied (or
-canceled) in the next version of the layout
-
-The following commands insert modifications to the set of proposed role changes
-for the next layout version (but they do not create the new layout immediately):
-
-```bash
-garage layout assign [...]
-garage layout remove [...]
-```
-
-The following command can be used to inspect the layout that is currently set in the cluster
-and the changes proposed for the next layout version, if any:
-
-```bash
-garage layout show
-```
-
-The following commands create a new layout with the specified version number,
-that either takes into account the proposed changes or cancels them:
-
-```bash
-garage layout apply --version <new_version_number>
-garage layout revert --version <new_version_number>
-```
-
-The version number of the new layout to create must be 1 + the version number
-of the previous layout that existed in the cluster.  The `apply` and `revert`
-commands will fail otherwise.
-
-## Warnings about Garage cluster layout management
-
-**Warning: never make several calls to `garage layout apply` or `garage layout
-revert` with the same value of the `--version` flag. Doing so can lead to the
-creation of several different layouts with the same version number, in which
-case your Garage cluster will become inconsistent until fixed.** If a call to
-`garage layout apply` or `garage layout revert` has failed and `garage layout
-show` indicates that a new layout with the given version number has not been
-set in the cluster, then it is fine to call the command again with the same
-version number.
-
-If you are using the `garage` CLI by typing individual commands in your
-shell, you shouldn't have much issues as long as you run commands one after
-the other and take care of checking the output of `garage layout show`
-before applying any changes.
-
-If you are using the `garage` CLI to script layout changes, follow the following recommendations:
-
-- Make all of your `garage` CLI calls to the same RPC host. Do not use the
-  `garage` CLI to connect to individual nodes to send them each a piece of the
-  layout changes you are making, as the changes propagate asynchronously
-  between nodes and might not all be taken into account at the time when the
-  new layout is applied.
-
-- **Only call `garage layout apply` once**, and call it **strictly after** all
-  of the `layout assign` and `layout remove` commands have returned.

doc/book/reference-manual/s3-compatibility.md

@@ -75,16 +75,13 @@ but these endpoints are documented in [Red Hat Ceph Storage - Chapter 2. Ceph Ob
 
 | Endpoint                     | Garage                           | [Openstack Swift](https://docs.openstack.org/swift/latest/s3_compat.html) | [Ceph Object Gateway](https://docs.ceph.com/en/latest/radosgw/s3/) | [Riak CS](https://docs.riak.com/riak/cs/2.1.1/references/apis/storage/s3/index.html) | [OpenIO](https://docs.openio.io/latest/source/arch-design/s3_compliancy.html) |
 |------------------------------|----------------------------------|-----------------|---------------|---------|-----|
-| [AbortMultipartUpload](https://docs.aws.amazon.com/AmazonS3/latest/API/API_AbortMultipartUpload.html)         | ✅ Implemented                      |  ✅ | ✅ | ✅ | ✅ |
-| [CompleteMultipartUpload](https://docs.aws.amazon.com/AmazonS3/latest/API/API_CompleteMultipartUpload.html)   | ✅ Implemented (see details below)                      | ✅ | ✅ | ✅ | ✅ |
-| [CreateMultipartUpload](https://docs.aws.amazon.com/AmazonS3/latest/API/API_CreateMultipartUpload.html)        | ✅ Implemented                      | ✅| ✅ | ✅ | ✅ |
-| [ListMultipartUpload](https://docs.aws.amazon.com/AmazonS3/latest/API/API_ListMultipartUpload.html)          | ✅ Implemented                      | ✅ | ✅ | ✅ | ✅ |
-| [ListParts](https://docs.aws.amazon.com/AmazonS3/latest/API/API_ListParts.html)                    | ✅ Implemented                      | ✅ | ✅ | ✅ | ✅ |
-| [UploadPart](https://docs.aws.amazon.com/AmazonS3/latest/API/API_UploadPart.html)                  | ✅ Implemented (see details below)                      | ✅ | ✅| ✅ | ✅ |
-| [UploadPartCopy](https://docs.aws.amazon.com/AmazonS3/latest/API/API_UploadPartCopy.html)               | ✅ Implemented                      | ✅ | ✅ | ✅ | ✅ |
-
-Our implementation of Multipart Upload is currently a bit more restrictive than Amazon's one in some edge cases.
-For more information, please refer to our [issue tracker](https://git.deuxfleurs.fr/Deuxfleurs/garage/issues/204).
+| [AbortMultipartUpload](https://docs.aws.amazon.com/AmazonS3/latest/API/API_AbortMultipartUpload.html)         | ✅ Implemented  |  ✅ | ✅ | ✅ | ✅ |
+| [CompleteMultipartUpload](https://docs.aws.amazon.com/AmazonS3/latest/API/API_CompleteMultipartUpload.html)   | ✅ Implemented  | ✅ | ✅ | ✅ | ✅ |
+| [CreateMultipartUpload](https://docs.aws.amazon.com/AmazonS3/latest/API/API_CreateMultipartUpload.html)        | ✅ Implemented | ✅| ✅ | ✅ | ✅ |
+| [ListMultipartUpload](https://docs.aws.amazon.com/AmazonS3/latest/API/API_ListMultipartUpload.html)          | ✅ Implemented   | ✅ | ✅ | ✅ | ✅ |
+| [ListParts](https://docs.aws.amazon.com/AmazonS3/latest/API/API_ListParts.html)                    | ✅ Implemented             | ✅ | ✅ | ✅ | ✅ |
+| [UploadPart](https://docs.aws.amazon.com/AmazonS3/latest/API/API_UploadPart.html)                  | ✅ Implemented             | ✅ | ✅| ✅ | ✅ |
+| [UploadPartCopy](https://docs.aws.amazon.com/AmazonS3/latest/API/API_UploadPartCopy.html)               | ✅ Implemented        | ✅ | ✅ | ✅ | ✅ |
 
 ### Website endpoints
 
@@ -127,15 +124,22 @@ If you need this feature, please [share your use case in our dedicated issue](ht
 
 | Endpoint                     | Garage                           | [Openstack Swift](https://docs.openstack.org/swift/latest/s3_compat.html) | [Ceph Object Gateway](https://docs.ceph.com/en/latest/radosgw/s3/) | [Riak CS](https://docs.riak.com/riak/cs/2.1.1/references/apis/storage/s3/index.html) | [OpenIO](https://docs.openio.io/latest/source/arch-design/s3_compliancy.html) |
 |------------------------------|----------------------------------|-----------------|---------------|---------|-----|
-| [DeleteBucketLifecycle](https://docs.aws.amazon.com/AmazonS3/latest/API/API_DeleteBucketLifecycle.html) | ❌ Missing | ❌| ✅| ❌| ✅|
-| [GetBucketLifecycleConfiguration](https://docs.aws.amazon.com/AmazonS3/latest/API/API_GetBucketLifecycleConfiguration.html) | ❌ Missing | ❌| ✅ | ❌| ✅|
-| [PutBucketLifecycleConfiguration](https://docs.aws.amazon.com/AmazonS3/latest/API/API_PutBucketLifecycleConfiguration.html) | ❌ Missing | ❌| ✅ | ❌| ✅|
+| [DeleteBucketLifecycle](https://docs.aws.amazon.com/AmazonS3/latest/API/API_DeleteBucketLifecycle.html) | ✅ Implemented | ❌| ✅| ❌| ✅|
+| [GetBucketLifecycleConfiguration](https://docs.aws.amazon.com/AmazonS3/latest/API/API_GetBucketLifecycleConfiguration.html) | ✅ Implemented | ❌| ✅ | ❌| ✅|
+| [PutBucketLifecycleConfiguration](https://docs.aws.amazon.com/AmazonS3/latest/API/API_PutBucketLifecycleConfiguration.html) | ⚠ Partially implemented (see below) | ❌| ✅ | ❌| ✅|
 | [GetBucketVersioning](https://docs.aws.amazon.com/AmazonS3/latest/API/API_GetBucketVersioning.html)          | ❌ Stub (see below)       | ✅| ✅ | ❌| ✅|
 | [ListObjectVersions](https://docs.aws.amazon.com/AmazonS3/latest/API/API_ListObjectVersions.html) | ❌ Missing | ❌| ✅ | ❌| ✅|
 | [PutBucketVersioning](https://docs.aws.amazon.com/AmazonS3/latest/API/API_PutBucketVersioning.html) | ❌ Missing | ❌| ✅| ❌| ✅|
 
+**PutBucketLifecycleConfiguration:** The only actions supported are
+`AbortIncompleteMultipartUpload` and `Expiration` (without the
+`ExpiredObjectDeleteMarker` field).  All other operations are dependent on
+either bucket versionning or storage classes which Garage currently does not
+implement. The deprecated `Prefix` member directly in the the `Rule`
+structure/XML tag is not supported, specified prefixes must be inside the
+`Filter` structure/XML tag.
 
-**GetBucketVersioning:** Stub implementation (Garage does not yet support versionning so this always returns "versionning not enabled").
+**GetBucketVersioning:** Stub implementation which always returns "versionning not enabled", since Garage does not yet support bucket versionning.
 
 ### Replication endpoints
 

doc/book/working-documents/_index.md

@@ -1,6 +1,6 @@
 +++
 title = "Working Documents"
-weight = 8
+weight = 90
 sort_by = "weight"
 template = "documentation.html"
 +++

doc/book/working-documents/migration-09.md

@@ -0,0 +1,72 @@
++++
+title = "Migrating from 0.8 to 0.9"
+weight = 12
++++
+
+**This guide explains how to migrate to 0.9 if you have an existing 0.8 cluster.
+We don't recommend trying to migrate to 0.9 directly from 0.7 or older.**
+
+This migration procedure has been tested on several clusters without issues.
+However, it is still a *critical procedure* that might cause issues.
+**Make sure to back up all your data before attempting it!**
+
+You might also want to read our [general documentation on upgrading Garage](@/documentation/operations/upgrading.md).
+
+The following are **breaking changes** in Garage v0.9 that require your attention when migrating:
+
+- LMDB is now the default metadata db engine and Sled is deprecated. If you were using Sled, make sure to specify `db_engine = "sled"` in your configuration file, or take the time to [convert your database](https://garagehq.deuxfleurs.fr/documentation/reference-manual/configuration/#db-engine-since-v0-8-0).
+
+- Capacity values are now in actual byte units. The translation from the old layout will assign 1 capacity = 1Gb by default, which might be wrong for your cluster. This does not cause any data to be moved around, but you might want to re-assign correct capacity values post-migration.
+
+- Multipart uploads that were started in Garage v0.8 will not be visible in Garage v0.9 and will have to be restarted from scratch.
+
+- Changes to the admin API: some `v0/` endpoints have been replaced by `v1/` counterparts with updated/uniformized syntax. All other endpoints have also moved to `v1/` by default, without syntax changes, but are still available under `v0/` for compatibility.
+
+
+## Simple migration procedure (takes cluster offline for a while)
+
+The migration steps are as follows:
+
+1. Disable API and web access. You may do this by stopping your reverse proxy or by commenting out
+   the `api_bind_addr` values in your `config.toml` file and restarting Garage.
+2. Do `garage repair --all-nodes --yes tables` and `garage repair --all-nodes --yes blocks`,
+   check the logs and check that all data seems to be synced correctly between
+   nodes. If you have time, do additional checks (`versions`, `block_refs`, etc.)
+3. Check that the block resync queue and Merkle queue are empty:
+   run `garage stats -a` to query them or inspect metrics in the Grafana dashboard.
+4. Turn off Garage v0.8
+5. **Backup the metadata folder of all your nodes!** For instance, use the following command
+	if your metadata directory is `/var/lib/garage/meta`: `cd /var/lib/garage ; tar -acf meta-v0.8.tar.zst meta/`
+6. Install Garage v0.9
+7. Update your configuration file if necessary.
+8. Turn on Garage v0.9
+9. Do `garage repair --all-nodes --yes tables` and `garage repair --all-nodes --yes blocks`.
+   Wait for a full table sync to run.
+10. Your upgraded cluster should be in a working state. Re-enable API and Web
+    access and check that everything went well.
+11. Monitor your cluster in the next hours to see if it works well under your production load, report any issue.
+12. You might want to assign correct capacity values to all your nodes. Doing so might cause data to be moved
+    in your cluster, which should also be monitored carefully.
+
+## Minimal downtime migration procedure
+
+The migration to Garage v0.9 can be done with almost no downtime,
+by restarting all nodes at once in the new version.
+
+The migration steps are as follows:
+
+1. Do `garage repair --all-nodes --yes tables` and `garage repair --all-nodes --yes blocks`,
+   check the logs and check that all data seems to be synced correctly between
+   nodes. If you have time, do additional checks (`versions`, `block_refs`, etc.)
+
+2. Turn off each node individually; back up its metadata folder (see above); turn it back on again.
+   This will allow you to take a backup of all nodes without impacting global cluster availability.
+   You can do all nodes of a single zone at once as this does not impact the availability of Garage.
+
+3. Prepare your binaries and configuration files for Garage v0.9
+
+4. Shut down all v0.8 nodes simultaneously, and restart them all simultaneously in v0.9.
+   Use your favorite deployment tool (Ansible, Kubernetes, Nomad) to achieve this as fast as possible.
+   Garage v0.9 should be in a working state as soon as it starts.
+
+5. Proceed with repair and monitoring as described in steps 9-12 above.

doc/drafts/admin-api.md

@@ -52,11 +52,11 @@ Returns an HTTP status 200 if the node is ready to answer user's requests,
 and an HTTP status 503 (Service Unavailable) if there are some partitions
 for which a quorum of nodes is not available.
 A simple textual message is also returned in a body with content-type `text/plain`.
-See `/v0/health` for an API that also returns JSON output.
+See `/v1/health` for an API that also returns JSON output.
 
