- We always recalculate the reference count of a block before deleting
it locally, to make sure that it is indeed zero.
- If we had to fetch a remote block but we were not able to get it,
check that refcount is indeed > 0.
- Repair procedure that checks everything
ci/woodpecker/push/debug Pipeline was successfulDetails
ci/woodpecker/pr/debug Pipeline was successfulDetails
This page of the AWS docs indicate that Content-Type should be part of
the CanonicalHeaders (and therefore SignedHeaders) strings in signature
calculation:
https://docs.aws.amazon.com/AmazonS3/latest/API/sig-v4-header-based-auth.html
However, testing with Minio Client revealed that it did not sign the
Content-Type header, and therefore we broke CI by expecting it to be
signed. With this commit, we don't mandate Content-Type to be signed
anymore, for better compatibility with the ecosystem. Testing against
the official behavior of S3 on AWS has not been done.
ci/woodpecker/pr/debug Pipeline was successfulDetails
For some users, this might be their first time being interacting with
the `env_logger` crate.
As such, they might not be aware that less verbose log levels exist.
Some might not want to log every incoming request, for example.
This commit also adds syntax hints to the code-fence for bash for better
syntax highlighting of that section, and repeats itself multiple times,
that `info` is, in fact, the default.
No changes to the recommendation of log levels were made.
continuous-integration/drone/pr Build is passingDetails
Having all Engine enum variants conditional causes compilation errors
when *none* of the DB engine features is enabled. This is not an issue
for full garage build, but affects crates that use garage_db as
dependency.
Change all variants to be present at all times. It solves compilation
errors and also allows us to better differentiate between invalid DB
engine name and engine with support not compiled in current binary.
continuous-integration/drone/pr Build is passingDetails
Use optional DB open overrides for both input and output database.
Duplicating the same override flag for input/output would result in too
many, too long flags. It would be too costly for very rare edge-case
where converting between same DB engine, just with different flags.
Because overrides flags for different engines are disjoint and we are
preventing conversion between same input/ouput DB engine, we can have
only one set.
The override flag will be passed either to input or output, based on
engine type it belongs to. It will never be passed to both of them and
cause unwelcome surprise to user.
continuous-integration/drone/push Build is passingDetails
continuous-integration/drone/pr Build is passingDetails
This minor release includes the following improvements and fixes:
New features:
- Configuration: make LMDB's `map_size` configurable and make `block_size` and `sled_cache_capacity` expressable as strings (such as `10M`) (#628, #630)
- Add support for binding to Unix sockets for the S3, K2V, Admin and Web API servers (#640)
- Move the `convert_db` command into the main Garage binary (#645)
- Add support for specifying RPC secret and admin tokens as environment variables (#643)
- Add `allow_world_readable_secrets` option to config file (#663, #685)
Bug fixes:
- Use `statvfs` instead of mount list to determine free space in metadata/data directories (#611, #631)
- Add missing casts to fix 32-bit build (#632)
- Fix error when none of the HTTP servers (S3/K2V/Admin/Web) is started and fix shutdown hang (#613, #633)
- Add missing CORS headers to PostObject response (#609, #656)
- Monitoring: finer histogram boundaries in Prometheus exported metrics (#531, #686)
Other:
- Documentation improvements (#641)
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Merge tag 'v0.8.5' into sync-08-09
Garage v0.8.5
This minor release includes the following improvements and fixes:
New features:
- Configuration: make LMDB's `map_size` configurable and make `block_size` and `sled_cache_capacity` expressable as strings (such as `10M`) (#628, #630)
- Add support for binding to Unix sockets for the S3, K2V, Admin and Web API servers (#640)
- Move the `convert_db` command into the main Garage binary (#645)
- Add support for specifying RPC secret and admin tokens as environment variables (#643)
- Add `allow_world_readable_secrets` option to config file (#663, #685)
Bug fixes:
- Use `statvfs` instead of mount list to determine free space in metadata/data directories (#611, #631)
- Add missing casts to fix 32-bit build (#632)
- Fix error when none of the HTTP servers (S3/K2V/Admin/Web) is started and fix shutdown hang (#613, #633)
- Add missing CORS headers to PostObject response (#609, #656)
- Monitoring: finer histogram boundaries in Prometheus exported metrics (#531, #686)
Other:
- Documentation improvements (#641)
continuous-integration/drone/pr Build is passingDetails
Sometimes, the secret files permissions checks gets in the way. It's
by no mean complete, it doesn't take the Posix ACLs into account among
other things. Correctly checking the ACLs would be too involving (see
#658 (comment))
and would likely still fail in some weird chmod settings.
