412 lines
12 KiB
Rust
412 lines
12 KiB
Rust
//! This module deals with graph algorithms.
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//! It is used in layout.rs to build the partition to node assignation.
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use rand::prelude::SliceRandom;
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use std::cmp::{max, min};
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use std::collections::HashMap;
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use std::collections::VecDeque;
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/// Vertex data structures used in all the graphs used in layout.rs.
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/// usize parameters correspond to node/zone/partitions ids.
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/// To understand the vertex roles below, please refer to the formal description
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/// of the layout computation algorithm.
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#[derive(Clone, Copy, Debug, PartialEq, Eq, Hash)]
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pub enum Vertex {
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Source,
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Pup(usize), // The vertex p+ of partition p
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Pdown(usize), // The vertex p- of partition p
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PZ(usize, usize), // The vertex corresponding to x_(partition p, zone z)
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N(usize), // The vertex corresponding to node n
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Sink,
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}
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/// Edge data structure for the flow algorithm.
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#[derive(Clone, Copy, Debug)]
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pub struct FlowEdge {
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cap: u32, // flow maximal capacity of the edge
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flow: i32, // flow value on the edge
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dest: usize, // destination vertex id
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rev: usize, // index of the reversed edge (v, self) in the edge list of vertex v
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}
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/// Edge data structure for the detection of negative cycles.
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#[derive(Clone, Copy, Debug)]
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pub struct WeightedEdge {
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w: i32, // weight of the edge
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dest: usize,
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}
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pub trait Edge: Clone + Copy {}
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impl Edge for FlowEdge {}
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impl Edge for WeightedEdge {}
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/// Struct for the graph structure. We do encapsulation here to be able to both
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/// provide user friendly Vertex enum to address vertices, and to use internally usize
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/// indices and Vec instead of HashMap in the graph algorithm to optimize execution speed.
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pub struct Graph<E: Edge> {
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vertex_to_id: HashMap<Vertex, usize>,
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id_to_vertex: Vec<Vertex>,
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// The graph is stored as an adjacency list
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graph: Vec<Vec<E>>,
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}
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pub type CostFunction = HashMap<(Vertex, Vertex), i32>;
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impl<E: Edge> Graph<E> {
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pub fn new(vertices: &[Vertex]) -> Self {
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let mut map = HashMap::<Vertex, usize>::new();
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for (i, vert) in vertices.iter().enumerate() {
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map.insert(*vert, i);
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}
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Graph::<E> {
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vertex_to_id: map,
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id_to_vertex: vertices.to_vec(),
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graph: vec![Vec::<E>::new(); vertices.len()],
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}
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}
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fn get_vertex_id(&self, v: &Vertex) -> Result<usize, String> {
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self.vertex_to_id
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.get(v)
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.cloned()
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.ok_or_else(|| format!("The graph does not contain vertex {:?}", v))
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}
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}
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impl Graph<FlowEdge> {
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/// This function adds a directed edge to the graph with capacity c, and the
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/// corresponding reversed edge with capacity 0.
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pub fn add_edge(&mut self, u: Vertex, v: Vertex, c: u32) -> Result<(), String> {
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let idu = self.get_vertex_id(&u)?;
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let idv = self.get_vertex_id(&v)?;
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if idu == idv {
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return Err("Cannot add edge from vertex to itself in flow graph".into());
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}
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let rev_u = self.graph[idu].len();
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let rev_v = self.graph[idv].len();
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self.graph[idu].push(FlowEdge {
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cap: c,
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dest: idv,
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flow: 0,
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rev: rev_v,
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});
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self.graph[idv].push(FlowEdge {
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cap: 0,
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dest: idu,
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flow: 0,
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rev: rev_u,
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});
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Ok(())
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}
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/// This function returns the list of vertices that receive a positive flow from
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/// vertex v.
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pub fn get_positive_flow_from(&self, v: Vertex) -> Result<Vec<Vertex>, String> {
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let idv = self.get_vertex_id(&v)?;
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let mut result = Vec::<Vertex>::new();
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for edge in self.graph[idv].iter() {
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if edge.flow > 0 {
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result.push(self.id_to_vertex[edge.dest]);
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}
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}
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Ok(result)
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}
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/// This function returns the value of the flow incoming to v.
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pub fn get_inflow(&self, v: Vertex) -> Result<i32, String> {
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let idv = self.get_vertex_id(&v)?;
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let mut result = 0;
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for edge in self.graph[idv].iter() {
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result += max(0, self.graph[edge.dest][edge.rev].flow);
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}
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Ok(result)
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}
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/// This function returns the value of the flow outgoing from v.
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pub fn get_outflow(&self, v: Vertex) -> Result<i32, String> {
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let idv = self.get_vertex_id(&v)?;
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let mut result = 0;
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for edge in self.graph[idv].iter() {
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result += max(0, edge.flow);
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}
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Ok(result)
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}
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/// This function computes the flow total value by computing the outgoing flow
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/// from the source.