 ### Cluster operations
 
-#### GetClusterStatus `GET /v0/status`
+#### GetClusterStatus `GET /v1/status`
 
 Returns the cluster's current status in JSON, including:
 
@@ -70,86 +70,112 @@ Example response body:
 ```json
 {
   "node": "ec79480e0ce52ae26fd00c9da684e4fa56658d9c64cdcecb094e936de0bfe71f",
-  "garage_version": "git:v0.8.0",
-  "knownNodes": {
-    "ec79480e0ce52ae26fd00c9da684e4fa56658d9c64cdcecb094e936de0bfe71f": {
+  "garageVersion": "git:v0.9.0-dev",
+  "garageFeatures": [
+    "k2v",
+    "sled",
+    "lmdb",
+    "sqlite",
+    "metrics",
+    "bundled-libs"
+  ],
+  "rustVersion": "1.68.0",
+  "dbEngine": "LMDB (using Heed crate)",
+  "knownNodes": [
+    {
+      "id": "ec79480e0ce52ae26fd00c9da684e4fa56658d9c64cdcecb094e936de0bfe71f",
       "addr": "10.0.0.11:3901",
-      "is_up": true,
-      "last_seen_secs_ago": 9,
+      "isUp": true,
+      "lastSeenSecsAgo": 9,
       "hostname": "node1"
     },
-    "4a6ae5a1d0d33bf895f5bb4f0a418b7dc94c47c0dd2eb108d1158f3c8f60b0ff": {
+    {
+      "id": "4a6ae5a1d0d33bf895f5bb4f0a418b7dc94c47c0dd2eb108d1158f3c8f60b0ff",
       "addr": "10.0.0.12:3901",
-      "is_up": true,
-      "last_seen_secs_ago": 1,
+      "isUp": true,
+      "lastSeenSecsAgo": 1,
       "hostname": "node2"
     },
-    "23ffd0cdd375ebff573b20cc5cef38996b51c1a7d6dbcf2c6e619876e507cf27": {
+    {
+      "id": "23ffd0cdd375ebff573b20cc5cef38996b51c1a7d6dbcf2c6e619876e507cf27",
       "addr": "10.0.0.21:3901",
-      "is_up": true,
-      "last_seen_secs_ago": 7,
+      "isUp": true,
+      "lastSeenSecsAgo": 7,
       "hostname": "node3"
     },
-    "e2ee7984ee65b260682086ec70026165903c86e601a4a5a501c1900afe28d84b": {
+    {
+      "id": "e2ee7984ee65b260682086ec70026165903c86e601a4a5a501c1900afe28d84b",
       "addr": "10.0.0.22:3901",
-      "is_up": true,
-      "last_seen_secs_ago": 1,
+      "isUp": true,
+      "lastSeenSecsAgo": 1,
       "hostname": "node4"
     }
-  },
+  ],
   "layout": {
     "version": 12,
-    "roles": {
-      "ec79480e0ce52ae26fd00c9da684e4fa56658d9c64cdcecb094e936de0bfe71f": {
+    "roles": [
+      {
+        "id": "ec79480e0ce52ae26fd00c9da684e4fa56658d9c64cdcecb094e936de0bfe71f",
         "zone": "dc1",
-        "capacity": 4,
+        "capacity": 10737418240,
         "tags": [
           "node1"
         ]
       },
-      "4a6ae5a1d0d33bf895f5bb4f0a418b7dc94c47c0dd2eb108d1158f3c8f60b0ff": {
+      {
+        "id": "4a6ae5a1d0d33bf895f5bb4f0a418b7dc94c47c0dd2eb108d1158f3c8f60b0ff",
         "zone": "dc1",
-        "capacity": 6,
+        "capacity": 10737418240,
         "tags": [
           "node2"
         ]
       },
-      "23ffd0cdd375ebff573b20cc5cef38996b51c1a7d6dbcf2c6e619876e507cf27": {
+      {
+        "id": "23ffd0cdd375ebff573b20cc5cef38996b51c1a7d6dbcf2c6e619876e507cf27",
         "zone": "dc2",
-        "capacity": 10,
+        "capacity": 10737418240,
         "tags": [
           "node3"
         ]
       }
-    },
-    "stagedRoleChanges": {
-      "e2ee7984ee65b260682086ec70026165903c86e601a4a5a501c1900afe28d84b": {
+    ],
+    "stagedRoleChanges": [
+      {
+        "id": "e2ee7984ee65b260682086ec70026165903c86e601a4a5a501c1900afe28d84b",
+        "remove": false,
         "zone": "dc2",
-        "capacity": 5,
+        "capacity": 10737418240,
         "tags": [
           "node4"
         ]
       }
-    }
+      {
+        "id": "23ffd0cdd375ebff573b20cc5cef38996b51c1a7d6dbcf2c6e619876e507cf27",
+        "remove": true,
+        "zone": null,
+        "capacity": null,
+        "tags": null,
+      }
+    ]
   }
 }
 ```
 
-#### GetClusterHealth `GET /v0/health`
+#### GetClusterHealth `GET /v1/health`
 
 Returns the cluster's current health in JSON format, with the following variables:
 
-- `status`: one of `Healthy`, `Degraded` or `Unavailable`:
-  - Healthy: Garage node is connected to all storage nodes
-  - Degraded: Garage node is not connected to all storage nodes, but a quorum of write nodes is available for all partitions
-  - Unavailable: a quorum of write nodes is not available for some partitions
-- `known_nodes`: the number of nodes this Garage node has had a TCP connection to since the daemon started
-- `connected_nodes`: the nubmer of nodes this Garage node currently has an open connection to
-- `storage_nodes`: the number of storage nodes currently registered in the cluster layout
-- `storage_nodes_ok`: the number of storage nodes to which a connection is currently open
+- `status`: one of `healthy`, `degraded` or `unavailable`:
+  - healthy: Garage node is connected to all storage nodes
+  - degraded: Garage node is not connected to all storage nodes, but a quorum of write nodes is available for all partitions
+  - unavailable: a quorum of write nodes is not available for some partitions
+- `knownNodes`: the number of nodes this Garage node has had a TCP connection to since the daemon started
+- `connectedNodes`: the nubmer of nodes this Garage node currently has an open connection to
+- `storageNodes`: the number of storage nodes currently registered in the cluster layout
+- `storageNodesOk`: the number of storage nodes to which a connection is currently open
 - `partitions`: the total number of partitions of the data (currently always 256)
-- `partitions_quorum`: the number of partitions for which a quorum of write nodes is available
-- `partitions_all_ok`: the number of partitions for which we are connected to all storage nodes responsible of storing it
+- `partitionsQuorum`: the number of partitions for which a quorum of write nodes is available
+- `partitionsAllOk`: the number of partitions for which we are connected to all storage nodes responsible of storing it
 
 Contrarily to `GET /health`, this endpoint always returns a 200 OK HTTP response code.
 
@@ -157,18 +183,18 @@ Example response body:
 
 ```json
 {
-    "status": "Degraded",
-    "known_nodes": 3,
-    "connected_nodes": 2,
-    "storage_nodes": 3,
-    "storage_nodes_ok": 2,
-    "partitions": 256,
-    "partitions_quorum": 256,
-    "partitions_all_ok": 0
+  "status": "degraded",
+  "knownNodes": 3,
+  "connectedNodes": 3,
+  "storageNodes": 4,
+  "storageNodesOk": 3,
+  "partitions": 256,
+  "partitionsQuorum": 256,
+  "partitionsAllOk": 64
 }
 ```
 
-#### ConnectClusterNodes `POST /v0/connect`
+#### ConnectClusterNodes `POST /v1/connect`
 
 Instructs this Garage node to connect to other Garage nodes at specified addresses.
 
@@ -198,7 +224,7 @@ Example response:
 ]
 ```
 
-#### GetClusterLayout `GET /v0/layout`
+#### GetClusterLayout `GET /v1/layout`
 
 Returns the cluster's current layout in JSON, including:
 
@@ -212,42 +238,54 @@ Example response body:
 ```json
 {
   "version": 12,
-  "roles": {
-    "ec79480e0ce52ae26fd00c9da684e4fa56658d9c64cdcecb094e936de0bfe71f": {
+  "roles": [
+    {
+      "id": "ec79480e0ce52ae26fd00c9da684e4fa56658d9c64cdcecb094e936de0bfe71f",
       "zone": "dc1",
-      "capacity": 4,
+      "capacity": 10737418240,
       "tags": [
         "node1"
       ]
     },
-    "4a6ae5a1d0d33bf895f5bb4f0a418b7dc94c47c0dd2eb108d1158f3c8f60b0ff": {
+    {
+      "id": "4a6ae5a1d0d33bf895f5bb4f0a418b7dc94c47c0dd2eb108d1158f3c8f60b0ff",
       "zone": "dc1",
-      "capacity": 6,
+      "capacity": 10737418240,
       "tags": [
         "node2"
       ]
     },
-    "23ffd0cdd375ebff573b20cc5cef38996b51c1a7d6dbcf2c6e619876e507cf27": {
+    {
+      "id": "23ffd0cdd375ebff573b20cc5cef38996b51c1a7d6dbcf2c6e619876e507cf27",
       "zone": "dc2",
-      "capacity": 10,
+      "capacity": 10737418240,
       "tags": [
         "node3"
       ]
     }
-  },
-  "stagedRoleChanges": {
-    "e2ee7984ee65b260682086ec70026165903c86e601a4a5a501c1900afe28d84b": {
+  ],
+  "stagedRoleChanges": [
+    {
+      "id": "e2ee7984ee65b260682086ec70026165903c86e601a4a5a501c1900afe28d84b",
+      "remove": false,
       "zone": "dc2",
-      "capacity": 5,
+      "capacity": 10737418240,
       "tags": [
         "node4"
       ]
     }
-  }
+    {
+      "id": "23ffd0cdd375ebff573b20cc5cef38996b51c1a7d6dbcf2c6e619876e507cf27",
+      "remove": true,
+      "zone": null,
+      "capacity": null,
+      "tags": null,
+    }
+  ]
 }
 ```
 
-#### UpdateClusterLayout `POST /v0/layout`
+#### UpdateClusterLayout `POST /v1/layout`
 
 Send modifications to the cluster layout. These modifications will
 be included in the staged role changes, visible in subsequent calls
@@ -259,8 +297,9 @@ the layout.
 Request body format:
 
 ```json
-{
-  <node_id>: {
+[
+  {
+    "id": <node_id>,
     "capacity": <new_capacity>,
     "zone": <new_zone>,
     "tags": [
@@ -268,17 +307,22 @@ Request body format:
       ...
     ]
   },
-  <node_id_to_remove>: null,
-  ...
-}
+  {
+    "id": <node_id_to_remove>,
+    "remove": true
+  }
+]
 ```
 
 Contrary to the CLI that may update only a subset of the fields
 `capacity`, `zone` and `tags`, when calling this API all of these
 values must be specified.
 
+This returns the new cluster layout with the proposed staged changes,
+as returned by GetClusterLayout.
 
-#### ApplyClusterLayout `POST /v0/layout/apply`
+
+#### ApplyClusterLayout `POST /v1/layout/apply`
 
 Applies to the cluster the layout changes currently registered as
 staged layout changes.
@@ -295,7 +339,10 @@ Similarly to the CLI, the body must include the version of the new layout
 that will be created, which MUST be 1 + the value of the currently
 existing layout in the cluster.
 
-#### RevertClusterLayout `POST /v0/layout/revert`
+This returns the message describing all the calculations done to compute the new
+layout, as well as the description of the layout as returned by GetClusterLayout.
+
+#### RevertClusterLayout `POST /v1/layout/revert`
 
 Clears all of the staged layout changes.
 
@@ -313,10 +360,13 @@ Similarly to the CLI, the body must include the incremented
 version number, which MUST be 1 + the value of the currently
 existing layout in the cluster.
 
+This returns the new cluster layout with all changes reverted,
+as returned by GetClusterLayout.
+
 
 ### Access key operations
 
-#### ListKeys `GET /v0/key`
+#### ListKeys `GET /v1/key`
 
 Returns all API access keys in the cluster.
 
@@ -335,34 +385,8 @@ Example response:
 ]
 ```
 
-#### CreateKey `POST /v0/key`
-
-Creates a new API access key.
-
-Request body format:
-
-```json
-{
-    "name": "NameOfMyKey"
-}
-```
-
-#### ImportKey `POST /v0/key/import`
-
-Imports an existing API key.
-
-Request body format:
-
-```json
-{
-    "accessKeyId": "GK31c2f218a2e44f485b94239e",
-    "secretAccessKey": "b892c0665f0ada8a4755dae98baa3b133590e11dae3bcc1f9d769d67f16c3835",
-    "name": "NameOfMyKey"
-}
-```
-
-#### GetKeyInfo `GET /v0/key?id=<acces key id>`
-#### GetKeyInfo `GET /v0/key?search=<pattern>`
+#### GetKeyInfo `GET /v1/key?id=<acces key id>`
+#### GetKeyInfo `GET /v1/key?search=<pattern>`
 
 Returns information about the requested API access key.
 
@@ -370,6 +394,9 @@ If `id` is set, the key is looked up using its exact identifier (faster).
 If `search` is set, the key is looked up using its name or prefix
 of identifier (slower, all keys are enumerated to do this).
 
+Optionnally, the query parameter `showSecretKey=true` can be set to reveal the
+associated secret access key.
+
 Example response:
 
 ```json
@@ -433,11 +460,40 @@ Example response:
 }
 ```
 
-#### DeleteKey `DELETE /v0/key?id=<acces key id>`
+#### CreateKey `POST /v1/key`
 
-Deletes an API access key.
+Creates a new API access key.
 
-#### UpdateKey `POST /v0/key?id=<acces key id>`
+Request body format:
+
+```json
+{
+    "name": "NameOfMyKey"
+}
+```
+
+This returns the key info, including the created secret key,
+in the same format as the result of GetKeyInfo.
+
+#### ImportKey `POST /v1/key/import`
+
+Imports an existing API key.
+This will check that the imported key is in the valid format, i.e.
+is a key that could have been generated by Garage.
+
+Request body format:
+
+```json
+{
+    "accessKeyId": "GK31c2f218a2e44f485b94239e",
+    "secretAccessKey": "b892c0665f0ada8a4755dae98baa3b133590e11dae3bcc1f9d769d67f16c3835",
+    "name": "NameOfMyKey"
+}
+```
+
+This returns the key info in the same format as the result of GetKeyInfo.
+
+#### UpdateKey `POST /v1/key?id=<acces key id>`
 
 Updates information about the specified API access key.
 
@@ -453,14 +509,20 @@ Request body format:
 }
 ```
 
-All fields (`name`, `allow` and `deny`) are optionnal.
+All fields (`name`, `allow` and `deny`) are optional.
 If they are present, the corresponding modifications are applied to the key, otherwise nothing is changed.
 The possible flags in `allow` and `deny` are: `createBucket`.
 
+This returns the key info in the same format as the result of GetKeyInfo.
+
+#### DeleteKey `DELETE /v1/key?id=<acces key id>`
+
+Deletes an API access key.
+
 
 ### Bucket operations
 
-#### ListBuckets `GET /v0/bucket`
+#### ListBuckets `GET /v1/bucket`
 
 Returns all storage buckets in the cluster.
 
@@ -502,8 +564,8 @@ Example response:
 ]
 ```
 
-#### GetBucketInfo `GET /v0/bucket?id=<bucket id>`
-#### GetBucketInfo `GET /v0/bucket?globalAlias=<alias>`
+#### GetBucketInfo `GET /v1/bucket?id=<bucket id>`
+#### GetBucketInfo `GET /v1/bucket?globalAlias=<alias>`
 
 Returns information about the requested storage bucket.
 
@@ -535,7 +597,10 @@ Example response:
         ],
         "objects": 14827,
         "bytes": 13189855625,
-        "unfinshedUploads": 0,
+        "unfinishedUploads": 1,
+        "unfinishedMultipartUploads": 1,
+        "unfinishedMultipartUploadParts": 11,
+        "unfinishedMultipartUploadBytes": 41943040,
         "quotas": {
             "maxSize": null,
             "maxObjects": null
@@ -543,7 +608,7 @@ Example response:
 }
 ```
 
-#### CreateBucket `POST /v0/bucket`
+#### CreateBucket `POST /v1/bucket`
 
 Creates a new storage bucket.
 
@@ -583,13 +648,7 @@ or no alias at all.
 Technically, you can also specify both `globalAlias` and `localAlias` and that would create
 two aliases, but I don't see why you would want to do that.
 
-#### DeleteBucket `DELETE /v0/bucket?id=<bucket id>`
-
-Deletes a storage bucket. A bucket cannot be deleted if it is not empty.
-
-Warning: this will delete all aliases associated with the bucket!
-
-#### UpdateBucket `PUT /v0/bucket?id=<bucket id>`
+#### UpdateBucket `PUT /v1/bucket?id=<bucket id>`
 
 Updates configuration of the given bucket.
 
@@ -609,7 +668,7 @@ Request body format:
 }
 ```
 
-All fields (`websiteAccess` and `quotas`) are optionnal.
+All fields (`websiteAccess` and `quotas`) are optional.
 If they are present, the corresponding modifications are applied to the bucket, otherwise nothing is changed.
 
 In `websiteAccess`: if `enabled` is `true`, `indexDocument` must be specified.
@@ -621,9 +680,16 @@ In `quotas`: new values of `maxSize` and `maxObjects` must both be specified, or
 to remove the quotas. An absent value will be considered the same as a `null`. It is not possible
 to change only one of the two quotas.
 
+#### DeleteBucket `DELETE /v1/bucket?id=<bucket id>`
+
+Deletes a storage bucket. A bucket cannot be deleted if it is not empty.
+
+Warning: this will delete all aliases associated with the bucket!
+
+
 ### Operations on permissions for keys on buckets
 
-#### BucketAllowKey `POST /v0/bucket/allow`
+#### BucketAllowKey `POST /v1/bucket/allow`
 
 Allows a key to do read/write/owner operations on a bucket.
 