We're adding a new configuration file key allowing the user to disable
this permission check altogether.
The (already existing) env variable counterpart always take precedence
to this config file option. That's useful in cases where the
configuration file is static and cannot be easily altered.
Fixes #658
Co-authored-by: Florian Klink <flokli@flokli.de>
continuous-integration/drone/pr Build is passingDetails
This is still a bit confusing, as normally the flake.defaultNix attrset
gets exposed via a top-level default.nix, but at least it brings us
closer to that.
continuous-integration/drone/pr Build is passingDetails
Compiling garage_db v0.8.2 (garage-0.8.2/src/db)
error: cannot find macro `warn` in this scope
--> src/db/lmdb_adapter.rs:352:2
|
352 | warn!("LMDB is not recommended on 32-bit systems, database size will be limited");
| ^^^^
|
= help: consider importing this macro:
tracing::warn
= note: `warn` is in scope, but it is an attribute: `#[warn]`
error: could not compile `garage_db` due to previous error
continuous-integration/drone/pr Build is passingDetails
* Renamed my_garage_service -> garage-s3-service.
* Defined a web service for port 3902.
* Added a garage-s3 router.
* Pointed website definition at web service.
* Use the /health endpoint for loadBalancer health check.
* Renamed gzip_compress to just compression as traefik v3 will also do
brotli compression.
DaemonSet is a k8s resource that schedules one instance per node,
which is useful for some garage deployment use cases, including
managing garage nodes using k8s node labels
continuous-integration/drone/push Build is passingDetails
As discussed in the chat yesterday, I want to propose to disable the ingress per default.
The motivation behind this change is, that per default the ingress is "misconfigured"
meaning it can not work with the default values and requires a user of the chart to
add additional configuration. When installing the chart per default, I would not
expect to already expose garage publicly without my explicit configuration to do so
Commenting the ingressClass resource also allows for relying only on
annotations - otherwise the ingressClass would be always set to nginx
or require a user to override it with ingressClass: null
A small change on top, I've added the ability to specify user defined labels per ingress
The default values forces people to create an ingress resources,
where per default an ingress is not necessary to start garage.
If someone wants to utilize an ingress, he would need to define
the values for the ingress either way, so enabling the ingress
explicitly makes more sense, then requiring it to be disabled per default
continuous-integration/drone/push Build is failingDetails
continuous-integration/drone/pr Build is failingDetails
- self.node_id_vec was not properly updated when the previous ring was empty
- ClusterLayout::merge was not considering changes in the layout parameters
- have consistent error return types
- store the zone redundancy in a Lww
- print the error and message in the CLI (TODO: for the server Api, should msg be returned in the body response?)
It takes as paramters the replication factor and the zone redundancy, computes the
largest partition size reachable with these constraints, and among the possible
assignation with this partition size, it computes the one that moves the least number
of partitions compared to the previous assignation.
This computation uses graph algorithms defined in graph_algo.rs
continuous-integration/drone/push Build is failingDetails
The function now computes an optimal assignation (with respect to partition size) that minimizes the distance to the former assignation, using flow algorithms.
This commit was written by Mendes Oulamara <mendes.oulamara@pm.me>
2022-05-01 09:54:19 +02:00
592 changed files with 112369 additions and 16251 deletions
Nextcloud will now make Garage encrypt files at rest in the storage bucket.
