379 lines
12 KiB
Rust
379 lines
12 KiB
Rust
/*
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* This module deals with graph algorithm in complete bipartite
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* graphs. It is used in layout.rs to build the partition to node
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* assignation.
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* */
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use std::cmp::{min,max};
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use std::collections::VecDeque;
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use rand::prelude::SliceRandom;
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//Graph data structure for the flow algorithm.
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#[derive(Clone,Copy,Debug)]
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struct EdgeFlow{
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c : i32,
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flow : i32,
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v : usize,
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rev : usize,
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}
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//Graph data structure for the detection of positive cycles.
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#[derive(Clone,Copy,Debug)]
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struct WeightedEdge{
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w : i32,
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u : usize,
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v : usize,
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}
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/* This function takes two matchings (old_match and new_match) in a
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* complete bipartite graph. It returns a matching that has the
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* same degree as new_match at every vertex, and that is as close
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* as possible to old_match.
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* */
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pub fn optimize_matching( old_match : &Vec<Vec<usize>> ,
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new_match : &Vec<Vec<usize>> ,
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nb_right : usize )
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-> Vec<Vec<usize>> {
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let nb_left = old_match.len();
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let ed = WeightedEdge{w:-1,u:0,v:0};
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let mut edge_vec = vec![ed ; nb_left*nb_right];
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//We build the complete bipartite graph structure, represented
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//by the list of all edges.
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for i in 0..nb_left {
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for j in 0..nb_right{
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edge_vec[i*nb_right + j].u = i;
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edge_vec[i*nb_right + j].v = nb_left+j;
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}
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}
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for i in 0..edge_vec.len() {
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//We add the old matchings
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if old_match[edge_vec[i].u].contains(&(edge_vec[i].v-nb_left)) {
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edge_vec[i].w *= -1;
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}
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//We add the new matchings
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if new_match[edge_vec[i].u].contains(&(edge_vec[i].v-nb_left)) {
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(edge_vec[i].u,edge_vec[i].v) =
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(edge_vec[i].v,edge_vec[i].u);
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edge_vec[i].w *= -1;
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}
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}
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//Now edge_vec is a graph where edges are oriented LR if we
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//can add them to new_match, and RL otherwise. If
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//adding/removing them makes the matching closer to old_match
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//they have weight 1; and -1 otherwise.
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//We shuffle the edge list so that there is no bias depending in
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//partitions/zone label in the triplet dispersion
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let mut rng = rand::thread_rng();
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edge_vec.shuffle(&mut rng);
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//Discovering and flipping a cycle with positive weight in this
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//graph will make the matching closer to old_match.
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//We use Bellman Ford algorithm to discover positive cycles
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loop{
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if let Some(cycle) = positive_cycle(&edge_vec, nb_left, nb_right) {
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for i in cycle {
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//We flip the edges of the cycle.
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(edge_vec[i].u,edge_vec[i].v) =
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(edge_vec[i].v,edge_vec[i].u);
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edge_vec[i].w *= -1;
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}
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}
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else {
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//If there is no cycle, we return the optimal matching.
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break;
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}
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}
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//The optimal matching is build from the graph structure.
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let mut matching = vec![Vec::<usize>::new() ; nb_left];
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for e in edge_vec {
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if e.u > e.v {
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matching[e.v].push(e.u-nb_left);
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}
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}
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matching
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}
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//This function finds a positive cycle in a bipartite wieghted graph.
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fn positive_cycle( edge_vec : &Vec<WeightedEdge>, nb_left : usize,
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nb_right : usize) -> Option<Vec<usize>> {
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let nb_side_min = min(nb_left, nb_right);
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let nb_vertices = nb_left+nb_right;
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let weight_lowerbound = -((nb_left +nb_right) as i32) -1;
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let mut accessed = vec![false ; nb_left];
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//We try to find a positive cycle accessible from the left
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//vertex i.