@@ -644,7 +710,7 @@ Request body format:
 Flags in `permissions` which have the value `true` will be activated.
 Other flags will remain unchanged.
 
-#### BucketDenyKey `POST /v0/bucket/deny`
+#### BucketDenyKey `POST /v1/bucket/deny`
 
 Denies a key from doing read/write/owner operations on a bucket.
 
@@ -668,19 +734,19 @@ Other flags will remain unchanged.
 
 ### Operations on bucket aliases
 
-#### GlobalAliasBucket `PUT /v0/bucket/alias/global?id=<bucket id>&alias=<global alias>`
+#### GlobalAliasBucket `PUT /v1/bucket/alias/global?id=<bucket id>&alias=<global alias>`
 
 Empty body. Creates a global alias for a bucket.
 
-#### GlobalUnaliasBucket `DELETE /v0/bucket/alias/global?id=<bucket id>&alias=<global alias>`
+#### GlobalUnaliasBucket `DELETE /v1/bucket/alias/global?id=<bucket id>&alias=<global alias>`
 
 Removes a global alias for a bucket.
 
-#### LocalAliasBucket `PUT /v0/bucket/alias/local?id=<bucket id>&accessKeyId=<access key ID>&alias=<local alias>`
+#### LocalAliasBucket `PUT /v1/bucket/alias/local?id=<bucket id>&accessKeyId=<access key ID>&alias=<local alias>`
 
 Empty body. Creates a local alias for a bucket in the namespace of a specific access key.
 
-#### LocalUnaliasBucket `DELETE /v0/bucket/alias/local?id=<bucket id>&accessKeyId<access key ID>&alias=<local alias>`
+#### LocalUnaliasBucket `DELETE /v1/bucket/alias/local?id=<bucket id>&accessKeyId<access key ID>&alias=<local alias>`
 
 Removes a local alias for a bucket in the namespace of a specific access key.
 

doc/optimal_layout_report/.gitignore

@@ -0,0 +1,13 @@
+optimal_layout.aux
+optimal_layout.log
+optimal_layout.synctex.gz
+optimal_layout.bbl
+optimal_layout.blg
+
+geodistrib.aux
+geodistrib.bbl
+geodistrib.blg
+geodistrib.log
+geodistrib.out
+geodistrib.synctex.gz
+