These files will not be readable by an S3 client that has credentials to the
bucket but doesn't also know the secret encryption key.
### External Storage
**From the GUI.** Activate the "External storage support" app from the "Applications" page (click on your account icon on the top right corner of your screen to display the menu). Go to your parameters page (also located below your account icon). Click on external storage (or the corresponding translation in your language).
@ -128,20 +176,24 @@ In other words, Peertube is only responsible of the "control plane" and offload
In return, this system is a bit harder to configure.
We show how it is still possible to configure Garage with Peertube, allowing you to spread the load and the bandwidth usage on the Garage cluster.
Starting from version 5.0, Peertube also supports improving the security for private videos by not exposing them directly
but relying on a single control point in the Peertube instance. This is based on S3 per-object and prefix ACL, which are not currently supported
in Garage, so this feature is unsupported. While this technically impedes security for private videos, it is not a blocking issue and could be
a reasonable trade-off for some instances.
### Create resources in Garage
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.
We need two buckets, one for normal videos (named peertube-video) and one for webtorrent videos (named peertube-playlist).
```bash
garage bucket create peertube-video
garage bucket create peertube-videos
garage bucket create peertube-playlist
```
@ -195,6 +247,11 @@ object_storage:
max_upload_part: 2GB
proxy:
# You may enable this feature, yet it will not provide any security benefit, so
# you should rather benefit from Garage public endpoint for all videos
proxify_private_files: false
streaming_playlists:
bucket_name: 'peertube-playlist'
@ -206,7 +263,7 @@ object_storage:
# Same settings but for webtorrent videos
videos:
bucket_name: 'peertube-video'
bucket_name: 'peertube-videos'
prefix: ''
# You must fill this field to make Peertube use our reverse proxy/website logic
A cookbook, when you cook, is a collection of recipes.
Similarly, Garage's cookbook contains a collection of recipes that are known to works well!
Similarly, Garage's cookbook contains a collection of recipes that are known to work well!
This chapter could also be referred as "Tutorials" or "Best practices".
- **[Multi-node deployment](@/documentation/cookbook/real-world.md):** This page will walk you through all of the necessary
@ -16,6 +16,10 @@ This chapter could also be referred as "Tutorials" or "Best practices".
source in case a binary is not provided for your architecture, or if you want to
hack with us!
- **[Binary packages](@/documentation/cookbook/binary-packages.md):** This page
lists the different platforms that provide ready-built software packages for
Garage.
- **[Integration with Systemd](@/documentation/cookbook/systemd.md):** This page explains how to run Garage
as a Systemd service (instead of as a Docker container).
@ -26,6 +30,10 @@ This chapter could also be referred as "Tutorials" or "Best practices".
- **[Configuring a reverse-proxy](@/documentation/cookbook/reverse-proxy.md):** This page explains how to configure a reverse-proxy to add TLS support to your S3 api endpoint.
- **[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.
- **[Deploying on Kubernetes](@/documentation/cookbook/kubernetes.md):** This page explains how to deploy Garage on Kubernetes using our Helm chart.
- **[Deploying with Ansible](@/documentation/cookbook/ansible.md):** This page lists available Ansible roles developed by the community to deploy Garage.
- **[Monitoring Garage](@/documentation/cookbook/monitoring.md)** This page
explains the Prometheus metrics available for monitoring the Garage
@ -21,7 +21,7 @@ You can configure Garage as a gateway on all nodes that will consume your S3 API
The instructions are similar to a regular node, the only option that is different is while configuring the node, you must set the `--gateway` parameter:
This will allow anyone to scrape Prometheus metrics by fetching
`http://localhost:3093/metrics`. If you want to restrict access
`http://localhost:3903/metrics`. If you want to restrict access
to the exported metrics, set the `metrics_token` configuration value
to a bearer token to be used when fetching the metrics endpoint.