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for i in 0..nb_left{
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if accessed[i] {
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continue;
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}
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let mut weight =vec![weight_lowerbound ; nb_vertices];
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let mut prev =vec![ edge_vec.len() ; nb_vertices];
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weight[i] = 0;
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//We compute largest weighted paths from i.
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//Since the graph is bipartite, any simple cycle has length
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//at most 2*nb_side_min. In the general Bellman-Ford
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//algorithm, the bound here is the number of vertices. Since
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//the number of partitions can be much larger than the
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//number of nodes, we optimize that.
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for _ in 0..(2*nb_side_min) {
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for j in 0..edge_vec.len() {
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let e = edge_vec[j];
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if weight[e.v] < weight[e.u]+e.w {
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weight[e.v] = weight[e.u]+e.w;
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prev[e.v] = j;
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}
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}
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}
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//We update the accessed table
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for i in 0..nb_left {
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if weight[i] > weight_lowerbound {
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accessed[i] = true;
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}
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}
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//We detect positive cycle
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for e in edge_vec {
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if weight[e.v] < weight[e.u]+e.w {
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//it means e is on a path branching from a positive cycle
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let mut was_seen = vec![false ; nb_vertices];
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let mut curr = e.u;
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//We track back with prev until we reach the cycle.
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while !was_seen[curr]{
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was_seen[curr] = true;
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curr = edge_vec[prev[curr]].u;
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}
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//Now curr is on the cycle. We collect the edges ids.
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let mut cycle = Vec::<usize>::new();
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cycle.push(prev[curr]);
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let mut cycle_vert = edge_vec[prev[curr]].u;
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while cycle_vert != curr {
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cycle.push(prev[cycle_vert]);
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cycle_vert = edge_vec[prev[cycle_vert]].u;
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}
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return Some(cycle);
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}
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}
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}
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None
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}
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// This function takes two arrays of capacity and computes the
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// maximal matching in the complete bipartite graph such that the
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// left vertex i is matched to left_cap_vec[i] right vertices, and
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// the right vertex j is matched to right_cap_vec[j] left vertices.
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// To do so, we use Dinic's maximum flow algorithm.
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pub fn dinic_compute_matching( left_cap_vec : Vec<u32>,
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right_cap_vec : Vec<u32>) -> Vec< Vec<usize> >
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{
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let mut graph = Vec::<Vec::<EdgeFlow> >::new();
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let ed = EdgeFlow{c:0,flow:0,v:0, rev:0};
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// 0 will be the source
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graph.push(vec![ed ; left_cap_vec.len()]);
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for i in 0..left_cap_vec.len()
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{
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graph[0][i].c = left_cap_vec[i] as i32;
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graph[0][i].v = i+2;
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graph[0][i].rev = 0;
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}
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//1 will be the sink
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graph.push(vec![ed ; right_cap_vec.len()]);
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for i in 0..right_cap_vec.len()
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{
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graph[1][i].c = right_cap_vec[i] as i32;
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graph[1][i].v = i+2+left_cap_vec.len();
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graph[1][i].rev = 0;
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}
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//we add left vertices
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for i in 0..left_cap_vec.len() {
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graph.push(vec![ed ; 1+right_cap_vec.len()]);
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graph[i+2][0].c = 0; //directed
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graph[i+2][0].v = 0;
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graph[i+2][0].rev = i;
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for j in 0..right_cap_vec.len() {
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graph[i+2][j+1].c = 1;
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graph[i+2][j+1].v = 2+left_cap_vec.len()+j;
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graph[i+2][j+1].rev = i+1;
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}
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}
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//we add right vertices
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for i in 0..right_cap_vec.len() {
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let lft_ln = left_cap_vec.len();
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graph.push(vec![ed ; 1+lft_ln]);
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graph[i+lft_ln+2][0].c = graph[1][i].c;
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graph[i+lft_ln+2][0].v = 1;
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graph[i+lft_ln+2][0].rev = i;
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for j in 0..left_cap_vec.len() {
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graph[i+2+lft_ln][j+1].c = 0; //directed
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graph[i+2+lft_ln][j+1].v = j+2;
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graph[i+2+lft_ln][j+1].rev = i+1;
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}
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}
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//To ensure the dispersion of the triplets generated by the
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//assignation, we shuffle the neighbours of the nodes. Hence,
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//left vertices do not consider the right ones in the same order.
