/*
* This module deals with graph algorithm in complete bipartite
* graphs. It is used in layout.rs to build the partition to node
* assignation.
* */
use std::cmp::{min,max};
use std::collections::VecDeque;
use rand::prelude::SliceRandom;
//Graph data structure for the flow algorithm.
#[derive(Clone,Copy,Debug)]
struct EdgeFlow{
c : i32,
flow : i32,
v : usize,
rev : usize,
}
//Graph data structure for the detection of positive cycles.
#[derive(Clone,Copy,Debug)]
struct WeightedEdge{
w : i32,
u : usize,
v : usize,
}
/* This function takes two matchings (old_match and new_match) in a
* complete bipartite graph. It returns a matching that has the
* same degree as new_match at every vertex, and that is as close
* as possible to old_match.
* */
pub fn optimize_matching( old_match : &Vec<Vec<usize>> ,
new_match : &Vec<Vec<usize>> ,
nb_right : usize )
-> Vec<Vec<usize>> {
let nb_left = old_match.len();
let ed = WeightedEdge{w:-1,u:0,v:0};
let mut edge_vec = vec![ed ; nb_left*nb_right];
//We build the complete bipartite graph structure, represented
//by the list of all edges.
for i in 0..nb_left {
for j in 0..nb_right{
edge_vec[i*nb_right + j].u = i;
edge_vec[i*nb_right + j].v = nb_left+j;
}
}
for i in 0..edge_vec.len() {
//We add the old matchings
if old_match[edge_vec[i].u].contains(&(edge_vec[i].v-nb_left)) {
edge_vec[i].w *= -1;
}
//We add the new matchings
if new_match[edge_vec[i].u].contains(&(edge_vec[i].v-nb_left)) {
(edge_vec[i].u,edge_vec[i].v) =
(edge_vec[i].v,edge_vec[i].u);
edge_vec[i].w *= -1;
}
}
//Now edge_vec is a graph where edges are oriented LR if we
//can add them to new_match, and RL otherwise. If
//adding/removing them makes the matching closer to old_match
//they have weight 1; and -1 otherwise.
//We shuffle the edge list so that there is no bias depending in
//partitions/zone label in the triplet dispersion
let mut rng = rand::thread_rng();
edge_vec.shuffle(&mut rng);
//Discovering and flipping a cycle with positive weight in this
//graph will make the matching closer to old_match.
//We use Bellman Ford algorithm to discover positive cycles
loop{
if let Some(cycle) = positive_cycle(&edge_vec, nb_left, nb_right) {
for i in cycle {
//We flip the edges of the cycle.
(edge_vec[i].u,edge_vec[i].v) =
(edge_vec[i].v,edge_vec[i].u);
edge_vec[i].w *= -1;
}
}
else {
//If there is no cycle, we return the optimal matching.
break;
}
}
//The optimal matching is build from the graph structure.
let mut matching = vec![Vec::<usize>::new() ; nb_left];
for e in edge_vec {
if e.u > e.v {
matching[e.v].push(e.u-nb_left);
}
}
matching
}
//This function finds a positive cycle in a bipartite wieghted graph.
fn positive_cycle( edge_vec : &Vec<WeightedEdge>, nb_left : usize,
nb_right : usize) -> Option<Vec<usize>> {
let nb_side_min = min(nb_left, nb_right);
let nb_vertices = nb_left+nb_right;
let weight_lowerbound = -((nb_left +nb_right) as i32) -1;
let mut accessed = vec![false ; nb_left];
//We try to find a positive cycle accessible from the left
//vertex i.
for i in 0..nb_left{
if accessed[i] {
continue;
}
let mut weight =vec![weight_lowerbound ; nb_vertices];
let mut prev =vec![ edge_vec.len() ; nb_vertices];
weight[i] = 0;
//We compute largest weighted paths from i.
//Since the graph is bipartite, any simple cycle has length
//at most 2*nb_side_min. In the general Bellman-Ford
//algorithm, the bound here is the number of vertices. Since
//the number of partitions can be much larger than the
//number of nodes, we optimize that.
for _ in 0..(2*nb_side_min) {
for j in 0..edge_vec.len() {
let e = edge_vec[j];
if weight[e.v] < weight[e.u]+e.w {
weight[e.v] = weight[e.u]+e.w;
prev[e.v] = j;
}
}
}
//We update the accessed table
for i in 0..nb_left {
if weight[i] > weight_lowerbound {
accessed[i] = true;
}
}
//We detect positive cycle
for e in edge_vec {
if weight[e.v] < weight[e.u]+e.w {
//it means e is on a path branching from a positive cycle
let mut was_seen = vec![false ; nb_vertices];
let mut curr = e.u;
//We track back with prev until we reach the cycle.
while !was_seen[curr]{
was_seen[curr] = true;
curr = edge_vec[prev[curr]].u;
}
//Now curr is on the cycle. We collect the edges ids.
let mut cycle = Vec::<usize>::new();
cycle.push(prev[curr]);
let mut cycle_vert = edge_vec[prev[curr]].u;
while cycle_vert != curr {
cycle.push(prev[cycle_vert]);
cycle_vert = edge_vec[prev[cycle_vert]].u;
}
return Some(cycle);
}
}
}
None
}
// This function takes two arrays of capacity and computes the
// maximal matching in the complete bipartite graph such that the
// left vertex i is matched to left_cap_vec[i] right vertices, and
// the right vertex j is matched to right_cap_vec[j] left vertices.
