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