doc/optimal_layout_report/geodistrib.tex

@@ -0,0 +1,317 @@
+\documentclass[]{article}
+
+\usepackage{amsmath,amssymb}
+\usepackage{amsthm}
+
+\usepackage{stmaryrd}
+
+\usepackage{graphicx,xcolor}
+\usepackage{hyperref}
+
+\usepackage{algorithm,algpseudocode,float} 
+
+\renewcommand\thesubsubsection{\Alph{subsubsection})}
+
+\newtheorem{proposition}{Proposition}
+
+%opening
+\title{An algorithm for geo-distributed and redundant storage in Garage}
+\author{Mendes Oulamara \\ \emph{mendes@deuxfleurs.fr}}
+\date{}
+
+\begin{document}
+
+\maketitle
+
+\begin{abstract}
+	Garage
+\end{abstract}
+
+\section{Introduction}
+
+Garage\footnote{\url{https://garagehq.deuxfleurs.fr/}} is an open-source distributed object storage service tailored for self-hosting. It was designed by the Deuxfleurs association\footnote{\url{https://deuxfleurs.fr/}} to enable small structures (associations, collectives, small companies) to share storage resources to reliably self-host their data, possibly with old and non-reliable machines.
+
+To achieve these reliability and availability goals, the data is broken into \emph{partitions} and every partition is replicated over 3 different machines (that we call \emph{nodes}). When the data is queried, a consensus algorithm allows to fetch it from one of the nodes. A \emph{replication factor} of 3 ensures the best guarantees in the consensus algorithm \cite{ADD RREF}, but this parameter can be different.
+
+Moreover, if the nodes are spread over different \emph{zones} (different houses, offices, cities\dots), we can ask the data to be replicated over nodes belonging to different zones, to improve the storage robustness against zone failure (such as power outage). To do so, we set a \emph{redundancy parameter}, that is no more than the replication factor, and we ask that any partition is replicated over this number of zones at least.
+
+In this work, we propose a repartition algorithm that, given the nodes specifications and the replication and redundancy parameters, computes an optimal assignation of partitions to nodes. We say that the assignation is optimal in the sense that it maximizes the size of the partitions, and hence the effective storage capacity of the system.
+
+Moreover, when a former assignation exists, which is not optimal anymore due to nodes or zones updates, our algorithm computes a new optimal assignation that minimizes the amount of data to be transferred during the assignation update (the \emph{transfer load}).
+
+We call the set of nodes cooperating to store the data a \emph{cluster}, and a description of the nodes, zones and the assignation of partitions to nodes a \emph{cluster layout}
+
+\subsection{Notations}
+
+Let $k$ be some fixed parameter value, typically 8, that we call the ``partition bits''.
+Every object to be stored in the system is split into data blocks of fixed size. We compute a hash $h(\mathbf{b})$ of every such block $\mathbf{b}$, and we define the $k$ last bits of this hash to be the partition number $p(\mathbf{b})$ of the block. This label can take $P=2^k$ different values, and hence there are $P$ different partitions. We denote $\mathbf{P}$ the set of partition labels (i.e. $\mathbf{P}=\llbracket1,P\rrbracket$).
+
+We are given a set $\mathbf{N}$ of $N$ nodes  and a set  $\mathbf{Z}$ of $Z$ zones. Every node $n$ has a non-negative storage capacity $c_n\ge 0$ and belongs to a zone $z_n\in \mathbf{Z}$. We are also given a replication parameter $\rho_\mathbf{N}$ and a redundancy parameter $\rho_\mathbf{Z}$ such that  $1\le \rho_\mathbf{Z} \le \rho_\mathbf{N}$ (typical values would be $\rho_N=3$ and $\rho_Z=2$).
+
+Our goal is to compute an assignment $\alpha = (\alpha_p^1, \ldots, \alpha_p^{\rho_\mathbf{N}})_{p\in \mathbf{P}}$ such that every partition $p$ is associated to $\rho_\mathbf{N}$ distinct nodes $\alpha_p^1, \ldots, \alpha_p^{\rho_\mathbf{N}} \in \mathbf{N}$ and these nodes belong to at least $\rho_\mathbf{Z}$ distinct zones. Among the possible assignations, we choose one that \emph{maximizes} the effective storage capacity of the cluster. If the layout contained a previous assignment $\alpha'$, we \emph{minimize} the amount of data to transfer during the layout update by making $\alpha$ as close as possible to $\alpha'$. These maximization and minimization are described more formally in the following section.
+
+\subsection{Optimization parameters}
+
+To link the effective storage capacity of the cluster to partition assignment, we make the following assumption:
+\begin{equation}
+	\tag{H1}
+	\text{\emph{All partitions have the same size $s$.}}
+\end{equation}
+This assumption is justified by the dispersion of the hashing function, when the number of partitions is small relative to the number of stored blocks.
+
+Every node $n$ wille store some number $p_n$ of partitions (it is the number of partitions $p$ such that $n$ appears in the $\alpha_p$). Hence the partitions stored by $n$ (and hence all partitions by our assumption) have there size bounded by $c_n/p_n$. This remark leads us to define the optimal size that we will want to maximize:
+
+\begin{equation}
+	\label{eq:optimal}
+	\tag{OPT}
+s^* = \min_{n \in N} \frac{c_n}{p_n}.
+\end{equation}
+
+When the capacities of the nodes are updated (this includes adding or removing a node), we want to update the assignment as well. However, transferring the data between nodes has a cost and we would like to limit the number of changes in the assignment. We make the following assumption:
+\begin{equation}
+	\tag{H2}
+	\text{\emph{Nodes updates happen rarely relatively to block operations.}}
+\end{equation}
+This assumption justifies that when we compute the new assignment $\alpha$, it is worth to optimize the partition size \eqref{eq:optimal} first, and then, among the possible optimal solution, to try to minimize the number of partition transfers. More formally, we minimize the distance between two assignments defined by
+\begin{equation}
+	d(\alpha, \alpha') := \#\{ (n,p) \in \mathbf{N}\times\mathbf{P} ~|~ n\in \alpha_p \triangle \alpha'_p \}
+\end{equation}
+where the symmetric difference $\alpha_p \triangle \alpha'_p$ denotes the nodes appearing in one of the assignations but not in both.
+
+\section{Computation of an optimal assignment}
+
+The algorithm that we propose takes as inputs the cluster layout parameters $\mathbf{N}$, $\mathbf{Z}$, $\mathbf{P}$, $(c_n)_{n\in \mathbf{N}}$, $\rho_\mathbf{N}$, $\rho_\mathbf{Z}$, that we defined in the introduction, together with the former assignation $\alpha'$ (if any). The computation of the new optimal assignation $\alpha^*$ is done in three successive steps that will be detailed in the following sections. The first step computes the largest partition size $s^*$ that an assignation can achieve. The second step computes an optimal candidate assignment $\alpha$ that achieves $s^*$ and a heuristic is used in the computation to make it hopefully close to $\alpha'$. The third steps modifies $\alpha$ iteratively to reduces $d(\alpha, \alpha')$ and yields an assignation $\alpha^*$ achieving $s^*$, and minimizing $d(\cdot, \alpha')$ among such assignations.
+
+We will explain in the next section how to represent an assignment $\alpha$ by a flow $f$ on a weighted graph $G$ to enable the use of flow and graph algorithms. The main function of the algorithm can be written as follows.
+
+\subsubsection*{Algorithm}
+
+\begin{algorithmic}[1]
+	\Function{Compute Layout}{$\mathbf{N}$, $\mathbf{Z}$, $\mathbf{P}$, $(c_n)_{n\in \mathbf{N}}$, $\rho_\mathbf{N}$, $\rho_\mathbf{Z}$, $\alpha'$}
+	\State $s^* \leftarrow$ \Call{Compute Partition Size}{$\mathbf{N}$, $\mathbf{Z}$, $\mathbf{P}$, $(c_n)_{n\in \mathbf{N}}$, $\rho_\mathbf{N}$, $\rho_\mathbf{Z}$}
+	\State $G \leftarrow G(s^*)$
+	\State $f \leftarrow$ 	\Call{Compute Candidate Assignment}{$G$, $\alpha'$}
+	\State $f^* \leftarrow$ 	\Call{Minimize transfer load}{$G$, $f$, $\alpha'$}
+	\State Build $\alpha^*$ from $f^*$
+	\State \Return $\alpha^*$
+	\EndFunction
+\end{algorithmic}
+
+\subsubsection*{Complexity}
+As we will see in the next sections, the worst case complexity of this algorithm is $O(P^2 N^2)$. The minimization of transfer load is the most expensive step, and it can run with a timeout since it is only an optimization step. Without this step (or with a smart timeout), the worst cas complexity can be $O((PN)^{3/2}\log C)$ where $C$ is the total storage capacity of the cluster. 
+
+\subsection{Determination of the partition size $s^*$}
+
+We will represent an assignment $\alpha$ as a flow in a specific graph $G$. We will not compute the optimal partition size $s^*$ a priori, but we will determine it by dichotomy, as the largest size $s$ such that the maximal flow achievable on $G=G(s)$ has value $\rho_\mathbf{N}P$. We will assume that the capacities are given in a small enough unit (say, Megabytes), and we will determine $s^*$ at the precision of the given unit.
+
+Given some candidate size value $s$, we describe the oriented weighted graph $G=(V,E)$ with vertex set $V$ arc set $E$ (see Figure \ref{fig:flowgraph}).
+
+The set of vertices $V$ contains the source $\mathbf{s}$, the sink $\mathbf{t}$, vertices 
+$\mathbf{p^+, p^-}$ for every partition $p$, vertices $\mathbf{x}_{p,z}$ for every partition $p$ and zone $z$, and vertices $\mathbf{n}$ for every node $n$. 
+
+The set of arcs $E$ contains:
+\begin{itemize}
+	\item ($\mathbf{s}$,$\mathbf{p}^+$, $\rho_\mathbf{Z}$) for every partition $p$;
+	\item ($\mathbf{s}$,$\mathbf{p}^-$, $\rho_\mathbf{N}-\rho_\mathbf{Z}$) for every partition $p$;
+	\item ($\mathbf{p}^+$,$\mathbf{x}_{p,z}$, 1) for every partition $p$ and zone $z$;
+	\item ($\mathbf{p}^-$,$\mathbf{x}_{p,z}$, $\rho_\mathbf{N}-\rho_\mathbf{Z}$) for every partition $p$ and zone $z$;
+	\item ($\mathbf{x}_{p,z}$,$\mathbf{n}$, 1) for every partition $p$, zone $z$ and node $n\in z$;
+	\item ($\mathbf{n}$, $\mathbf{t}$, $\lfloor c_n/s \rfloor$) for every node $n$.
+\end{itemize}
+
+\begin{figure}
+	\centering
+	\includegraphics[width=\linewidth]{figures/flow_graph_param}
+	\caption{An example of graph $G(s)$. Arcs are oriented from left to right, and unlabeled arcs have capacity 1. In this example, nodes $n_1,n_2,n_3$ belong to zone $z_1$, and nodes $n_4,n_5$ belong to zone $z_2$.}
+	\label{fig:flowgraph}
+\end{figure}
+
+In the following complexity calculations, we will use the number of vertices and edges of $G$. Remark from now that $\# V = O(PZ)$ and $\# E = O(PN)$.
+
+\begin{proposition}
+	An assignment $\alpha$ is realizable with partition size $s$ and the redundancy constraints $(\rho_\mathbf{N},\rho_\mathbf{Z})$ if and only if there exists a maximal flow function $f$ in $G$ with total flow $\rho_\mathbf{N}P$, such that the arcs ($\mathbf{x}_{p,z}$,$\mathbf{n}$, 1) used are exactly those for which $p$ is associated to $n$ in $\alpha$.
+\end{proposition}
+\begin{proof}
+	Given such flow $f$, we can reconstruct a candidate $\alpha$. In $f$, the flow passing through $\mathbf{p^+}$ and $\mathbf{p^-}$ is $\rho_\mathbf{N}$, and since the outgoing capacity of every $\mathbf{x}_{p,z}$ is 1, every partition is associated to $\rho_\mathbf{N}$ distinct nodes. The fraction $\rho_\mathbf{Z}$ of the flow passing through every $\mathbf{p^+}$ must be spread over as many distinct zones as every arc outgoing from $\mathbf{p^+}$ has capacity 1. So the reconstructed $\alpha$ verifies the redundancy constraints. For every node $n$, the flow between $\mathbf{n}$ and $\mathbf{t}$ corresponds to the number of partitions associated to $n$. By construction of $f$, this does not exceed $\lfloor c_n/s \rfloor$. We assumed that the partition size is $s$, hence this association does not exceed the storage capacity of the nodes.
+	
+	In the other direction, given an assignment $\alpha$, one can similarly check that the facts that $\alpha$ respects the redundancy constraints, and the storage capacities of the nodes, are necessary condition to construct a maximal flow function $f$.
+\end{proof}
+
+\textbf{Implementation remark:} In the flow algorithm, while exploring the graph, we explore the neighbours of every vertex in a random order to heuristically spread the associations between nodes and partitions.
+
+\subsubsection*{Algorithm}
+With this result mind, we can describe the first step of our algorithm. All divisions are supposed to be integer divisions.
+\begin{algorithmic}[1]
+	\Function{Compute Partition Size}{$\mathbf{N}$, $\mathbf{Z}$, $\mathbf{P}$, $(c_n)_{n\in \mathbf{N}}$, $\rho_\mathbf{N}$, $\rho_\mathbf{Z}$}
+	
+	\State Build the graph $G=G(s=1)$
+	\State $ f \leftarrow$ \Call{Maximal flow}{$G$}
+	\If{$f.\mathrm{total flow} < \rho_\mathbf{N}P$}
+	
+	\State \Return Error: capacities too small or constraints too strong.
+	\EndIf
+	
+	\State $s^- \leftarrow 1$
+	\State $s^+ \leftarrow 1+\frac{1}{\rho_\mathbf{N}}\sum_{n \in \mathbf{N}} c_n$
+	
+	\While{$s^-+1 < s^+$}
+	\State Build the graph $G=G(s=(s^-+s^+)/2)$
+	\State $ f \leftarrow$ \Call{Maximal flow}{$G$}
+	\If{$f.\mathrm{total flow} < \rho_\mathbf{N}P$}
+	\State $s^+ \leftarrow (s^- + s^+)/2$
+	\Else
+	\State $s^- \leftarrow (s^- + s^+)/2$
+	\EndIf
+	\EndWhile
+	
+	\State \Return $s^-$
+	\EndFunction
+\end{algorithmic}
+
+\subsubsection*{Complexity}
+
+To compute the maximal flow, we use Dinic's algorithm. Its complexity on general graphs is $O(\#V^2 \#E)$, but on graphs with edge capacity bounded by a constant, it turns out to be $O(\#E^{3/2})$. The graph $G$ does not fall in this case since the capacities of the arcs incoming to $\mathbf{t}$ are far from bounded. However, the proof of this complexity function works readily for graphs where we only ask the edges \emph{not} incoming to the sink $\mathbf{t}$ to have their capacities bounded by a constant. One can find the proof of this claim in \cite[Section 2]{even1975network}.
+The dichotomy adds a logarithmic factor $\log (C)$ where $C=\sum_{n \in \mathbf{N}} c_n$ is the total capacity of the cluster. The total complexity of this first function is hence 
+$O(\#E^{3/2}\log C ) = O\big((PN)^{3/2} \log C\big)$.
+
+\subsubsection*{Metrics}
+We can display the discrepancy between the computed $s^*$ and the best size we could have hoped for the given total capacity, that is $C/\rho_\mathbf{N}$.
+
+\subsection{Computation of a candidate assignment}
+
+Now that we have the optimal partition size $s^*$, to compute a candidate assignment it would be enough to compute a maximal flow function $f$ on $G(s^*)$. This is what we do if there is no former assignation $\alpha'$.
+
+If there is some $\alpha'$, we add a step that will heuristically help to obtain a candidate $\alpha$ closer to $\alpha'$. We fist compute a flow function $\tilde{f}$ that uses only the partition-to-node associations appearing in $\alpha'$. Most likely, $\tilde{f}$ will not be a maximal flow of $G(s^*)$. In Dinic's algorithm, we can start from a non maximal flow function and then discover improving paths. This is what we do by starting from $\tilde{f}$. The hope\footnote{This is only a hope, because one can find examples where the construction of $f$ from $\tilde{f}$ produces an assignment $\alpha$ that is not as close as possible to $\alpha'$.} is that the final flow function $f$ will tend to keep the associations appearing in $\tilde{f}$.
+
+More formally, we construct the graph $G_{|\alpha'}$ from $G$ by removing all the arcs $(\mathbf{x}_{p,z},\mathbf{n}, 1)$ where $p$ is not associated to $n$ in $\alpha'$. We compute a maximal flow function $\tilde{f}$ in $G_{|\alpha'}$. The flow $\tilde{f}$ is also a valid (most likely non maximal) flow function on $G$. We compute a maximal flow function $f$ on $G$ by starting Dinic's algorithm on $\tilde{f}$.
+
+\subsubsection*{Algorithm}
+\begin{algorithmic}[1]
+	\Function{Compute Candidate Assignment}{$G$, $\alpha'$}
+	\State Build the graph $G_{|\alpha'}$
+	\State $ \tilde{f} \leftarrow$ \Call{Maximal flow}{$G_{|\alpha'}$}
+	\State $ f \leftarrow$ \Call{Maximal flow from flow}{$G$, $\tilde{f}$}
+	\State \Return $f$
+	\EndFunction
+\end{algorithmic}
+
+~ 
+
+\textbf{Remark:} The function ``Maximal flow'' can be just seen as the function ``Maximal flow from flow'' called with the zero flow function as starting flow.
+
+\subsubsection*{Complexity}
+With the considerations of the last section, we have the complexity of the Dinic's algorithm $O(\#E^{3/2}) = O((PN)^{3/2})$.
+
+\subsubsection*{Metrics}
+
+We can display the flow value of $\tilde{f}$, which is an upper bound of the distance between $\alpha$ and $\alpha'$. It might be more a Debug level display than Info.
+
+\subsection{Minimization of the transfer load}
+
+Now that we have a candidate flow function $f$, we want to modify it to make its corresponding assignation $\alpha$ as close as possible to $\alpha'$. Denote by $f'$ the maximal flow corresponding to $\alpha'$, and let $d(f, \alpha')=d(f, f'):=d(\alpha,\alpha')$\footnote{It is the number of arcs of type $(\mathbf{x}_{p,z},\mathbf{n})$ saturated in one flow and not in the other.}. 
+We want to build a sequence $f=f_0, f_1, f_2 \dots$ of maximal flows such that $d(f_i, \alpha')$ decreases as $i$ increases. The distance being a non-negative integer, this sequence of flow functions must be finite. We now explain how to find some improving $f_{i+1}$ from $f_i$.
+
+For any maximal flow $f$ in $G$, we define the oriented weighted graph $G_f=(V, E_f)$ as follows. The vertices of $G_f$ are the same as the vertices of $G$. $E_f$ contains the arc $(v_1,v_2, w)$ between vertices $v_1,v_2\in V$ with weight $w$ if and only if the arc $(v_1,v_2)$ is not saturated in $f$ (i.e. $c(v_1,v_2)-f(v_1,v_2) \ge 1$, we also consider reversed arcs). The weight $w$ is: 
+\begin{itemize}
+	\item  $-1$  if $(v_1,v_2)$ is of type $(\mathbf{x}_{p,z},\mathbf{n})$ or $(\mathbf{x}_{p,z},\mathbf{n})$ and is saturated in only one of the two flows $f,f'$;
+	\item $+1$  if $(v_1,v_2)$ is of type $(\mathbf{x}_{p,z},\mathbf{n})$ or $(\mathbf{x}_{p,z},\mathbf{n})$ and is saturated in either both or none of the two flows $f,f'$;
+	\item $0$ otherwise.
+\end{itemize}
+
+If $\gamma$ is a simple cycle of arcs in $G_f$, we define its weight $w(\gamma)$ as the sum of the weights of its arcs. We can add $+1$ to the value of $f$ on the arcs of $\gamma$, and by construction of $G_f$ and the fact that $\gamma$ is a cycle, the function that we get is still a valid flow function on $G$, it is maximal as it has the same flow value as $f$. We denote this new function $f+\gamma$.
+
+\begin{proposition}
+	Given a maximal flow $f$ and a simple cycle $\gamma$ in $G_f$, we have $d(f+\gamma, f') - d(f,f') = w(\gamma)$.
+\end{proposition}
+\begin{proof}
+	Let $X$ be the set of arcs of type $(\mathbf{x}_{p,z},\mathbf{n})$. Then we can express $d(f,f')$ as
+	\begin{align*}
+		d(f,f') & = \#\{e\in X ~|~ f(e)\neq f'(e)\}
+		= \sum_{e\in X} 1_{f(e)\neq f'(e)} \\
+		& = \frac{1}{2}\big( \#X + \sum_{e\in X} 1_{f(e)\neq f'(e)} - 1_{f(e)= f'(e)} \big).
+	\end{align*}
+	We can express the cycle weight as
+	\begin{align*}
+		w(\gamma) &  = \sum_{e\in X, e\in \gamma} - 1_{f(e)\neq f'(e)} + 1_{f(e)= f'(e)}.
+	\end{align*}
+	Remark that since we passed on unit of flow in $\gamma$ to construct $f+\gamma$, we have for any $e\in X$, $f(e)=f'(e)$ if and only if $(f+\gamma)(e) \neq f'(e)$.
+	Hence
+	\begin{align*}
+		w(\gamma) &  = \frac{1}{2}(w(\gamma) + w(\gamma)) \\
+		&= \frac{1}{2} \Big( 
+		\sum_{e\in X, e\in \gamma} - 1_{f(e)\neq f'(e)} +  1_{f(e)= f'(e)} \\
+		& \qquad +
+		\sum_{e\in X, e\in \gamma} 1_{(f+\gamma)(e)\neq f'(e)} +  1_{(f+\gamma)(e)= f'(e)}
+		\Big).
+	\end{align*}
+	Plugging this in the previous equation, we find that 
+	$$d(f,f')+w(\gamma) = d(f+\gamma, f').$$
+\end{proof}
+
+This result suggests that given some flow $f_i$, we just need to find a negative cycle $\gamma$ in $G_{f_i}$ to construct $f_{i+1}$ as $f_i+\gamma$. The following proposition ensures that this greedy strategy reaches an optimal flow. 
+
+\begin{proposition}
+	For any maximal flow $f$, $G_f$ contains a negative cycle if and only if there exists a maximal flow $f^*$ in $G$ such that $d(f^*, f') < d(f, f')$.
+\end{proposition}
+\begin{proof}
+	Suppose that there is such flow $f^*$. Define the oriented multigraph $M_{f,f^*}=(V,E_M)$ with the same vertex set $V$ as in $G$, and for every $v_1,v_2 \in V$, $E_M$ contains $(f^*(v_1,v_2) - f(v_1,v_2))_+$  copies of the arc $(v_1,v_2)$. For every vertex $v$, its total degree (meaning its outer degree minus its inner degree) is equal to 
+	\begin{align*}
+		\deg v & = \sum_{u\in V} (f^*(v,u) - f(v,u))_+ - \sum_{u\in V} (f^*(u,v) - f(u,v))_+  \\
+		& = \sum_{u\in V} f^*(v,u) - f(v,u) = \sum_{u\in V} f^*(v,u) - \sum_{u\in V}  f(v,u).
+	\end{align*}
+	The last two sums are zero for any inner vertex since $f,f^*$ are flows, and they are equal on the source and sink since the two flows are both maximal and have hence the same value. Thus, $\deg v = 0$ for every vertex $v$.
+	
+	This implies that the multigraph $M_{f,f^*}$ is the union of disjoint simple cycles. $f$ can be transformed into $f^*$ by pushing a mass 1 along all these cycles in any order. Since $d(f^*, f')<d(f,f')$, there must exists one of these simple cycles $\gamma$ with $d(f+\gamma, f') < d(f, f')$. Finally, since we can push a mass in $f$ along $\gamma$, it must appear in $G_f$. Hence $\gamma$ is a cycle of $G_f$ with negative weight.
+\end{proof}
+
+In the next section we describe the corresponding algorithm. Instead of discovering only one cycle, we are allowed to discover a set $\Gamma$ of disjoint negative cycles. 
+
+\subsubsection*{Algorithm}
+\begin{algorithmic}[1]
+	\Function{Minimize transfer load}{$G$, $f$, $\alpha'$}
+	\State Build the graph $G_f$
+	\State $\Gamma \leftarrow$ \Call{Detect Negative Cycles}{$G_f$}
+	\While{$\Gamma \neq \emptyset$}
+	\ForAll{$\gamma \in \Gamma$}
+	\State $f \leftarrow f+\gamma$
+	\EndFor
+	\State Update $G_f$
+	\State $\Gamma \leftarrow$ \Call{Detect Negative Cycles}{$G_f$}
+	\EndWhile
+	\State \Return $f$
+	\EndFunction
+\end{algorithmic}
+
+\subsubsection*{Complexity}
+The distance $d(f,f')$ is bounded by the maximal number of differences in the associated assignment. If these assignment are totally disjoint, this distance is $2\rho_N P$. At every iteration of the While loop, the distance decreases, so there is at most $O(\rho_N P) = O(P)$ iterations.
+
+The detection of negative cycle is done with the Bellman-Ford algorithm, whose complexity should normally be $O(\#E\#V)$. In our case, it amounts to $O(P^2ZN)$. Multiplied by the complexity of the outer loop, it amounts to $O(P^3ZN)$ which is a lot when the number of partitions and nodes starts to be large. To avoid that, we adapt the Bellman-Ford algorithm.
+
+The Bellman-Ford algorithm runs $\#V$ iterations of an outer loop, and an inner loop over $E$. The idea is to compute the shortest paths from a source vertex $v$ to all other vertices. After $k$ iterations of the outer loop, the algorithm has computed all shortest path of length at most $k$. All simple paths have length at most $\#V-1$, so if there is an update in the last iteration of the loop, it means that there is a negative cycle in the graph. The observation that will enable us to improve the complexity is the following:
+
+\begin{proposition}
+	In the graph $G_f$ (and $G$), all simple paths have a length at most $4N$.
+\end{proposition}
+\begin{proof}
+	Since $f$ is a maximal flow, there is no outgoing edge from $\mathbf{s}$ in $G_f$. One can thus check than any simple path of length 4 must contain at least two node of type $\mathbf{n}$. Hence on a path, at most 4 arcs separate two successive nodes of type $\mathbf{n}$. 
+\end{proof}
+
+Thus, in the absence of negative cycles, shortest paths in $G_f$ have length at most $4N$. So we can do only $4N+1$ iterations of the outer loop in the Bellman-Ford algorithm. This makes the complexity of the detection of one set of cycle to be $O(N\#E) = O(N^2 P)$.
+
+With this improvement, the complexity of the whole algorithm is, in the worst case, $O(N^2P^2)$. However, since we detect several cycles at once and we start with a flow that might be close to the previous one, the number of iterations of the outer loop might be smaller in practice.
+
+
+
+\subsubsection*{Metrics}
+We can display the node and zone utilization ratio, by dividing the flow passing through them divided by their outgoing capacity. In particular, we can pinpoint saturated nodes and zones (i.e. used at their full potential).
+
+We can display the distance to the previous assignment, and the number of partition transfers.
+
+
+\bibliography{optimal_layout} 
+\bibliographystyle{ieeetr}
+
+\end{document}
+
+
+

doc/optimal_layout_report/optimal_layout.bib

@@ -0,0 +1,11 @@
+
+@article{even1975network,
+	title={Network flow and testing graph connectivity},
+	author={Even, Shimon and Tarjan, R Endre},
+	journal={SIAM journal on computing},
+	volume={4},
+	number={4},
+	pages={507--518},
+	year={1975},
+	publisher={SIAM}
+}