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let mut rng = rand::thread_rng();
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for i in 0..graph.len() {
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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..graph[i].len() {
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let target_v = graph[i][j].v;
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let target_rev = graph[i][j].rev;
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graph[target_v][target_rev].rev = j;
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}
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}
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let nb_vertices = graph.len();
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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![-1; nb_vertices];
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let mut fifo = VecDeque::new();
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fifo.push_back((0,0));
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while !fifo.is_empty() {
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if let Some((id,lvl)) = fifo.pop_front(){
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if level[id] == -1 {
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level[id] = lvl;
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for e in graph[id].iter(){
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if e.c-e.flow > 0{
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fifo.push_back((e.v,lvl+1));
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}
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}
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}
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}
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}
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if level[1] == -1 {
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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 = VecDeque::new();
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let flow_upper_bound;
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if let Some(x) = left_cap_vec.iter().max() {
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flow_upper_bound=*x as i32;
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}
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else {
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flow_upper_bound = 0;
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assert!(false);
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}
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lifo.push_back((0,flow_upper_bound));
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loop
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{
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if let Some((id_tmp, f_tmp)) = lifo.back() {
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let id = *id_tmp;
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let f = *f_tmp;
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if id == 1 {
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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_back();
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while !lifo.is_empty() {
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if let Some((id,_)) = lifo.pop_back(){
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let nbd=next_nbd[id];
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graph[id][nbd].flow += f;
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let id_v = graph[id][nbd].v;
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let nbd_v = graph[id][nbd].rev;
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graph[id_v][nbd_v].flow -= f;
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}
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}
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lifo.push_back((0,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 >= graph[id].len() {
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//There is nothing to explore from id anymore
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lifo.pop_back();
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if let Some((parent, _)) = lifo.back(){
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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(f,graph[id][nbd].c
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- graph[id][nbd].flow);
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if level[graph[id][nbd].v] <= level[id] ||
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new_flow == 0 {
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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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//otherwise, we send flow to nbd.
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lifo.push_back((graph[id][nbd].v, new_flow));
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}
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else {
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break;
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}
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}
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}
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//We return the association
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let assoc_table = (0..left_cap_vec.len()).map(
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|id| graph[id+2].iter()
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.filter(|e| e.flow > 0)
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.map( |e| e.v-2-left_cap_vec.len())
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.collect()).collect();
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//consistency check
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//it is a flow
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for i in 3..graph.len(){
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assert!( graph[i].iter().map(|e| e.flow).sum::<i32>() == 0);
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for e in graph[i].iter(){
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assert!(e.flow + graph[e.v][e.rev].flow == 0);
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}
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}
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//it solves the matching problem
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for i in 0..left_cap_vec.len(){
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assert!(left_cap_vec[i] as i32 ==
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graph[i+2].iter().map(|e| max(0,e.flow)).sum::<i32>());
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}
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for i in 0..right_cap_vec.len(){
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assert!(right_cap_vec[i] as i32 ==
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graph[i+2+left_cap_vec.len()].iter()
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.map(|e| max(0,e.flow)).sum::<i32>());
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}
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assoc_table
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}
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#[cfg(test)]
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mod tests {
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use super::*;
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#[test]
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fn test_flow() {
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let left_vec = vec![3;8];
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let right_vec = vec![0,4,8,4,8];
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//There are asserts in the function that computes the flow
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let _ = dinic_compute_matching(left_vec, right_vec);
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}
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//maybe add tests relative to the matching optilization ?
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}
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