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pub fn get_flow_value(&mut self) -> Result<i32, String> {
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self.get_outflow(Vertex::Source)
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}
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/// This function shuffles the order of the edge lists. It keeps the ids of the
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/// reversed edges consistent.
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fn shuffle_edges(&mut self) {
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let mut rng = rand::thread_rng();
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for i in 0..self.graph.len() {
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self.graph[i].shuffle(&mut rng);
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// We need to update the ids of the reverse edges.
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for j in 0..self.graph[i].len() {
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let target_v = self.graph[i][j].dest;
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let target_rev = self.graph[i][j].rev;
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self.graph[target_v][target_rev].rev = j;
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}
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}
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}
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/// Computes an upper bound of the flow on the graph
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pub fn flow_upper_bound(&self) -> Result<u32, String> {
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let idsource = self.get_vertex_id(&Vertex::Source)?;
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let mut flow_upper_bound = 0;
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for edge in self.graph[idsource].iter() {
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flow_upper_bound += edge.cap;
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}
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Ok(flow_upper_bound)
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}
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/// This function computes the maximal flow using Dinic's algorithm. It starts with
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/// the flow values already present in the graph. So it is possible to add some edge to
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/// the graph, compute a flow, add other edges, update the flow.
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pub fn compute_maximal_flow(&mut self) -> Result<(), String> {
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let idsource = self.get_vertex_id(&Vertex::Source)?;
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let idsink = self.get_vertex_id(&Vertex::Sink)?;
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let nb_vertices = self.graph.len();
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let flow_upper_bound = self.flow_upper_bound()?;
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// To ensure the dispersion of the associations generated by the
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// assignation, we shuffle the neighbours of the nodes. Hence,
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// the vertices do not consider their neighbours in the same order.
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self.shuffle_edges();
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// We run Dinic's max flow algorithm
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loop {
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// We build the level array from Dinic's algorithm.
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let mut level = vec![None; nb_vertices];
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let mut fifo = VecDeque::new();
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fifo.push_back((idsource, 0));
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while let Some((id, lvl)) = fifo.pop_front() {
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if level[id] == None {
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// it means id has not yet been reached
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level[id] = Some(lvl);
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for edge in self.graph[id].iter() {
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if edge.cap as i32 - edge.flow > 0 {
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fifo.push_back((edge.dest, lvl + 1));
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}
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}
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}
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}
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if level[idsink] == None {
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// There is no residual flow
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break;
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}
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// Now we run DFS respecting the level array
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let mut next_nbd = vec![0; nb_vertices];
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let mut lifo = Vec::new();
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lifo.push((idsource, flow_upper_bound));
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while let Some((id, f)) = lifo.last().cloned() {
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if id == idsink {
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// The DFS reached the sink, we can add a
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// residual flow.
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lifo.pop();
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while let Some((id, _)) = lifo.pop() {
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let nbd = next_nbd[id];
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self.graph[id][nbd].flow += f as i32;
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let id_rev = self.graph[id][nbd].dest;
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let nbd_rev = self.graph[id][nbd].rev;
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self.graph[id_rev][nbd_rev].flow -= f as i32;
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}
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lifo.push((idsource, flow_upper_bound));
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continue;
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}
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// else we did not reach the sink
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let nbd = next_nbd[id];
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if nbd >= self.graph[id].len() {
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// There is nothing to explore from id anymore
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lifo.pop();
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if let Some((parent, _)) = lifo.last() {
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next_nbd[*parent] += 1;
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}
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continue;
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}
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// else we can try to send flow from id to its nbd
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let new_flow = min(
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f as i32,
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self.graph[id][nbd].cap as i32 - self.graph[id][nbd].flow,
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) as u32;
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if new_flow == 0 {
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next_nbd[id] += 1;
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continue;
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}
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if let (Some(lvldest), Some(lvlid)) = (level[self.graph[id][nbd].dest], level[id]) {
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if lvldest <= lvlid {
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// We cannot send flow to nbd.
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next_nbd[id] += 1;
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continue;
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}
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}
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// otherwise, we send flow to nbd.
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lifo.push((self.graph[id][nbd].dest, new_flow));
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}
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}
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Ok(())
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}
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/// This function takes a flow, and a cost function on the edges, and tries to find an
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/// equivalent flow with a better cost, by finding improving overflow cycles. It uses
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/// as subroutine the Bellman Ford algorithm run up to path_length.
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/// We assume that the cost of edge (u,v) is the opposite of the cost of (v,u), and
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/// only one needs to be present in the cost function.