@ -49,258 +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
### Metrics of the API endpoints
#### `api_admin_request_counter` (counter)
Counts the number of requests to a given endpoint of the administration API. Example:
@ -19,8 +19,13 @@ 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.
Running behind a NAT is likely to be possible but hasn't been tested for the latest version (TODO).
- 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.
- This guide will assume you are using Docker containers to deploy Garage on each node.
Garage can also be run independently, for instance as a [Systemd service](@/documentation/cookbook/systemd.md).
@ -38,7 +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
@ -48,9 +53,9 @@ to store 2 TB of data in total.
### Best practices
- If you have fast dedicated networking between all your nodes, and are planing to store
very large files, bump the `block_size` configuration parameter to 10 MB
(`block_size = 10485760`).
- If you have reasonably fast networking between all your nodes, and are planing to store
mostly large files, bump the `block_size` configuration parameter to 10 MB
(`block_size = "10M"`).
- Garage stores its files in two locations: it uses a metadata directory to store frequently-accessed
small metadata items, and a data directory to store data blocks of uploaded objects.
@ -63,36 +68,42 @@ to store 2 TB of data in total.
EXT4 is not recommended as it has more strict limitations on the number of inodes,
which might cause issues with Garage when large numbers of objects are stored.
- If you only have an HDD and no SSD, it's fine to put your metadata alongside the data
on the same drive. Having lots of RAM for your kernel to cache the metadata will
help a lot with performance. Make sure to use the LMDB database engine,
instead of Sled, which suffers from quite bad performance degradation on HDDs.
Sled is still the default for legacy reasons, but is not recommended anymore.
- Servers with multiple HDDs are supported natively by Garage without resorting
to RAID, see [our dedicated documentation page](@/documentation/operations/multi-hdd.md).
- 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
EXT4 or XFS.
verification on its own, so it is better to use a robust filesystem such as
BTRFS or ZFS. Users have reported that when using the LMDB database engine
(the default), database files have a tendency of becoming corrupted after an
unclean shutdown (e.g. a power outage), so you should take regular snapshots
to be able to recover from such a situation. This can be done using Garage's
built-in automatic snapshotting (since v0.9.4), or by using filesystem level
snapshots. If you cannot do so, you might want to switch to Sqlite which is
more robust.
- 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.
- LMDB is the fastest and most tested database engine, but it has the following
weaknesses: 1/ data files are not architecture-independent, you cannot simply
move a Garage metadata directory between nodes running different architectures,
and 2/ LMDB is not suited for 32-bit platforms. Sqlite is a viable alternative
if any of these are of concern.
- If you only have an HDD and no SSD, it's fine to put your metadata alongside
the data on the same drive, but then consider your filesystem choice wisely
(see above). Having lots of RAM for your kernel to cache the metadata will
help a lot with performance. The default LMDB database engine is the most
tested and has good performance.
## 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. `v1.0.0`) and not the `latest` tag.
For this example, we will use the latest published version at the time of the writing which is `v1.0.0` 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:v1.0.0
```
## Deploying and configuring Garage
@ -109,14 +120,15 @@ especially you must consider the following folders/files:
this folder will be your main data storage and must be on a large storage (e.g. large HDD)
A valid `/etc/garage/garage.toml` for our cluster would look as follows:
A valid `/etc/garage.toml` for our cluster would look as follows:
```toml
metadata_dir = "/var/lib/garage/meta"
data_dir = "/var/lib/garage/data"
db_engine = "lmdb"
metadata_auto_snapshot_interval = "6h"
replication_mode = "3"
replication_factor = 3
compression_level = 2
@ -157,12 +169,13 @@ 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:v1.0.0
```
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:
@ -170,7 +183,7 @@ 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:v1.0.0
network_mode: "host"
restart: unless-stopped
volumes:
@ -179,12 +192,14 @@ 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
## Controlling the daemon
The `garage` binary has two purposes:
- it acts as a daemon when launched with `garage server`
@ -193,6 +208,12 @@ 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 <containername> /garage"
```
You can test your `garage` CLI utility by running a simple command such as:
```bash
@ -236,7 +257,7 @@ You can then instruct nodes to connect to one another as follows:
@ -210,10 +235,15 @@ To add a new website, add the following declaration to your Traefik configuratio
```toml
[http.routers]
[http.routers.garage-s3]
rule = "Host(`s3.example.org`)"
service = "garage-s3-service"
entryPoints = ["websecure"]
[http.routers.my_website]
rule = "Host(`yoururl.example.org`)"
service = "my_garage_service"
entryPoints = ["web"]
service = "garage-web-service"
entryPoints = ["websecure"]
```
Enable HTTPS access to your website with the following configuration section ([documentation](https://doc.traefik.io/traefik/https/overview/)):
@ -226,7 +256,7 @@ Enable HTTPS access to your website with the following configuration section ([d
...