doc/optimal_layout_report/optimal_layout.tex

@@ -0,0 +1,709 @@
+\documentclass[]{article}
+
+\usepackage{amsmath,amssymb}
+\usepackage{amsthm}
+
+\usepackage{graphicx,xcolor}
+
+\usepackage{algorithm,algpseudocode,float} 
+
+\renewcommand\thesubsubsection{\Alph{subsubsection})}
+
+\newtheorem{proposition}{Proposition}
+
+%opening
+\title{Optimal partition assignment in Garage}
+\author{Mendes}
+
+\begin{document}
+
+\maketitle
+
+\section{Introduction}
+
+\subsection{Context}
+
+Garage is an open-source distributed storage service blablabla$\dots$
+
+Every object to be stored in the system falls in a partition given by the last $k$ bits of its hash. There are $P=2^k$ partitions. Every partition will be stored on distinct nodes of the system. The goal of the assignment of partitions to nodes is to ensure (nodes and zone) redundancy and to be as efficient as possible.
+
+\subsection{Formal description of the problem}
+
+We are given a set of nodes $\mathbf{N}$ and a set of zones $\mathbf{Z}$. Every node $n$ has a non-negative storage capacity $c_n\ge 0$ and belongs to a zone $z\in \mathbf{Z}$. We are also given a number of partition $P>0$ (typically $P=256$).
+
+We would like to compute an assignment of nodes to partitions. We will impose some redundancy constraints to this assignment, and under these constraints, we want our system to have the largest storage capacity possible. To link storage capacity to partition assignment, we make the following assumption:
+\begin{equation}
+	\tag{H1}
+	\text{\emph{All partitions have the same size $s$.}}
+\end{equation}
+This assumption is justified by the dispersion of the hashing function, when the number of partitions is small relative to the number of stored large objects.
+
+Every node $n$ wille store some number $k_n$ of partitions. Hence the partitions stored by $n$ (and hence all partitions by our assumption) have there size bounded by $c_n/k_n$. This remark leads us to define the optimal size that we will want to maximize:
+
+\begin{equation}
+	\label{eq:optimal}
+	\tag{OPT}
+s^* = \min_{n \in N} \frac{c_n}{k_n}.
+\end{equation}
+
+When the capacities of the nodes are updated (this includes adding or removing a node), we want to update the assignment as well. However, transferring the data between nodes has a cost and we would like to limit the number of changes in the assignment. We make the following assumption:
+\begin{equation}
+	\tag{H2}
+	\text{\emph{Updates of capacity happens rarely relatively to object storing.}}
+\end{equation}
+This assumption justifies that when we compute the new assignment, it is worth to optimize the partition size \eqref{eq:optimal} first, and then, among the possible optimal solution, to try to minimize the number of partition transfers.
+
+For now, in the following, we ask the following redundancy constraint:
+
+\textbf{Parametric node and zone redundancy:} Given two integer parameters $1\le \rho_\mathbf{Z} \le \rho_\mathbf{N}$, we ask every partition to be stored on $\rho_\mathbf{N}$ distinct nodes, and these nodes must belong to at least $\rho_\mathbf{Z}$ distinct zones.
+
+
+\textbf{Mode 3-strict:}  every partition needs to be assignated to three nodes belonging to three different zones.
+
+\textbf{Mode 3:} every partition needs to be assignated to three nodes. We try to spread the three nodes over different zones as much as possible.
+
+\textbf{Warning:} This is a working document written incrementaly. The last version of the algorithm is the \textbf{parametric assignment} described in the next section.
+
+
+\section{Computation of a parametric assignment}
+\textbf{Attention : }We change notations in this section.
+
+Notations : let $P$ be the number of partitions, $N$ the number of nodes, $Z$ the number of zones. Let $\mathbf{P,N,Z}$ be the label sets of, respectively, partitions, nodes and zones.
+Let $s^*$ be the largest partition size achievable with the redundancy constraints. Let $(c_n)_{n\in \mathbf{N}}$ be the storage capacity of every node. 
+
+In this section, we propose a third specification of the problem. The user inputs two redundancy parameters $1\le \rho_\mathbf{Z} \le \rho_\mathbf{N}$. We compute an assignment $\alpha = (\alpha_p^1, \ldots, \alpha_p^{\rho_\mathbf{N}})_{p\in \mathbf{P}}$ such that every partition $p$ is associated to $\rho_\mathbf{N}$ distinct nodes $\alpha_p^1, \ldots, \alpha_p^{\rho_\mathbf{N}}$ and these nodes belong to at least $\rho_\mathbf{Z}$ distinct zones.
+
+If the layout contained a previous assignment $\alpha'$, we try to minimize the amount of data to transfer during the layout update by making $\alpha$ as close as possible to $\alpha'$.
+
+In the following subsections, we describe the successive steps of the algorithm we propose to compute $\alpha$.
+
+\subsubsection*{Algorithm}
+
+\begin{algorithmic}[1]
+	\Function{Compute Layout}{$\mathbf{N}$, $\mathbf{Z}$, $\mathbf{P}$, $(c_n)_{n\in \mathbf{N}}$, $\rho_\mathbf{N}$, $\rho_\mathbf{Z}$, $\alpha'$}
+	\State $s^* \leftarrow$ \Call{Compute Partition Size}{$\mathbf{N}$, $\mathbf{Z}$, $\mathbf{P}$, $(c_n)_{n\in \mathbf{N}}$, $\rho_\mathbf{N}$, $\rho_\mathbf{Z}$}
+	\State $G \leftarrow G(s^*)$
+	\State $f \leftarrow$ 	\Call{Compute Candidate Assignment}{$G$, $\alpha'$}
+	\State $f^* \leftarrow$ 	\Call{Minimize transfer load}{$G$, $f$, $\alpha'$}
+	\State Build $\alpha^*$ from $f^*$
+	\State \Return $\alpha^*$
+	\EndFunction
+\end{algorithmic}
+
+\subsubsection*{Complexity}
+As we will see in the next sections, the worst case complexity of this algorithm is $O(P^2 N^2)$. The minimization of transfer load is the most expensive step, and it can run with a timeout since it is only an optimization step. Without this step (or with a smart timeout), the worst cas complexity can be $O((PN)^{3/2}\log C)$ where $C$ is the total storage capacity of the cluster. 
+
+\subsection{Determination of the partition size $s^*$}
+
+Again, we will represent an assignment $\alpha$ as a flow in a specific graph $G$. We will not compute the optimal partition size $s^*$ a priori, but we will determine it by dichotomy, as the largest size $s$ such that the maximal flow achievable on $G=G(s)$ has value $\rho_\mathbf{N}P$. We will assume that the capacities are given in a small enough unit (say, Megabytes), and we will determine $s^*$ at the precision of the given unit.
+
+Given some candidate size value $s$, we describe the oriented weighted graph $G=(V,E)$ with vertex set $V$ arc set $E$.
+
+The set of vertices $V$ contains the source $\mathbf{s}$, the sink $\mathbf{t}$, vertices 
+$\mathbf{p^+, p^-}$ for every partition $p$, vertices $\mathbf{x}_{p,z}$ for every partition $p$ and zone $z$, and vertices $\mathbf{n}$ for every node $n$. 
+
+The set of arcs $E$ contains:
+\begin{itemize}
+	\item ($\mathbf{s}$,$\mathbf{p}^+$, $\rho_\mathbf{Z}$) for every partition $p$;
+	\item ($\mathbf{s}$,$\mathbf{p}^-$, $\rho_\mathbf{N}-\rho_\mathbf{Z}$) for every partition $p$;
+	\item ($\mathbf{p}^+$,$\mathbf{x}_{p,z}$, 1) for every partition $p$ and zone $z$;
+	\item ($\mathbf{p}^-$,$\mathbf{x}_{p,z}$, $\rho_\mathbf{N}-\rho_\mathbf{Z}$) for every partition $p$ and zone $z$;
+	\item ($\mathbf{x}_{p,z}$,$\mathbf{n}$, 1) for every partition $p$, zone $z$ and node $n\in z$;
+	\item ($\mathbf{n}$, $\mathbf{t}$, $\lfloor c_n/s \rfloor$) for every node $n$.
+\end{itemize}
+
+In the following complexity calculations, we will use the number of vertices and edges of $G$. Remark from now that $\# V = O(PZ)$ and $\# E = O(PN)$.
+
+\begin{proposition}
+	An assignment $\alpha$ is realizable with partition size $s$ and the redundancy constraints $(\rho_\mathbf{N},\rho_\mathbf{Z})$ if and only if there exists a maximal flow function $f$ in $G$ with total flow $\rho_\mathbf{N}P$, such that the arcs ($\mathbf{x}_{p,z}$,$\mathbf{n}$, 1) used are exactly those for which $p$ is associated to $n$ in $\alpha$.
+\end{proposition}
+\begin{proof}
+	Given such flow $f$, we can reconstruct a candidate $\alpha$. In $f$, the flow passing through $\mathbf{p^+}$ and $\mathbf{p^-}$ is $\rho_\mathbf{N}$, and since the outgoing capacity of every $\mathbf{x}_{p,z}$ is 1, every partition is associated to $\rho_\mathbf{N}$ distinct nodes. The fraction $\rho_\mathbf{Z}$ of the flow passing through every $\mathbf{p^+}$ must be spread over as many distinct zones as every arc outgoing from $\mathbf{p^+}$ has capacity 1. So the reconstructed $\alpha$ verifies the redundancy constraints. For every node $n$, the flow between $\mathbf{n}$ and $\mathbf{t}$ corresponds to the number of partitions associated to $n$. By construction of $f$, this does not exceed $\lfloor c_n/s \rfloor$. We assumed that the partition size is $s$, hence this association does not exceed the storage capacity of the nodes.
+	
+	In the other direction, given an assignment $\alpha$, one can similarly check that the facts that $\alpha$ respects the redundancy constraints, and the storage capacities of the nodes, are necessary condition to construct a maximal flow function $f$.
+\end{proof}
+
+\textbf{Implementation remark:} In the flow algorithm, while exploring the graph, we explore the neighbours of every vertex in a random order to heuristically spread the association between nodes and partitions.
+
+\subsubsection*{Algorithm}
+With this result mind, we can describe the first step of our algorithm. All divisions are supposed to be integer division.
+\begin{algorithmic}[1]
+	\Function{Compute Partition Size}{$\mathbf{N}$, $\mathbf{Z}$, $\mathbf{P}$, $(c_n)_{n\in \mathbf{N}}$, $\rho_\mathbf{N}$, $\rho_\mathbf{Z}$}
+	
+	\State Build the graph $G=G(s=1)$
+	\State $ f \leftarrow$ \Call{Maximal flow}{$G$}
+	\If{$f.\mathrm{total flow} < \rho_\mathbf{N}P$}
+	
+	\State \Return Error: capacities too small or constraints too strong.
+	\EndIf
+	
+	\State $s^- \leftarrow 1$
+	\State $s^+ \leftarrow 1+\frac{1}{\rho_\mathbf{N}}\sum_{n \in \mathbf{N}} c_n$
+	
+	\While{$s^-+1 < s^+$}
+	\State Build the graph $G=G(s=(s^-+s^+)/2)$
+	\State $ f \leftarrow$ \Call{Maximal flow}{$G$}
+	\If{$f.\mathrm{total flow} < \rho_\mathbf{N}P$}
+	\State $s^+ \leftarrow (s^- + s^+)/2$
+	\Else
+	\State $s^- \leftarrow (s^- + s^+)/2$
+	\EndIf
+	\EndWhile
+	
+	\State \Return $s^-$
+	\EndFunction
+\end{algorithmic}
+
+\subsubsection*{Complexity}
+
+To compute the maximal flow, we use Dinic's algorithm. Its complexity on general graphs is $O(\#V^2 \#E)$, but on graphs with edge capacity bounded by a constant, it turns out to be $O(\#E^{3/2})$. The graph $G$ does not fall in this case since the capacities of the arcs incoming to $\mathbf{t}$ are far from bounded. However, the proof of this complexity works readily for graph where we only ask the edges \emph{not} incoming to the sink $\mathbf{t}$ to have their capacities bounded by a constant. One can find the proof of this claim in \cite[Section 2]{even1975network}.
+The dichotomy adds a logarithmic factor $\log (C)$ where $C=\sum_{n \in \mathbf{N}} c_n$ is the total capacity of the cluster. The total complexity of this first function is hence 
+$O(\#E^{3/2}\log C ) = O\big((PN)^{3/2} \log C\big)$.
+
+\subsubsection*{Metrics}
+We can display the discrepancy between the computed $s^*$ and the best size we could hope for a given total capacity, that is $C/\rho_\mathbf{N}$.
+
+\subsection{Computation of a candidate assignment}
+
+Now that we have the optimal partition size $s^*$, to compute a candidate assignment, it would be enough to compute a maximal flow function $f$ on $G(s^*)$. This is what we do if there was no previous assignment $\alpha'$.
+
+If there was some $\alpha'$, we add a step that will heuristically help to obtain a candidate $\alpha$ closer to $\alpha'$. to do so, we fist compute a flow function $\tilde{f}$ that uses only the partition-to-node association appearing in $\alpha'$. Most likely, $\tilde{f}$ will not be a maximal flow of $G(s^*)$. In Dinic's algorithm, we can start from a non maximal flow function and then discover improving paths. This is what we do in starting from $\tilde{f}$. The hope\footnote{This is only a hope, because one can find examples where the construction of $f$ from $\tilde{f}$ produces an assignment $\alpha$ that is not as close as possible to $\alpha'$.} is that the final flow function $f$ will tend to keep the associations appearing in $\tilde{f}$.
+
+More formally, we construct the graph $G_{|\alpha'}$ from $G$ by removing all the arcs $(\mathbf{x}_{p,z},\mathbf{n}, 1)$ where $p$ is not associated to $n$ in $\alpha'$. We compute a maximal flow function $\tilde{f}$ in $G_{|\alpha'}$. $\tilde{f}$ is also a valid (most likely non maximal) flow function in $G$. We compute a maximal flow function $f$ on $G$ by starting Dinic's algorithm on $\tilde{f}$.
+
+\subsubsection*{Algorithm}
+\begin{algorithmic}[1]
+	\Function{Compute Candidate Assignment}{$G$, $\alpha'$}
+	\State Build the graph $G_{|\alpha'}$
+	\State $ \tilde{f} \leftarrow$ \Call{Maximal flow}{$G_{|\alpha'}$}
+	\State $ f \leftarrow$ \Call{Maximal flow from flow}{$G$, $\tilde{f}$}
+	\State \Return $f$
+	\EndFunction
+\end{algorithmic}
+
+\textbf{Remark:} The function ``Maximal flow'' can be just seen as the function ``Maximal flow from flow'' called with the zero flow function as starting flow.
+
+\subsubsection*{Complexity}
+From the consideration of the last section, we have the complexity of the Dinic's algorithm $O(\#E^{3/2}) = O((PN)^{3/2})$.
+
+\subsubsection*{Metrics}
+
+We can display the flow value of $\tilde{f}$, which is an upper bound of the distance between $\alpha$ and $\alpha'$. It might be more a Debug level display than Info.
+
+\subsection{Minimization of the transfer load}
+
+Now that we have a candidate flow function $f$, we want to modify it to make its associated assignment as close as possible to $\alpha'$. Denote by $f'$ the maximal flow associated to $\alpha'$, and let $d(f, f')$ be distance between the associated assignments\footnote{It is the number of arcs of type $(\mathbf{x}_{p,z},\mathbf{n})$ saturated in one flow and not in the other.}. 
+We want to build a sequence $f=f_0, f_1, f_2 \dots$ of maximal flows such that $d(f_i, \alpha')$ decreases as $i$ increases. The distance being a non-negative integer, this sequence of flow functions must be finite. We now explain how to find some improving $f_{i+1}$ from $f_i$.
+
+For any maximal flow $f$ in $G$, we define the oriented weighted graph $G_f=(V, E_f)$ as follows. The vertices of $G_f$ are the same as the vertices of $G$. $E_f$ contains the arc $(v_1,v_2, w)$ between vertices $v_1,v_2\in V$ with weight $w$ if and only if the arc $(v_1,v_2)$ is not saturated in $f$ (i.e. $c(v_1,v_2)-f(v_1,v_2) \ge 1$, we also consider reversed arcs). The weight $w$ is: 
+\begin{itemize}
+	\item  $-1$  if $(v_1,v_2)$ is of type $(\mathbf{x}_{p,z},\mathbf{n})$ or $(\mathbf{x}_{p,z},\mathbf{n})$ and is saturated in only one of the two flows $f,f'$;
+	\item $+1$  if $(v_1,v_2)$ is of type $(\mathbf{x}_{p,z},\mathbf{n})$ or $(\mathbf{x}_{p,z},\mathbf{n})$ and is saturated in either both or none of the two flows $f,f'$;
+	\item $0$ otherwise.
+\end{itemize}
+
+If $\gamma$ is a simple cycle of arcs in $G_f$, we define its weight $w(\gamma)$ as the sum of the weights of its arcs. We can add $+1$ to the value of $f$ on the arcs of $\gamma$, and by construction of $G_f$ and the fact that $\gamma$ is a cycle, the function that we get is still a valid flow function on $G$, it is maximal as it has the same flow value as $f$. We denote this new function $f+\gamma$.
+
+\begin{proposition}
+	Given a maximal flow $f$ and a simple cycle $\gamma$ in $G_f$, we have $d(f+\gamma, f') - d(f,f') = w(\gamma)$.
+\end{proposition}
+\begin{proof}
+	Let $X$ be the set of arcs of type $(\mathbf{x}_{p,z},\mathbf{n})$. Then we can express $d(f,f')$ as
+	\begin{align*}
+		d(f,f') & = \#\{e\in X ~|~ f(e)\neq f'(e)\}
+		= \sum_{e\in X} 1_{f(e)\neq f'(e)} \\
+		& = \frac{1}{2}\big( \#X + \sum_{e\in X} 1_{f(e)\neq f'(e)} - 1_{f(e)= f'(e)} \big).
+	\end{align*}
+	We can express the cycle weight as
+	\begin{align*}
+		w(\gamma) &  = \sum_{e\in X, e\in \gamma} - 1_{f(e)\neq f'(e)} + 1_{f(e)= f'(e)}.
+	\end{align*}
+	Remark that since we passed on unit of flow in $\gamma$ to construct $f+\gamma$, we have for any $e\in X$, $f(e)=f'(e)$ if and only if $(f+\gamma)(e) \neq f'(e)$.
+	Hence
+	\begin{align*}
+		w(\gamma) &  = \frac{1}{2}(w(\gamma) + w(\gamma)) \\
+		&= \frac{1}{2} \Big( 
+		\sum_{e\in X, e\in \gamma} - 1_{f(e)\neq f'(e)} +  1_{f(e)= f'(e)} \\
+		& \qquad +
+		\sum_{e\in X, e\in \gamma} 1_{(f+\gamma)(e)\neq f'(e)} +  1_{(f+\gamma)(e)= f'(e)}
+		\Big).
+	\end{align*}
+	Plugging this in the previous equation, we find that 