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pub fn optimize_flow_with_cost(
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&mut self,
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cost: &CostFunction,
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path_length: usize,
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) -> Result<(), String> {
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// We build the weighted graph g where we will look for negative cycle
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let mut gf = self.build_cost_graph(cost)?;
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let mut cycles = gf.list_negative_cycles(path_length);
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while !cycles.is_empty() {
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// we enumerate negative cycles
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for c in cycles.iter() {
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for i in 0..c.len() {
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// We add one flow unit to the edge (u,v) of cycle c
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let idu = self.vertex_to_id[&c[i]];
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let idv = self.vertex_to_id[&c[(i + 1) % c.len()]];
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for j in 0..self.graph[idu].len() {
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// since idu appears at most once in the cycles, we enumerate every
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// edge at most once.
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let edge = self.graph[idu][j];
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if edge.dest == idv {
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self.graph[idu][j].flow += 1;
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self.graph[idv][edge.rev].flow -= 1;
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break;
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}
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}
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}
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}
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gf = self.build_cost_graph(cost)?;
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cycles = gf.list_negative_cycles(path_length);
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}
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Ok(())
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}
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/// Construct the weighted graph G_f from the flow and the cost function
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fn build_cost_graph(&self, cost: &CostFunction) -> Result<Graph<WeightedEdge>, String> {
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let mut g = Graph::<WeightedEdge>::new(&self.id_to_vertex);
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let nb_vertices = self.id_to_vertex.len();
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for i in 0..nb_vertices {
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for edge in self.graph[i].iter() {
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if edge.cap as i32 - edge.flow > 0 {
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// It is possible to send overflow through this edge
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let u = self.id_to_vertex[i];
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let v = self.id_to_vertex[edge.dest];
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if cost.contains_key(&(u, v)) {
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g.add_edge(u, v, cost[&(u, v)])?;
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} else if cost.contains_key(&(v, u)) {
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g.add_edge(u, v, -cost[&(v, u)])?;
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} else {
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g.add_edge(u, v, 0)?;
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}
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}
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}
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}
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Ok(g)
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}
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}
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impl Graph<WeightedEdge> {
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/// This function adds a single directed weighted edge to the graph.
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pub fn add_edge(&mut self, u: Vertex, v: Vertex, w: i32) -> Result<(), String> {
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let idu = self.get_vertex_id(&u)?;
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let idv = self.get_vertex_id(&v)?;
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self.graph[idu].push(WeightedEdge { w, dest: idv });
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Ok(())
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}
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/// This function lists the negative cycles it manages to find after path_length
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/// iterations of the main loop of the Bellman-Ford algorithm. For the classical
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/// algorithm, path_length needs to be equal to the number of vertices. However,
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/// for particular graph structures like in our case, the algorithm is still correct
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/// when path_length is the length of the longest possible simple path.
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/// See the formal description of the algorithm for more details.
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fn list_negative_cycles(&self, path_length: usize) -> Vec<Vec<Vertex>> {
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let nb_vertices = self.graph.len();
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// We start with every vertex at distance 0 of some imaginary extra -1 vertex.
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let mut distance = vec![0; nb_vertices];
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// The prev vector collects for every vertex from where does the shortest path come
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let mut prev = vec![None; nb_vertices];
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for _ in 0..path_length + 1 {
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for id in 0..nb_vertices {
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for e in self.graph[id].iter() {
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if distance[id] + e.w < distance[e.dest] {
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distance[e.dest] = distance[id] + e.w;
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prev[e.dest] = Some(id);
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}
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}
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}
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}
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// If self.graph contains a negative cycle, then at this point the graph described
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// by prev (which is a directed 1-forest/functional graph)
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// must contain a cycle. We list the cycles of prev.
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let cycles_prev = cycles_of_1_forest(&prev);
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// Remark that the cycle in prev is in the reverse order compared to the cycle
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// in the graph. Thus the .rev().
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return cycles_prev
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.iter()
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.map(|cycle| {
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cycle
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.iter()
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.rev()
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.map(|id| self.id_to_vertex[*id])
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.collect()
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})
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.collect();
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}
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}
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/// This function returns the list of cycles of a directed 1 forest. It does not
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/// check for the consistency of the input.
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fn cycles_of_1_forest(forest: &[Option<usize>]) -> Vec<Vec<usize>> {
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let mut cycles = Vec::<Vec<usize>>::new();
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let mut time_of_discovery = vec![None; forest.len()];
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for t in 0..forest.len() {
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let mut id = t;
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// while we are on a valid undiscovered node
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while time_of_discovery[id] == None {
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time_of_discovery[id] = Some(t);
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if let Some(i) = forest[id] {
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id = i;
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} else {
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break;
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}
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}
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if forest[id] != None && time_of_discovery[id] == Some(t) {
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// We discovered an id that we explored at this iteration t.
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// It means we are on a cycle
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let mut cy = vec![id; 1];
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let mut id2 = id;
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while let Some(id_next) = forest[id2] {
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id2 = id_next;
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if id2 != id {
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cy.push(id2);
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} else {
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break;
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}
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}
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cycles.push(cy);
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}
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}
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cycles
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}
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