```
### Adding gzip compression
### Adding compression
Add the following configuration section [to compress response](https://doc.traefik.io/traefik/middlewares/http/compress/) using [gzip](https://developer.mozilla.org/en-US/docs/Glossary/GZip_compression) before sending them to the client:
@ -234,10 +264,10 @@ Add the following configuration section [to compress response](https://doc.traef
[http.routers]
[http.routers.my_website]
...
middlewares = ["gzip_compress"]
middlewares = ["compression"]
...
[http.middlewares]
[http.middlewares.gzip_compress.compress]
[http.middlewares.compression.compress]
```
### Add caching response
@ -262,27 +292,54 @@ Traefik's caching middleware is only available on [entreprise version](https://d
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
[on-demand TLS](https://caddyserver.com/docs/automatic-https#on-demand-tls), by
which one can configure the webserver to provision TLS certificates when a
client first connects to it.
In order to prevent an attack vector whereby domains are simply pointed at your
webserver and certificates are requested for them - Caddy can be configured to
ask Garage if a domain is authorized for web hosting, before it then requests
a TLS certificate.
This 'check' endpoint, which is on the admin port (3903 by default), can be
configured in Caddy's global section as follows:
```caddy
{
...
on_demand_tls {
ask http://localhost:3903/check
interval 2m
burst 5
}
...
}
```
The host section can then be configured with (note that this uses the web
*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:
Garage is a stateful clustered application, where all nodes are communicating together and share data structures.
It makes upgrade more difficult than stateless applications so you must be more careful when upgrading.
On a new version release, there is 2 possibilities:
- protocols and data structures remained the same ➡️ this is a **straightforward upgrade**
- protocols or data structures changed ➡️ this is an **advanced upgrade**
You can quickly now what type of update you will have to operate by looking at the version identifier.
Following the [SemVer ](https://semver.org/) terminology, if only the *patch* number changed, it will only need a straightforward upgrade.
Example: an upgrade from v0.6.0 from v0.6.1 is a straightforward upgrade.
If the *minor* or *major* number changed however, you will have to do an advanced upgrade. Example: from v0.6.1 to v0.7.0.
Migrations are designed to be run only between contiguous versions (from a *major*.*minor* perspective, *patches* can be skipped).
Example: migrations from v0.6.1 to v0.7.0 and from v0.6.0 to v0.7.0 are supported but migrations from v0.5.0 to v0.7.0 are not supported.
## Straightforward upgrades
Straightforward upgrades do not imply cluster downtime.
Before upgrading, you should still read [the changelog](https://git.deuxfleurs.fr/Deuxfleurs/garage/releases) and ideally test your deployment on a staging cluster before.
When you are ready, start by checking the health of your cluster.
You can force some checks with `garage repair`, we recommend at least running `garage repair --all-nodes --yes` that is very quick to run (less than a minute).
You will see that the command correctly terminated in the logs of your daemon.
Finally, you can simply upgrades nodes one by one.
For each node: stop it, install the new binary, edit the configuration if needed, restart it.
## Advanced upgrades
Advanced upgrades will imply cluster downtime.