+	$$d(f,f')+w(\gamma) = d(f+\gamma, f').$$
+\end{proof}
+
+This result suggests that given some flow $f_i$, we just need to find a negative cycle $\gamma$ in $G_{f_i}$ to construct $f_{i+1}$ as $f_i+\gamma$. The following proposition ensures that this greedy strategy reaches an optimal flow. 
+
+\begin{proposition}
+	For any maximal flow $f$, $G_f$ contains a negative cycle if and only if there exists a maximal flow $f^*$ in $G$ such that $d(f^*, f') < d(f, f')$.
+\end{proposition}
+\begin{proof}
+	Suppose that there is such flow $f^*$. Define the oriented multigraph $M_{f,f^*}=(V,E_M)$ with the same vertex set $V$ as in $G$, and for every $v_1,v_2 \in V$, $E_M$ contains $(f^*(v_1,v_2) - f(v_1,v_2))_+$  copies of the arc $(v_1,v_2)$. For every vertex $v$, its total degree (meaning its outer degree minus its inner degree) is equal to 
+	\begin{align*}
+		\deg v & = \sum_{u\in V} (f^*(v,u) - f(v,u))_+ - \sum_{u\in V} (f^*(u,v) - f(u,v))_+  \\
+		& = \sum_{u\in V} f^*(v,u) - f(v,u) = \sum_{u\in V} f^*(v,u) - \sum_{u\in V}  f(v,u).
+	\end{align*}
+	The last two sums are zero for any inner vertex since $f,f^*$ are flows, and they are equal on the source and sink since the two flows are both maximal and have hence the same value. Thus, $\deg v = 0$ for every vertex $v$.
+	
+	This implies that the multigraph $M_{f,f^*}$ is the union of disjoint simple cycles. $f$ can be transformed into $f^*$ by pushing a mass 1 along all these cycles in any order. Since $d(f^*, f')<d(f,f')$, there must exists one of these simple cycles $\gamma$ with $d(f+\gamma, f') < d(f, f')$. Finally, since we can push a mass in $f$ along $\gamma$, it must appear in $G_f$. Hence $\gamma$ is a cycle of $G_f$ with negative weight.
+\end{proof}
+
+In the next section we describe the corresponding algorithm. Instead of discovering only one cycle, we are allowed to discover a set $\Gamma$ of disjoint negative cycles. 
+
+\subsubsection*{Algorithm}
+\begin{algorithmic}[1]
+	\Function{Minimize transfer load}{$G$, $f$, $\alpha'$}
+	\State Build the graph $G_f$
+	\State $\Gamma \leftarrow$ \Call{Detect Negative Cycles}{$G_f$}
+	\While{$\Gamma \neq \emptyset$}
+	\ForAll{$\gamma \in \Gamma$}
+	\State $f \leftarrow f+\gamma$
+	\EndFor
+	\State Update $G_f$
+	\State $\Gamma \leftarrow$ \Call{Detect Negative Cycles}{$G_f$}
+	\EndWhile
+	\State \Return $f$
+	\EndFunction
+\end{algorithmic}
+
+\subsubsection*{Complexity}
+The distance $d(f,f')$ is bounded by the maximal number of differences in the associated assignment. If these assignment are totally disjoint, this distance is $2\rho_N P$. At every iteration of the While loop, the distance decreases, so there is at most $O(\rho_N P) = O(P)$ iterations.
+
+The detection of negative cycle is done with the Bellman-Ford algorithm, whose complexity should normally be $O(\#E\#V)$. In our case, it amounts to $O(P^2ZN)$. Multiplied by the complexity of the outer loop, it amounts to $O(P^3ZN)$ which is a lot when the number of partitions and nodes starts to be large. To avoid that, we adapt the Bellman-Ford algorithm.
+
+The Bellman-Ford algorithm runs $\#V$ iterations of an outer loop, and an inner loop over $E$. The idea is to compute the shortest paths from a source vertex $v$ to all other vertices. After $k$ iterations of the outer loop, the algorithm has computed all shortest path of length at most $k$. All simple paths have length at most $\#V-1$, so if there is an update in the last iteration of the loop, it means that there is a negative cycle in the graph. The observation that will enable us to improve the complexity is the following:
+
+\begin{proposition}
+	In the graph $G_f$ (and $G$), all simple paths have a length at most $4N$.
+\end{proposition}
+\begin{proof}
+	Since $f$ is a maximal flow, there is no outgoing edge from $\mathbf{s}$ in $G_f$. One can thus check than any simple path of length 4 must contain at least two node of type $\mathbf{n}$. Hence on a path, at most 4 arcs separate two successive nodes of type $\mathbf{n}$. 
+\end{proof}
+
+Thus, in the absence of negative cycles, shortest paths in $G_f$ have length at most $4N$. So we can do only $4N+1$ iterations of the outer loop in Bellman-Ford algorithm. This makes the complexity of the detection of one set of cycle to be $O(N\#E) = O(N^2 P)$.
+
+With this improvement, the complexity of the whole algorithm is, in the worst case, $O(N^2P^2)$. However, since we detect several cycles at once and we start with a flow that might be close to the previous one, the number of iterations of the outer loop might be smaller in practice.
+
+
+
+\subsubsection*{Metrics}
+We can display the node and zone utilization ratio, by dividing the flow passing through them divided by their outgoing capacity. In particular, we can pinpoint saturated nodes and zones (i.e. used at their full potential).
+
+We can display the distance to the previous assignment, and the number of partition transfers.
+
+
+
+
+
+\section{Properties of an optimal 3-strict assignment}
+
+\subsection{Optimal assignment}
+\label{sec:opt_assign}
+
+For every zone $z\in Z$, define the zone capacity $c_z = \sum_{v, z_v=z} c_v$ and define $C = \sum_v c_v = \sum_z c_z$.
+
+One can check that the best we could be doing to maximize $s^*$ would be to use the nodes proportionally to their capacity. This would yield $s^*=C/(3N)$. This is not possible because of (i) redundancy constraints and (ii) integer rounding but it gives and upper bound. 
+
+\subsubsection*{Optimal utilization}
+
+We call an \emph{utilization} a collection of non-negative integers $(n_v)_{v\in V}$ such that $\sum_v n_v = 3N$ and for every zone $z$, $\sum_{v\in z} n_v \le N$. We call such utilization \emph{optimal} if it maximizes $s^*$.
+
+We start by computing a node sub-utilization $(\hat{n}_v)_{v\in V}$ such that for every zone $z$, $\sum_{v\in z} \hat{n}_v \le N$ and we show that there is an optimal utilization respecting the constraints and such that $\hat{n}_v \le n_v$ for every node.
+
+Assume that there is a zone $z_0$ such that $c_{z_0}/C \ge 1/3$. Then for any $v\in z_0$, we define 
+$$\hat{n}_v = \left\lfloor\frac{c_v}{c_{z_0}}N\right\rfloor.$$ 
+This choice ensures for any such $v$ that
+$$
+\frac{c_v}{\hat{n}_v} \ge \frac{c_{z_0}}{N} \ge \frac{C}{3N}
+$$
+which is the universal upper bound on $s^*$. Hence any optimal utilization $(n_v)$ can be modified to another optimal utilization such that $n_v\ge \hat{n}_v$
+
+Because $z_0$ cannot store more than $N$ partition occurences, in any assignment, at least $2N$ partitions must be assignated to the zones $Z\setminus\{z_0\}$. Let $C_0 = C-c_{z_0}$. Suppose that there exists a zone $z_1\neq z_0$ such that $c_{z_1}/C_0 \ge 1/2$. Then, with the same argument as for $z_0$, we can define 
+$$\hat{n}_v = \left\lfloor\frac{c_v}{c_{z_1}}N\right\rfloor$$
+for every $v\in z_1$.
+
+Now we can assign the remaining partitions. Let $(\hat{N}, \hat{C})$ to be
+\begin{itemize}
+	\item $(3N,C)$ if we did not find any $z_0$;
+	\item $(2N,C-c_{z_0})$ if there was a $z_0$ but no $z_1$;
+	\item $(N,C-c_{z_0}-c_{z_1})$ if there was a $z_0$ and a $z_1$.
+\end{itemize}
+Then at least $\hat{N}$ partitions must be spread among the remaining zones. Hence $s^*$ is upper bounded by $\hat{C}/\hat{N}$ and without loss of generality, we can define, for every node that is not in $z_0$ nor $z_1$, 
+$$\hat{n}_v = \left\lfloor\frac{c_v}{\hat{C}}\hat{N}\right\rfloor.$$ 
+
+We constructed a sub-utilization $\hat{n}_v$. Now notice that $3N-\sum_v \hat{n}_v \le \# V$ where $\# V$ denotes the number of nodes. We can iteratively pick a node $v^*$ such that 
+\begin{itemize}
+	\item $\sum_{v\in z_{v^*}} \hat{n}_v < N$ where $z_{v^*}$ is the zone of $v^*$;
+	\item $v^*$ maximizes the quantity $c_v/(\hat{n}_v+1)$ among the vertices satisfying the first condition (i.e. not in a saturated zone).
+\end{itemize}
+We iterate these instructions until $\sum_v \hat{n}_v= 3N$, and at this stage we define $(n_v) = (\hat{n}_v)$. It is easy to prove by induction that at every step, there is an optimal utilization that is pointwise larger than $\hat{n}_v$, and in particular, that $(n_v)$ is optimal.
+
+\subsubsection*{Existence of an optimal assignment}
+
+As for now, the \emph{optimal utilization} that we obtained is just a vector of numbers and it is not clear that it can be realized as the utilization of some concrete assignment. Here is a way to get a concrete assignment.
+
+Define $3N$ tokens $t_1,\ldots, t_{3N}\in V$ as follows:
+\begin{itemize}
+	\item Enumerate the zones $z$ of $Z$ in any order;
+	\item enumerate the nodes $v$ of $z$ in any order;
+	\item repeat $n_v$ times the token $v$.
+\end{itemize} 
+Then for $1\le i \le N$, define the triplet $T_i$ to be 
+$(t_i, t_{i+N}, t_{i+2N})$. Since the same nodes of a zone appear contiguously, the three nodes of a triplet must belong to three distinct zones.
+
+However simple, this solution to go from an utilization to an assignment has the drawback of not spreading the triplets: a node will tend to be associated to the same two other nodes for many partitions. Hence, during data transfer, it will tend to use only two link, instead of spreading the bandwith use over many other links to other nodes. To achieve this goal, we will reframe the search of an assignment as a flow problem. and in the flow algorithm, we will introduce randomness in the order of exploration. This will be sufficient to obtain a good dispersion of the triplets.
+
+\begin{figure}
+	\centering
+	\includegraphics[width=0.9\linewidth]{figures/naive}
+	\caption{On the left, the creation of a concrete assignment with the naive approach of repeating tokens. On the right, the zones containing the nodes.}
+\end{figure}
+
+\subsubsection*{Assignment as a maximum flow problem}
+
+We describe the flow problem via its graph $(X,E)$ where $X$ is a set of vertices, and $E$ are directed weighted edges between the vertices. For every zone $z$, define $n_z=\sum_{v\in z} n_v$.
+
+The set of vertices $X$ contains the source $\mathbf{s}$ and the sink $\mathbf{t}$; a vertex $\mathbf{x}_z$ for every zone $z\in Z$, and a vertex $\mathbf{y}_i$ for every partition index $1\le i\le N$. 
+
+The set of edges $E$ contains
+\begin{itemize}
+	\item the edge $(\mathbf{s}, \mathbf{x}_z, n_z)$ for every zone $z\in Z$;
+	\item the edge $(\mathbf{x}_z, \mathbf{y}_i, 1)$ for every zone $z\in Z$ and partition $1\le i\le N$;
+	\item the edge $(\mathbf{y}_i, \mathbf{t}, 3)$ for every partition $1\le i\le N$.
+\end{itemize}
+
+\begin{figure}[b]
+	\centering
+	\includegraphics[width=0.6\linewidth]{figures/flow}
+	\caption{Flow problem to compute and optimal assignment.}
+\end{figure}
+
+We first show the equivalence between this problem and and the construction of an assignment. Given some optimal assignment $(n_v)$, define the flow $f:E\to \mathbb{N}$ that saturates every edge from $\mathbf{s}$ or to $\mathbf{t}$, takes value $1$ on the edge between $\mathbf{x}_z$ and $\mathbf{y}_i$ if partition $i$ is stored in some node of the zone $z$, and $0$ otherwise. One can easily check that $f$ thus defined is indeed a flow and is maximum.
+
+Reciprocally, by the existence of maximum flows constructed from optimal assignments, any maximum flow must saturate the edges linked to the source or the sink. It can only take value 0 or 1 on the other edge, and every partition vertex is associated to exactly three distinct zone vertices. Every zone is associated to exactly $n_z$ partitions.
+
+A maximum flow can be constructed using, for instance, Dinic's algorithm. This algorithm works by discovering augmenting path to iteratively increase the flow. During the exploration of the graph to find augmenting path, we can shuffle the order of enumeration of the neighbours to spread the associations between zones and partitions.
+
+Once we have such association, we can randomly distribute the $n_z$ edges picked for every zone $z$ to its nodes $v\in z$ such that every such $v$ gets $n_z$ edges. This defines an optimal assignment of partitions to nodes.
+
+
+\subsection{Minimal transfer}
+
+Assume that there was a previous assignment $(T'_i)_{1\le i\le N}$ corresponding to utilizations $(n'_v)_{v\in V}$. We would like the new computed assignment $(T_i)_{1\le i\le N}$ from some $(n_v)_{v\in V}$ to minimize the number of partitions that need to be transferred. We can imagine two different objectives corresponding to different hypotheses: 
+\begin{equation}
+	\tag{H3A}
+	\label{hyp:A}
+	\text{\emph{Transfers between different zones cost much more than inside a zone.}}
+\end{equation}
+\begin{equation}
+	\tag{H3B}
+	\label{hyp:B}
+	\text{\emph{Changing zone is not the largest cost when transferring a partition.}}
+\end{equation}
+
+In case $A$, our goal will be to minimize the number of changes of zone in the assignment of partitions to zone. More formally, we will maximize the quantity
+$$
+Q_Z := 
+\sum_{1\le i\le N} 
+\#\{z\in Z ~|~ z\cap T_i \neq \emptyset, z\cap T'_i \neq \emptyset \}
+.$$
+
+In case $B$, our goal will be to minimize the number of changes of nodes in the assignment of partitions to nodes. We will maximize the quantity
+$$
+Q_V :=
+\sum_{1\le i\le N} \#(T_i \cap T'_i).
+$$
+
+It is tempting to hope that there is a way to maximize both quantity, that having the least discrepancy in terms of nodes will lead to the least discrepancy in terms of zones. But this is actually wrong! We propose the following counter-example to convince the reader:
+
+We consider eight nodes $a, a', b, c, d, d', e, e'$ belonging to five different zones $\{a,a'\}, \{b\}, \{c\}, \{d,d'\}, \{e, e'\}$. We take three partitions ($N=3$), that are originally assigned with some utilization $(n'_v)_{v\in V}$ as follows:
+$$
+T'_1=(a,b,c) \qquad
+T'_2=(a',b,d) \qquad 
+T'_3=(b,c,e).
+$$
+This assignment, with updated utilizations $(n_v)_{v\in V}$ minimizes the number of zone changes:
+$$
+T_1=(d,b,c) \qquad
+T_2=(a,b,d) \qquad 
+T_3=(b,c,e').
+$$
+This one, with the same utilization, minimizes the number of node changes:
+$$
+T_1=(a,b,c) \qquad
+T_2=(e',b,d) \qquad 
+T_3=(b,c,d').
+$$
+One can check that in this case, it is impossible to minimize both the number of zone and node changes.
+
+Because of the redundancy constraint, we cannot use a greedy algorithm to just replace nodes in the triplets to try to get the new utilization rate: this could lead to blocking situation where there is still a hole to fill in a triplet but no available node satisfies the zone separation constraint. To circumvent this issue, we propose an algorithm based on finding cycles in a graph encoding of the assignment. As in section \ref{sec:opt_assign}, we can explore the neigbours in a random order in the graph algorithms, to spread the triplets distribution.
+
+
+\subsubsection{Minimizing the zone discrepancy}
+
+
+First, notice that, given an assignment of partitions to \emph{zones}, it is easy to deduce an assignment to \emph{nodes} that minimizes the number of transfers for this zone assignment: For every zone $z$ and every node $v\in z$, pick in any way a set $P_v$ of partitions that where assigned to $v$ in $T'$, to $z_v$ in $T$, with the cardinality of $P_v$ smaller than $n_v$. Once all these sets are chosen, complement the assignment to reach the right utilization for every node. If $\#P_v > n_v$, it means that all the partitions that could stay in $v$ (i.e. that were already in $v$ and are still assigned to its zone) do stay in $v$. If $\#P_v = n_v$, then $n_v$ partitions stay in $v$, which is the number of partitions that need to be in $v$ in the end. In both cases, we could not hope for better given the partition to zone assignment.
+
+Our goal now is to find a assignment of partitions to zones that minimizes the number of zone transfers. To do so we are going to represent an assignment as a graph.
+
+Let $G_T=(X,E_T)$ be the directed weighted graph with vertices $(\mathbf{x}_i)_{1\le i\le N}$ and $(\mathbf{y}_z)_{z\in Z}$. For any $1\le i\le N$ and $z\in Z$, $E_T$ contains  the arc:
+\begin{itemize}
+	\item $(\mathbf{x}_i, \mathbf{y}_z, +1)$, if $z$ appears in $T_i'$ and $T_i$;
+	\item $(\mathbf{x}_i, \mathbf{y}_z, -1)$, if $z$ appears in $T_i$ but not in $T'_i$;
+	\item $(\mathbf{y}_z, \mathbf{x}_i, -1)$, if $z$ appears in $T'_i$ but not in $T_i$;
+	\item $(\mathbf{y}_z, \mathbf{x}_i, +1)$, if $z$ does not appear in $T'_i$ nor in $T_i$.
+\end{itemize}
+In other words, the orientation of the arc encodes whether partition $i$ is stored in zone $z$ in the assignment $T$ and the weight $\pm 1$ encodes whether this corresponds to what happens in the assignment $T'$.
+
+\begin{figure}[t]
+	\centering