// To do so, we use Dinic's maximum flow algorithm.
pub fn dinic_compute_matching( left_cap_vec : Vec<u32>,
right_cap_vec : Vec<u32>) -> Vec< Vec<usize> >
{
let mut graph = Vec::<Vec::<EdgeFlow> >::new();
let ed = EdgeFlow{c:0,flow:0,v:0, rev:0};
// 0 will be the source
graph.push(vec![ed ; left_cap_vec.len()]);
for i in 0..left_cap_vec.len()
{
graph[0][i].c = left_cap_vec[i] as i32;
graph[0][i].v = i+2;
graph[0][i].rev = 0;
}
//1 will be the sink
graph.push(vec![ed ; right_cap_vec.len()]);
for i in 0..right_cap_vec.len()
{
graph[1][i].c = right_cap_vec[i] as i32;
graph[1][i].v = i+2+left_cap_vec.len();
graph[1][i].rev = 0;
}
//we add left vertices
for i in 0..left_cap_vec.len() {
graph.push(vec![ed ; 1+right_cap_vec.len()]);
graph[i+2][0].c = 0; //directed
graph[i+2][0].v = 0;
graph[i+2][0].rev = i;
for j in 0..right_cap_vec.len() {
graph[i+2][j+1].c = 1;
graph[i+2][j+1].v = 2+left_cap_vec.len()+j;
graph[i+2][j+1].rev = i+1;
}
}
//we add right vertices
for i in 0..right_cap_vec.len() {
let lft_ln = left_cap_vec.len();
graph.push(vec![ed ; 1+lft_ln]);
graph[i+lft_ln+2][0].c = graph[1][i].c;
graph[i+lft_ln+2][0].v = 1;
graph[i+lft_ln+2][0].rev = i;
for j in 0..left_cap_vec.len() {
graph[i+2+lft_ln][j+1].c = 0; //directed
graph[i+2+lft_ln][j+1].v = j+2;
graph[i+2+lft_ln][j+1].rev = i+1;
}
}
//To ensure the dispersion of the triplets generated by the
//assignation, we shuffle the neighbours of the nodes. Hence,
//left vertices do not consider the right ones in the same order.
let mut rng = rand::thread_rng();
for i in 0..graph.len() {
graph[i].shuffle(&mut rng);
//We need to update the ids of the reverse edges.
for j in 0..graph[i].len() {
let target_v = graph[i][j].v;
let target_rev = graph[i][j].rev;
graph[target_v][target_rev].rev = j;
}
}
let nb_vertices = graph.len();
//We run Dinic's max flow algorithm
loop{
//We build the level array from Dinic's algorithm.
let mut level = vec![-1; nb_vertices];
let mut fifo = VecDeque::new();
fifo.push_back((0,0));
while !fifo.is_empty() {
if let Some((id,lvl)) = fifo.pop_front(){
if level[id] == -1 {
level[id] = lvl;
for e in graph[id].iter(){
if e.c-e.flow > 0{
fifo.push_back((e.v,lvl+1));
}
}
}
}
}
if level[1] == -1 {
//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();
let flow_upper_bound;
if let Some(x) = left_cap_vec.iter().max() {
flow_upper_bound=*x as i32;
}
else {
flow_upper_bound = 0;
assert!(false);
}
lifo.push_back((0,flow_upper_bound));
loop
{
if let Some((id_tmp, f_tmp)) = lifo.back() {
let id = *id_tmp;
let f = *f_tmp;
if id == 1 {
//The DFS reached the sink, we can add a
//residual flow.
lifo.pop_back();
while !lifo.is_empty() {
if let Some((id,_)) = lifo.pop_back(){
let nbd=next_nbd[id];
graph[id][nbd].flow += f;
let id_v = graph[id][nbd].v;
let nbd_v = graph[id][nbd].rev;
graph[id_v][nbd_v].flow -= f;
}
}
lifo.push_back((0,flow_upper_bound));
continue;
}
//else we did not reach the sink
let nbd = next_nbd[id];
if nbd >= 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,graph[id][nbd].c
- graph[id][nbd].flow);
if level[graph[id][nbd].v] <= level[id] ||
new_flow == 0 {
//We cannot send flow to nbd.
next_nbd[id] += 1;
continue;
}
//otherwise, we send flow to nbd.
lifo.push_back((graph[id][nbd].v, new_flow));
}
else {
break;
}
}
}
//We return the association
let assoc_table = (0..left_cap_vec.len()).map(
|id| graph[id+2].iter()
.filter(|e| e.flow > 0)
.map( |e| e.v-2-left_cap_vec.len())
.collect()).collect();
//consistency check
//it is a flow
for i in 3..graph.len(){
assert!( graph[i].iter().map(|e| e.flow).sum::<i32>() == 0);
for e in graph[i].iter(){
assert!(e.flow + graph[e.v][e.rev].flow == 0);
}
}
//it solves the matching problem
for i in 0..left_cap_vec.len(){
assert!(left_cap_vec[i] as i32 ==
graph[i+2].iter().map(|e| max(0,e.flow)).sum::<i32>());
}
for i in 0..right_cap_vec.len(){
assert!(right_cap_vec[i] as i32 ==
graph[i+2+left_cap_vec.len()].iter()
.map(|e| max(0,e.flow)).sum::<i32>());
}
assoc_table
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn test_flow() {
let left_vec = vec![3;8];
let right_vec = vec![0,4,8,4,8];
//There are asserts in the function that computes the flow
let _ = dinic_compute_matching(left_vec, right_vec);
}
//maybe add tests relative to the matching optilization ?
}