Before upgrading, you must read [the changelog](https://git.deuxfleurs.fr/Deuxfleurs/garage/releases) and you must test your deployment on a staging cluster before.
From a high level perspective, an advanced upgrade looks like this:
1. Make sure the health of your cluster is good (see `garage repair`)
2. Disable API access (comment the configuration in your reverse proxy)
3. Check that your cluster is idle
4. Stop the whole cluster
5. Backup the metadata folder of all your nodes, so that you will be able to restore it quickly if the upgrade fails (blocks being immutable, they should not be impacted)
6. Install the new binary, update the configuration
7. Start the whole cluster
8. If needed, run the corresponding migration from `garage migrate`
9. Make sure the health of your cluster is good
10. Enable API access (uncomment the configuration in your reverse proxy)
11. Monitor your cluster while load comes back, check that all your applications are happy with this new version
We write guides for each advanced upgrade, they are stored under the "Working Documents" section of this documentation.
We love to talk and hear about Garage, that's why we keep a log here:
- [(en, 2023-01-18) Presentation of Garage with some details on CRDTs and data partitioning among nodes](https://git.deuxfleurs.fr/Deuxfleurs/garage/src/commit/4cff37397f626ef063dad29e5b5e97ab1206015d/doc/talks/2023-01-18-tocatta/talk.pdf)
- [(fr, 2022-11-19) De l'auto-hébergement à l'entre-hébergement : Garage, pour conserver ses données ensemble](https://git.deuxfleurs.fr/Deuxfleurs/garage/src/commit/4cff37397f626ef063dad29e5b5e97ab1206015d/doc/talks/2022-11-19-Capitole-du-Libre/pr%C3%A9sentation.pdf)
- [(en, 2022-06-23) General presentation of Garage](https://git.deuxfleurs.fr/Deuxfleurs/garage/src/commit/4cff37397f626ef063dad29e5b5e97ab1206015d/doc/talks/2022-06-23-stack/talk.pdf)
- [(fr, 2021-11-13, video) Garage : Mille et une façons de stocker vos données](https://video.tedomum.net/w/moYKcv198dyMrT8hCS5jz9) and [slides (html)](https://rfid.deuxfleurs.fr/presentations/2021-11-13/garage/) - during [RFID#1](https://rfid.deuxfleurs.fr/programme/2021-11-13/) event
- [(en, 2021-04-28) Distributed object storage is centralised](https://git.deuxfleurs.fr/Deuxfleurs/garage/raw/commit/b1f60579a13d3c5eba7f74b1775c84639ea9b51a/doc/talks/2021-04-28_spirals-team/talk.pdf)
- [(en, 2021-04-28) Distributed object storage is centralised](https://git.deuxfleurs.fr/Deuxfleurs/garage/src/commit/b1f60579a13d3c5eba7f74b1775c84639ea9b51a/doc/talks/2021-04-28_spirals-team/talk.pdf)
- [(fr, 2020-12-02) Garage : jouer dans la cour des grands quand on est un hébergeur associatif](https://git.deuxfleurs.fr/Deuxfleurs/garage/raw/commit/b1f60579a13d3c5eba7f74b1775c84639ea9b51a/doc/talks/2020-12-02_wide-team/talk.pdf)
*Did you write or talk about Garage? [Open a pull request](https://git.deuxfleurs.fr/Deuxfleurs/garage/) to add a link here!*
- [(fr, 2020-12-02) Garage : jouer dans la cour des grands quand on est un hébergeur associatif](https://git.deuxfleurs.fr/Deuxfleurs/garage/src/commit/b1f60579a13d3c5eba7f74b1775c84639ea9b51a/doc/talks/2020-12-02_wide-team/talk.pdf)
@ -12,7 +12,7 @@ as pictures, video, images, documents, etc., in a redundant multi-node
setting. S3 is versatile enough to also be used to publish a static
website.
Garage is an opinionated object storage solutoin, we focus on the following **desirable properties**:
Garage is an opinionated object storage solution, we focus on the following **desirable properties**:
- **Internet enabled**: made for multi-sites (eg. datacenters, offices, households, etc.) interconnected through regular Internet connections.