+	\begin{minipage}{.40\linewidth}
+		\centering
+		\includegraphics[width=.8\linewidth]{figures/mini_zone}
+	\end{minipage}
+	\begin{minipage}{.55\linewidth}
+		\centering
+		\includegraphics[width=.8\linewidth]{figures/mini_node}
+	\end{minipage}
+	\caption{On the left: the graph $G_T$ encoding an assignment to minimize the zone discrepancy. On the right: the graph $G_T$ encoding an assignment to minimize the node discrepancy.}
+\end{figure}
+
+
+Notice that at every partition, there are three outgoing arcs, and at every zone, there are $n_z$ incoming arcs. Moreover, if $w(e)$ is the weight of an arc $e$, define the weight of $G_T$ by
+\begin{align*}
+w(G_T) := \sum_{e\in E} w(e) &= \#Z \times N - 4 \sum_{1\le i\le N} \#\{z\in Z ~|~ z\cap T_i = \emptyset, z\cap T'_i \neq \emptyset\}	\\
+&=\#Z \times N - 4 \sum_{1\le i\le N} 3- \#\{z\in Z ~|~ z\cap T_i \neq \emptyset, z\cap T'_i \neq \emptyset\} \\
+&= (\#Z-12)N + 4 Q_Z.
+\end{align*}
+Hence maximizing $Q_Z$ is equivalent to maximizing $w(G_T)$.
+
+Assume that their exist some assignment $T^*$ with the same utilization $(n_v)_{v\in V}$. Define $G_{T^*}$ similarly and consider the set $E_\mathrm{Diff} = E_T \setminus E_{T^*}$ of arcs that appear only in $G_T$. Since all vertices have the same number of incoming arcs in $G_T$ and $G_{T^*}$, the vertices of the graph $(X, E_\mathrm{Diff})$ must all have the same number number of incoming and outgoing arrows. So $E_\mathrm{Diff}$ can be expressed as a union of disjoint cycles. Moreover, the edges of $E_\mathrm{Diff}$ must appear in $E_{T^*}$ with reversed orientation and opposite weight. Hence, we have
+$$
+ w(G_T) - w(G_{T^*}) = 2 \sum_{e\in E_\mathrm{Diff}} w(e). 
+$$
+Hence, if $T$ is not optimal, there exists some $T^*$ with $w(G_T) < w(G_{T^*})$, and by the considerations above, there must exist a cycle in $E_\mathrm{Diff}$, and hence in $G_T$, with negative weight. If we reverse the edges and weights along this cycle, we obtain some graph. Since we did not change the incoming degree of any vertex, this is the graph encoding of some valid assignment $T^+$ such that $w(G_{T^+}) > w(G_T)$. We can iterate this operation until there is no other assignment $T^*$ with larger weight, that is until we obtain an optimal assignment.
+
+
+
+\subsubsection{Minimizing the node discrepancy}
+
+We will follow an approach similar to the one where we minimize the zone discrepancy. Here we will directly obtain a node assignment from a graph encoding.
+
+Let $G_T=(X,E_T)$ be the directed weighted graph with vertices $(\mathbf{x}_i)_{1\le i\le N}$, $(\mathbf{y}_{z,i})_{z\in Z, 1\le i\le N}$ and $(\mathbf{u}_v)_{v\in V}$. For any $1\le i\le N$ and $z\in Z$, $E_T$ contains  the arc:
+\begin{itemize}
+	\item $(\mathbf{x}_i, \mathbf{y}_{z,i}, 0)$, if $z$ appears in $T_i$;
+	\item $(\mathbf{y}_{z,i}, \mathbf{x}_i, 0)$, if $z$ does not appear in $T_i$.
+\end{itemize}
+For any $1\le i\le N$ and $v\in V$, $E_T$ contains  the arc:
+\begin{itemize}
+	\item $(\mathbf{y}_{z_v,i}, \mathbf{u}_v, +1)$, if $v$ appears in $T_i'$ and $T_i$;
+	\item $(\mathbf{y}_{z_v,i}, \mathbf{u}_v, -1)$, if $v$ appears in $T_i$ but not in $T'_i$;
+	\item $(\mathbf{u}_v, \mathbf{y}_{z_v,i}, -1)$, if $v$ appears in $T'_i$ but not in $T_i$;
+	\item $(\mathbf{u}_v, \mathbf{y}_{z_v,i}, +1)$, if $v$ does not appear in $T'_i$ nor in $T_i$.
+\end{itemize}
+Every vertex $\mathbb{x}_i$ has outgoing degree 3, every vertex $\mathbf{y}_{z,v}$ has outgoing degree 1, and every vertex $\mathbf{u}_v$ has incoming degree $n_v$. 
+Remark that any graph respecting these degree constraints is the encoding of a valid assignment with utilizations $(n_v)_{v\in V}$, in particular no partition is stored in two nodes of the same zone.
+
+We define $w(G_T)$ similarly:
+\begin{align*}
+	w(G_T) := \sum_{e\in E_T} w(e) &= \#V \times N - 4\sum_{1\le i\le N} 3-\#(T_i\cap T'_i) \\
+	&= (\#V-12)N + 4Q_V.
+\end{align*}
+
+Exactly like in the previous section, the existence of an assignment with larger weight implies the existence of a negatively weighted cycle in $G_T$. Reversing this cycle gives us the encoding of a valid assignment with a larger weight. Iterating this operation yields an optimal assignment.
+
+
+\subsubsection{Linear combination of both criteria}
+
+In the graph $G_T$ defined in the previous section, instead of having weights $0$ and $\pm 1$, we could be having weights $\pm\alpha$ between $\mathbf{x}$ and  $\mathbf{y}$ vertices, and weights $\pm\beta$ between $\mathbf{y}$ and  $\mathbf{u}$ vertices, for some $\alpha,\beta>0$ (we have positive weight if the assignment corresponds to $T'$ and negative otherwise). Then
+\begin{align*}
+	w(G_T) &= \sum_{e\in E_T} w(e) = 
+	\alpha \big( (\#Z-12)N + 4 Q_Z\big) + 
+	\beta \big( (\#V-12)N + 4 Q_V\big) \\
+	&= \mathrm{const}+ 4(\alpha Q_Z + \beta Q_V).
+\end{align*}
+So maximizing the weight of such graph encoding would be equivalent to maximizing a linear combination of $Q_Z$ and $Q_V$.
+
+
+\subsection{Algorithm}
+We give a high level description of the algorithm to compute an optimal 3-strict assignment. The operations appearing at lines 1,2,4 are respectively described by Algorithms \ref{alg:util},\ref{alg:opt} and \ref{alg:mini}.
+
+
+
+\begin{algorithm}[H]
+	\caption{Optimal 3-strict assignment}
+	\label{alg:total}
+	\begin{algorithmic}[1]
+	\Function{Optimal 3-strict assignment}{$N$, $(c_v)_{v\in V}$, $T'$}
+		\State $(n_v)_{v\in V} \leftarrow$ \Call{Compute optimal utilization}{$N$, $(c_v)_{v\in V}$}
+		\State $(T_i)_{1\le i\le N} \leftarrow$ \Call{Compute candidate assignment}{$N$, $(n_v)_{v\in V}$}
+		\If {there was a previous assignment $T'$}
+			\State $T \leftarrow$ \Call{Minimization of transfers}{$(T_i)_{1\le i\le N}$, $(T'_i)_{1\le i\le N}$}
+		\EndIf
+		\State \Return $T$.
+	\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+We give some considerations of worst case complexity for these algorithms. In the following, we assume $N>\#V>\#Z$. The complexity of Algorithm \ref{alg:total} is $O(N^3\# Z)$ if we assume \eqref{hyp:A} and $O(N^3 \#Z \#V)$ if we assume \eqref{hyp:B}.
+
+Algorithm \ref{alg:util} can be implemented with complexity $O(\#V^2)$. The complexity of the function call at line \ref{lin:subutil} is $O(\#V)$. The difference between the sum of the subutilizations and $3N$ is at most the sum of the rounding errors when computing the $\hat{n}_v$. Hence it is bounded by $\#V$ and the loop at line \ref{lin:loopsub} is iterated at most $\#V$ times. Finding the minimizing $v$ at line \ref{lin:findmin} takes $O(\#V)$ operations (naively, we could also use a heap).
+
+Algorithm \ref{alg:opt} can be implemented with complexity $O(N^3\times \#Z)$. The flow graph has $O(N+\#Z)$ vertices and $O(N\times \#Z)$ edges. Dinic's algorithm has complexity $O(\#\mathrm{Vertices}^2\#\mathrm{Edges})$ hence in our case it is $O(N^3\times \#Z)$.
+ 
+Algorithm \ref{alg:mini} can be implented with complexity $O(N^3\# Z)$ under \eqref{hyp:A} and $O(N^3 \#Z \#V)$ under \eqref{hyp:B}.
+The graph $G_T$ has $O(N)$ vertices and $O(N\times \#Z)$ edges under assumption \eqref{hyp:A} and respectively $O(N\times \#Z)$ vertices and $O(N\times \#V)$ edges under assumption \eqref{hyp:B}. The loop at line \ref{lin:repeat} is iterated at most $N$ times since the distance between $T$ and $T'$ decreases at every iteration. Bellman-Ford algorithm has complexity $O(\#\mathrm{Vertices}\#\mathrm{Edges})$, which in our case amounts to $O(N^2\# Z)$ under \eqref{hyp:A} and $O(N^2 \#Z \#V)$ under \eqref{hyp:B}.
+ 
+\begin{algorithm}
+	\caption{Computation of the optimal utilization}
+	\label{alg:util}
+	\begin{algorithmic}[1]
+\Function{Compute optimal utilization}{$N$, $(c_v)_{v\in V}$}
+	\State $(\hat{n}_v)_{v\in V} \leftarrow $ \Call{Compute subutilization}{$N$, $(c_v)_{v\in V}$} \label{lin:subutil}
+	\While{$\sum_{v\in V} \hat{n}_v < 3N$} \label{lin:loopsub}
+	\State Pick $v\in V$ minimizing $\frac{c_v}{\hat{n}_v+1}$ and such that
+		$\sum_{v'\in z_v} \hat{n}_{v'} < N$ \label{lin:findmin}
+		\State $\hat{n}_v \leftarrow \hat{n}_v+1$
+	\EndWhile
+	\State \Return $(\hat{n}_v)_{v\in V}$
+\EndFunction
+\State
+
+\Function{Compute subutilization}{$N$, $(c_v)_{v\in V}$}
+		\State $R \leftarrow 3$
+\For{$v\in V$}
+\State $\hat{n}_v \leftarrow \mathrm{unset}$
+\EndFor
+\For{$z\in Z$}
+\State $c_z \leftarrow \sum_{v\in z} c_v$
+\EndFor
+\State $C \leftarrow \sum_{z\in Z} c_z$
+\While{$\exists z \in Z$ such that $R\times c_{z} > C$}
+\For{$v\in z$}
+\State $\hat{n}_v \leftarrow \left\lfloor \frac{c_v}{c_z} N \right\rfloor$
+\EndFor
+\State $C \leftarrow C-c_z$
+\State $R\leftarrow R-1$
+\EndWhile
+\For{$v\in V$}
+\If{$\hat{n}_v = \mathrm{unset}$}
+\State $\hat{n}_v \leftarrow \left\lfloor \frac{Rc_v}{C} N \right\rfloor$
+\EndIf
+\EndFor
+\State \Return $(\hat{n}_v)_{v\in V}$
+\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+\begin{algorithm}
+	\caption{Computation of a candidate assignment}
+	\label{alg:opt}
+	\begin{algorithmic}[1]
+		\Function{Compute candidate assignment}{$N$, $(n_v)_{v\in V}$}
+		\State Compute the flow graph $G$
+		\State Compute the maximal flow $f$ using Dinic's algorithm with randomized neighbours enumeration
+		\State Construct the assignment $(T_i)_{1\le i\le N}$ from $f$
+		\State \Return $(T_i)_{1\le i\le N}$
+		\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+
+\begin{algorithm}
+	\caption{Minimization of the number of transfers}
+	\label{alg:mini}
+	\begin{algorithmic}[1]
+		\Function{Minimization of transfers}{$(T_i)_{1\le i\le N}$, $(T'_i)_{1\le i\le N}$}
+		\State Construct the graph encoding $G_T$
+		\Repeat \label{lin:repeat}
+		\State Find a negative cycle $\gamma$ using Bellman-Ford algorithm on $G_T$
+		\State Reverse the orientations and weights of edges in $\gamma$
+		\Until{no negative cycle is found}
+		\State Update $(T_i)_{1\le i\le N}$ from $G_T$
+		\State \Return $(T_i)_{1\le i\le N}$
+		\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+\newpage
+
+\section{Computation of a 3-non-strict assignment}
+
+\subsection{Choices of optimality}
+
+In this mode, we primarily want to store every partition on three nodes, and only secondarily try to spread the nodes among different zone. So we make the choice of not taking the zone repartition in the criterion of optimality.
+
+We try to maximize $s^*$ defined in \eqref{eq:optimal}. So we can compute the optimal utilizations $(n_v)_{v\in V}$ with the only constraint that $n_v \le N$ for every node $v$. As in the previous section, we start with a sub-utilization proportional to $c_v$ (and capped at $N$), and we iteratively increase the $\hat{n}_v$ that is less than $N$ and maximizes the quantity $c_v/(\hat{n}_v+1)$, until the total sum is $3N$.
+
+\subsection{Computation of a candidate assignment}
+
+To compute a candidate assignment (that does not optimize zone spreading nor distance to a previous assignment yet), we can use the folowing flow problem.
+
+Define the oriented weighted graph $(X,E)$. The set of vertices $X$ contains the source $\mathbf{s}$, the sink $\mathbf{t}$, vertices 
+$\mathbf{x}_p, \mathbf{u}^+_p, \mathbf{u}^-_p$ for every partition $p$, vertices $\mathbf{y}_{p,z}$ for every partition $p$ and zone $z$, and vertices $\mathbf{z}_v$ for every node $v$. 
+
+The set of edges is composed of the following arcs:
+\begin{itemize}
+	\item ($\mathbf{s}$,$\mathbf{x}_p$, 3) for every partition $p$;
+	\item ($\mathbf{x}_p$,$\mathbf{u}^+_p$, 3) for every partition $p$;
+	\item ($\mathbf{x}_p$,$\mathbf{u}^-_p$, 2) for every partition $p$;
+	\item ($\mathbf{u}^+_p$,$\mathbf{y}_{p,z}$, 1) for every partition $p$ and zone $z$;
+	\item ($\mathbf{u}^-_p$,$\mathbf{y}_{p,z}$, 2) for every partition $p$ and zone $z$;
+	\item ($\mathbf{y}_{p,z}$,$\mathbf{z}_v$, 1) for every partition $p$, zone $z$ and node $v\in z$;
+	\item ($\mathbf{z}_v$, $\mathbf{t}$, $n_v$) for every node $v$;
+\end{itemize}
+
+One can check that any maximal flow in this graph corresponds to an assignment of partitions to nodes. In such a flow, all the arcs from $\mathbf{s}$ and to $\mathbf{t}$ are saturated. The arc from $\mathbf{y}_{p,z}$ to $\mathbf{z}_v$ is saturated if and only if $p$ is associated to~$v$. 
+Finally the flow from $\mathbf{x}_p$ to $\mathbf{y}_{p,z}$ can go either through $\mathbf{u}^+_p$ or $\mathbf{u}^-_p$.
+
+
+
+\subsection{Maximal spread and minimal transfers}
+Notice that if the arc $\mathbf{u}_p^+\mathbf{y}_{p,z}$ is not saturated but there is some flow in $\mathbf{u}_p^-\mathbf{y}_{p,z}$, then it is possible to transfer a unit of flow from the path  $\mathbf{x}_p\mathbf{u}_p^-\mathbf{y}_{p,z}$ to the path $\mathbf{x}_p\mathbf{u}_p^+\mathbf{y}_{p,z}$. So we can always find an equivalent maximal flow $f^*$ that uses the path through $\mathbf{u}_p^-$ only if the path through $\mathbf{u}_p^+$ is saturated.
+
+We will use this fact to consider the amount of flow going through the vertices $\mathbf{u}^+$ as a measure of how well the partitions are spread over nodes belonging to different zones. If the partition $p$ is associated to 3 different zones, then a flow of 3 will cross $\mathbf{u}_p^+$ in $f^*$ (i.e. a flow of 0 will cross $\mathbf{u}_p^+$). If $p$ is associated to two zones, a flow of $2$ will cross $\mathbf{u}_p^+$. If $p$ is associated to a single zone, a flow of $1$ will cross $\mathbf{u}_p^+$.
+
+Let $N_1, N_2, N_3$ be the number of partitions associated to respectively 1,2 and 3 distinct zones. We will optimize a linear combination of these variables using the discovery of positively weighted circuits in a graph.
+
+At the same step, we will also optimize the distance to a previous assignment $T'$. Let $\alpha> \beta> \gamma \ge 0$ be three parameters.
+
+Given the flow $f$, let $G_f=(X',E_f)$ be the multi-graph where $X' = X\setminus\{\mathbf{s},\mathbf{t}\}$. The set $E_f$ is composed of the arcs:
+\begin{itemize}
+\item As many arcs from $(\mathbf{x}_p, \mathbf{u}^+_p,\alpha), (\mathbf{x}_p, \mathbf{u}^+_p,\beta), (\mathbf{x}_p, \mathbf{u}^+_p,\gamma)$ (selected in this order) as there is flow crossing $\mathbf{u}^+_p$ in $f$;
+\item As many arcs from $(\mathbf{u}^+_p, \mathbf{x}_p,-\gamma), (\mathbf{u}^+_p, \mathbf{x}_p,-\beta), (\mathbf{u}^+_p, \mathbf{x}_p,-\alpha)$ (selected in this order) as there is flow crossing $\mathbf{u}^-_p$ in $f$;
+\item  As many copies of $(\mathbf{x}_p, \mathbf{u}^-_p,0)$ as there is flow through $\mathbf{u}^-_p$;
+\item  As many copies of $(\mathbf{u}^-_p,\mathbf{x}_p,0)$ so that the number of arcs between these two vertices is 2;
+\item $(\mathbf{u}^+_p,\mathbf{y}_{p,z}, 0)$ if the flow between these vertices is 1, and the opposite arc otherwise;
+\item as many copies of $(\mathbf{u}^-_p,\mathbf{y}_{p,z}, 0)$ as the flow between these vertices, and as many copies of the opposite arc as 2~$-$~the flow;
+\item $(\mathbf{y}_{p,z},\mathbf{z}_v, \pm1)$ if it is saturated in $f$, with $+1$ if $v\in T'_p$ and $-1$ otherwise;
+\item $(\mathbf{z}_v,\mathbf{y}_{p,z}, \pm1)$ if it is not saturated in $f$, with $+1$ if $v\notin T'_p$ and $-1$ otherwise.
+\end{itemize}
+To summarize, arcs are oriented left to right if they correspond to a presence of flow in $f$, and right to left if they correspond to an absence of flow. They are positively weighted if we want them to stay at their current state, and negatively if we want them to switch. Let us compute the weight of such graph.
+
+\begin{multline*}
+	w(G_f)  = \sum_{e\in E_f} w(e_f)  \\
+	= 
+	(\alpha - \beta -\gamma) N_1 + (\alpha +\beta - \gamma) N_2 + (\alpha+\beta+\gamma) N_3 
+	\\ + 
+	\#V\times N - 4 \sum_p 3-\#(T_p\cap T'_p) \\ 
+	=(\#V-12+\alpha-\beta-\gamma)\times N + 4Q_V + 2\beta N_2 + 2(\beta+\gamma) N_3 \\
+\end{multline*}
+
+As for the mode 3-strict, one can check that the difference of two such graphs corresponding to the same $(n_v)$ is always eulerian. Hence we can navigate in this class with the same greedy algorithm that discovers positive cycles and flips them.
+
+The function that we optimize is 
+$$
+2Q_V +  \beta N_2 + (\beta+\gamma) N_3.
+$$
+The choice of parameters $\beta$ and $\gamma$ should be lead by the following question: For $\beta$, where to put the tradeoff between zone dispersion and distance to the previous configuration? For $\gamma$, do we prefer to have more partitions spread between 2 zones, or have less between at least 2 zones but more between 3 zones.
+
+The quantity $Q_V$ varies between $0$ and $3N$, it should be of order $N$. The quantity $N_2+N_3$ should also be of order $N$ (it is exactly $N$ in the strict mode). So the two terms of the function are comparable. 
+
+
+\bibliography{optimal_layout} 
+\bibliographystyle{ieeetr}
+
+\end{document}
+
+
+