- **Self-contained & lightweight**: works everywhere and integrates well in existing environments to target [hyperconverged infrastructures](https://en.wikipedia.org/wiki/Hyper-converged_infrastructure).
@ -42,15 +42,11 @@ 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
- In the Drone continuous integration platform to store task logs
- As a Nix binary cache
- As a backup target using `rclone`
- 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`
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
**[IPFS](https://ipfs.io/):** IPFS has design goals radically different from Garage, we have [a blog post](@/blog/2022-ipfs/index.md) talking about it.
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)
- `garage repair block-rc`: checks that the reference counters for blocks are in sync with the actual number of non-deleted entries in the block reference table
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_factor`](@/documentation/reference-manual/configuration.md#replication_factor)).
The layout determines the correspondence between these partitions,
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:
an overview for which Garage versions are currently in use within a cluster.
## Minor upgrades
Minor upgrades do not imply cluster downtime.
Before upgrading, you should still read [the changelog](https://git.deuxfleurs.fr/Deuxfleurs/garage/releases) and ideally test your deployment on a staging cluster before.
When you are ready, start by checking the health of your cluster.
You can force some checks with `garage repair`, we recommend at least running `garage repair --all-nodes --yes tables` which is very quick to run (less than a minute).
You will see that the command correctly terminated in the logs of your daemon, or using `garage worker list` (the repair workers should be in the `Done` state).
Finally, you can simply upgrade nodes one by one.
For each node: stop it, install the new binary, edit the configuration if needed, restart it.
## Major upgrades
Major upgrades can be done with minimal downtime with a bit of preparation, but the simplest way is usually to put the cluster offline for the duration of the migration.
Before upgrading, you must read [the changelog](https://git.deuxfleurs.fr/Deuxfleurs/garage/releases) and you must test your deployment on a staging cluster before.
We write guides for each major upgrade, they are stored under the "Working Documents" section of this documentation.
### Major upgrades with full downtime
From a high level perspective, a major upgrade looks like this:
1. Disable API access (for instance in your reverse proxy, or by commenting the corresponding section in your Garage configuration file and restarting Garage)
2. Check that your cluster is idle
3. Make sure the health of your cluster is good (see `garage repair`)
4. Stop the whole cluster
5. Back up the metadata folder of all your nodes, so that you will be able to restore it if the upgrade fails (data blocks being immutable, they should not be impacted)
6. Install the new binary, update the configuration
7. Start the whole cluster
8. If needed, run the corresponding migration from `garage migrate`
9. Make sure the health of your cluster is good
10. Enable API access (reverse step 1)
11. Monitor your cluster while load comes back, check that all your applications are happy with this new version
### Major upgarades with minimal downtime
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.
If all nodes fail to stop and restart simultaneously, some nodes might be temporarily shut out from the cluster as nodes using different RPC protocol
versions are prevented to talk to one another.
The entire procedure would look something like this:
1. Make sure the health of your cluster is good (see `garage repair`)
2. Take each node offline individually to back up its metadata folder, bring them back online once the backup is done.
You can do all of the nodes in a single zone at once as that won't impact global cluster availability.
Do not try to make a backup of the metadata folder of a running node.
**Since Garage v0.9.4,** you can use the `garage meta snapshot --all` command
to take a simultaneous snapshot of the metadata database files of all your
nodes. This avoids the tedious process of having to take them down one by
one before upgrading. Be careful that if automatic snapshotting is enabled,
Garage only keeps the last two snapshots and deletes older ones, so you might
want to disable automatic snapshotting in your upgraded configuration file
until you have confirmed that the upgrade ran successfully. In addition to
snapshotting the metadata databases of your nodes, you should back-up at
least the `cluster_layout` file of one of your Garage instances (this file
should be the same on all nodes and you can copy it safely while Garage is
running).
3. Prepare your binaries and configuration files for the new Garage version
4. Restart all nodes simultaneously in the new version
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, 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.
# 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 abuse for on-demand TLS, Caddy developers 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 these 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 is 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:
- [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
These endpoints are defined on a dedicated [Redocly page](https://garagehq.deuxfleurs.fr/api/garage-admin-v0.html). You can also download its [OpenAPI specification](https://garagehq.deuxfleurs.fr/api/garage-admin-v0.yml).
@ -12,13 +12,15 @@ back up all your data before attempting it!**
Garage v0.8 introduces new data tables that allow the counting of objects in buckets in order to implement bucket quotas.
A manual migration step is required to first count objects in Garage buckets and populate these tables with accurate data.
## Simple migration procedure (takes cluster offline for a while)
The migration steps are as follows:
1. Disable API and web access. Garage v0.7 does not support disabling
these endpoints but you can change the port number or stop your reverse proxy for instance.
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 (`scrub`, `block_refs`, etc.)
nodes. If you have time, do additional checks (`versions`, `block_refs`, etc.)
3. Check that queues are empty: run `garage stats` to query them or inspect metrics in the Grafana dashboard.
4. Turn off Garage v0.7
5. **Backup the metadata folder of all your nodes!** For instance, use the following command
@ -32,3 +34,24 @@ The migration steps are as follows:
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.
## Minimal downtime migration procedure
The migration to Garage v0.8 can be done with almost no downtime,
by restarting all nodes at once in the new version. The only limitation with this
method is that bucket sizes and item counts will not be estimated correctly
until all nodes have had a chance to run their offline migration procedure.
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 v0.8
4. Shut down all v0.7 nodes simultaneously, and restart them all simultaneously in v0.8. Use your favorite deployment tool (Ansible, Kubernetes, Nomad) to achieve this as fast as possible.
5. At this point, Garage will indicate invalid values for the size and number of objects in each bucket (most likely, it will indicate zero). To fix this, take each node offline individually to do the offline migration step: `garage offline-repair --yes object_counters`. Again you can do all nodes of a single zone at once.
**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.
| `prefix` | `null` | Restrict items to poll to those whose sort keys start with this prefix |
| `start` | `null` | The sort key of the first item to poll |
| `end` | `null` | The sort key of the last item to poll (excluded) |
| `timeout` | 300 | The timeout before 304 NOT MODIFIED is returned if no value in the range is updated |
| `seenMarker` | `null` | An opaque string returned by a previous PollRange call, that represents items already seen |
The timeout can be set to any number of seconds, with a maximum of 600 seconds (10 minutes).
The response is either:
- A HTTP 304 NOT MODIFIED response with an empty body, if the timeout expired and no changes occurred
- A HTTP 200 response, indicating that some changes have occurred since the last PollRange call, in which case a JSON object is returned in the body with the following fields:
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 $c_n\ge0$ 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
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.
\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 $n$.
\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.
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 $\tilde{f}$.
\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. $c(v_1,v_2)-f(v_1,v_2)\ge1$, 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
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 $(f+\gamma)(e)\neq f'(e)$.
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
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}{$G$, $f$, $\alpha'$}
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.
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 $c_n\ge0$ 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 $\alpha$.
\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.
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 $\tilde{f}$.
\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. $c(v_1,v_2)-f(v_1,v_2)\ge1$, 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
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 $(f+\gamma)(e)\neq f'(e)$.
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
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}{$G$, $f$, $\alpha'$}
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 $c_{z_0}/C \ge1/3$. Then for any $v\in z_0$, we define
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 $c_{z_1}/C_0\ge1/2$. Then, with the same argument as for $z_0$, we can define
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 $z_1$,
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.
\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 $1\le i\le N$.
\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 $\pm1$ encodes whether this corresponds to what happens in the assignment $T'$.
\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
&=\#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
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.
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 $\pm1$, 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}.
\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}.
\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 $\alpha> \beta> \gamma\ge0$ 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.
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.