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+
+!*.txt
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doc/talks/2023-09-20-ocp/abstract.md

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+### (fr) Garage, un système de stockage de données géo-distribué léger et robuste
+
+Garage est un système de stockage de données léger, géo-distribué, qui
+implémente le protocole de stockage S3 de Amazon. Garage est destiné
+principalement à l'auto-hébergement sur du matériel courant d'occasion. À ce
+titre, il doit tolérer un grand nombre de pannes: coupures de courant, coupures
+de connexion Internet, pannes de machines, ... Il doit également être facile à
+déployer et à maintenir, afin de pouvoir être facilement utilisé par des
+amateurs ou des petites organisations.
+
+Cette présentation vous proposera un aperçu de Garage et du choix technique
+principal qui rend un système comme Garage possible: le refus d'utiliser des
+algorithmes de consensus, remplacés avantageusement par des méthodes à
+cohérence faible.  Notre modèle est fortement inspiré de la base de donnée
+Dynamo (DeCandia et al, 2007), et fait usage des types de données CRDT (Shapiro
+et al, 2011).  Nous exploreront comment ces méthodes s'appliquent à la
+construction de l'abstraction "stockage objet" dans un système distribué, et
+quelles autres abstractions peuvent ou ne peuvent pas être construites dans ce
+modèle.
+
+### (en) Garage, a lightweight and robust geo-distributed data storage system
+
+Garage is a lightweight geo-distributed data store that implements the Amazon
+S3 object storage protocol.  Garage is meant primarily for self-hosting at home
+on second-hand commodity hardware, meaning it has to tolerate a wide variety of
+failure scenarios such as power cuts, Internet disconnections and machine
+crashes. It also has to be easy to deploy and maintain, so that hobbyists and
+small organizations can use it without trouble. 
+
+This talk will present Garage and the key technical choice that made Garage
+possible: refusing to use consensus algorithms and using instead weak
+consistency methods, with a model that is loosely based on that of the Dynamo
+database (DeCandia et al, 2007) and that makes heavy use of conflict-free
+replicated data types (Shapiro et al, 2011).  We will explore how these methods
+are suited to building the "object store" abstraction in a distributed system,
+and what other abstractions are possible or impossible to build in this model.
+
+
+

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+    <text
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+         id="tspan4278"
+         style="fill:#000000;stroke-width:0.264583"
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+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
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+       cx="18.004833"
+       cy="21.341394"
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+         id="tspan3705"
+         style="stroke-width:0.264583;fill:#000000"
+         x="6.9525447"
+         y="13.702217">$write(\{a\})$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
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+         id="tspan3705-5"
+         style="fill:#000000;stroke-width:0.264583"
+         x="6.9525447"
+         y="65.261436">$read()$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
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+         id="tspan3748"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
+         y="24.042637">$\sqsupseteq \{a\} \to$ OK</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
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+         id="tspan3748-2"
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+         x="23.457415"
+         y="33.02087">$\sqsupseteq \{a\} \to$ OK</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
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+         id="tspan3748-6"
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+       id="path3663-6-6-4"
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+       cy="70.942116"
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+    <text
+       xml:space="preserve"
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+       id="text3750-7"><tspan
+         sodipodi:role="line"
+         id="tspan3748-65"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
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+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
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+         id="tspan4278"
+         style="fill:#000000;stroke-width:0.264583"
+         x="14.395845"
+         y="52.022549">return OK</tspan></text>
+  </g>
+</svg>

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+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
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+    <g
+       id="g1183"
+       transform="translate(51.996784,1.0317046)"
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+      <text
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+      <text
+         xml:space="preserve"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
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+           id="tspan1115-5"
+           style="stroke-width:0.264583;fill:#000000"
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+           y="40.841526">$\{a,b\}$</tspan></text>
+      <text
+         xml:space="preserve"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
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+           id="tspan1115-2"
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+    <path
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+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-7-2-8)"
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+       id="path1300-2-6-9-0" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-75)"
+       d="m 228.52843,61.091809 0.33313,-12.554874 m -0.33313,12.554874 0.33313,-12.554874"
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+       sodipodi:nodetypes="cc" />
+    <path
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+    <path
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+       sodipodi:nodetypes="cc" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-75-7-1)"
+       d="m 165.29305,32.445235 0.33313,-12.554874 m -0.33313,12.554874 0.33313,-12.554874"
+       id="path1300-2-2-6-9"
+       sodipodi:nodetypes="cc" />
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
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+    <circle
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+    <circle
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+       cy="67.645035"
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+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
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+    <circle
+       style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
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+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6"
+       cx="18.004833"
+       cy="21.341394"
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+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;stroke-width:0.264583;fill:#000000"
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+       y="13.702217"
+       id="text3707"><tspan
+         sodipodi:role="line"
+         id="tspan3705"
+         style="stroke-width:0.264583;fill:#000000"
+         x="6.9525447"
+         y="13.702217">$write(\{a\})$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="6.9525447"
+       y="65.261436"
+       id="text3707-1"><tspan
+         sodipodi:role="line"
+         id="tspan3705-5"
+         style="fill:#000000;stroke-width:0.264583"
+         x="6.9525447"
+         y="65.261436">$read()$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="23.457415"
+       y="24.042637"
+       id="text3750"><tspan
+         sodipodi:role="line"
+         id="tspan3748"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
+         y="24.042637">$\sqsupseteq \{a\} \to$ OK</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="23.457415"
+       y="33.02087"
+       id="text3750-3"><tspan
+         sodipodi:role="line"
+         id="tspan3748-2"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
+         y="33.02087">$\sqsupseteq \{a\} \to$ OK</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       x="23.457415"
+       y="41.972523"
+       id="text3750-0"><tspan
+         sodipodi:role="line"
+         id="tspan3748-6"
+         style="fill:#999999;stroke-width:0.264583"
+         x="23.457415"
+         y="41.972523">$\not\sqsupseteq \{a\}$</tspan></text>
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6-4"
+       cx="18.004833"
+       cy="70.942116"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="23.457415"
+       y="73.643356"
+       id="text3750-7"><tspan
+         sodipodi:role="line"
+         id="tspan3748-65"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
+         y="73.643356">$\to \{\}$</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
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+       y="52.022549"
+       id="text4280"><tspan
+         sodipodi:role="line"
+         id="tspan4278"
+         style="fill:#000000;stroke-width:0.264583"
+         x="14.395845"
+         y="52.022549">return OK</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="14.395845"
+       y="92.005798"
+       id="text4280-3"><tspan
+         sodipodi:role="line"
+         id="tspan4278-7"
+         style="fill:#000000;stroke-width:0.264583"
+         x="14.395845"
+         y="92.005798">return $\{\}\sqcup\{a\}=\{a\}$</tspan></text>
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-1-5"
+       cx="18.004833"
+       cy="81.007744"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="23.457415"
+       y="83.577797"
+       id="text3750-0-6"><tspan
+         sodipodi:role="line"
+         id="tspan3748-6-9"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
+         y="83.577797">$\to \{a\}$</tspan></text>
+  </g>
+</svg>

doc/talks/2023-09-20-ocp/assets/lattice8.svg

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+         id="tspan3748"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
+         y="24.042637">$\sqsupseteq \{a\} \to$ OK</tspan></text>
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+    <text
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+         id="tspan3748-6-9"
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