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General Weighted Matching
(Graph/generalweightedmatching.hpp)
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- Last update: 2023-01-16 20:41:46+09:00
- Include:
#include "Graph/generalweightedmatching.hpp" - Link: http://www.cs.kent.edu/~dragan/GraphAn/p23-galil.pdf
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#pragma once
// reference: http://www.cs.kent.edu/~dragan/GraphAn/p23-galil.pdf
class GeneralWeightedMatching{
struct E{int u,v; ll w;};
int n,m,in;
vector<vector<E>> G;
vector<int> mate,slack,root,par,isS,used;
vector<vector<int>> flower,belong;
vector<ll> dual;
queue<int> que;
ll dist(const E& e){return dual[e.u]+dual[e.v]-e.w;}
void update(int u,int v){
if(!slack[v] or dist(G[u][v])<dist(G[slack[v]][v]))slack[v]=u;
}
void recalc(int v){
slack[v]=0;
rep(i,1,n+1)if(G[i][v].w and root[i]!=v and isS[root[i]]==1)update(i,v);
}
void push(int v){
if(v<=n)que.push(v);
else for(auto& nxt:flower[v])push(nxt);
}
void set(int v,int rt){
root[v]=rt;
if(v>n)for(auto& nxt:flower[v])set(nxt,rt);
}
int findeven(int b,int v){
int pos=find(ALL(flower[b]),v)-flower[b].begin();
if(pos&1){
reverse(flower[b].begin()+1,flower[b].end());
pos=flower[b].size()-pos;
}
return pos;
}
void match(int u,int v){
mate[u]=G[u][v].v;
if(u>n){
int x=belong[u][G[u][v].u];
int pos=findeven(u,x);
rep(i,0,pos)match(flower[u][i],flower[u][i^1]);
match(x,v);
rotate(flower[u].begin(),flower[u].begin()+pos,flower[u].end());
}
}
void link(int u,int v){
for(;;){
int nv=root[mate[u]];
match(u,v);
if(!nv)break;
v=nv,u=root[par[nv]];
match(v,u);
}
}
void make(int u,int v,int lca){
int id=n+1;
while(id<=m and root[id])id++;
if(id>m)m++;
flower[id].clear();
rep(i,1,m+1)G[id][i].w=G[i][id].w=0;
rep(i,1,n+1)belong[id][i]=0;
isS[id]=1,dual[id]=0,mate[id]=mate[lca];
while(u!=lca){
flower[id].push_back(u);
u=root[mate[u]];
push(u);
flower[id].push_back(u);
u=root[par[u]];
}
flower[id].push_back(lca);
reverse(ALL(flower[id]));
while(v!=lca){
flower[id].push_back(v);
v=root[mate[v]];
push(v);
flower[id].push_back(v);
v=root[par[v]];
}
set(id,id);
for(auto& c:flower[id]){
rep(i,1,m+1)if(!G[id][i].w or dist(G[c][i])<dist(G[id][i])){
G[id][i]=G[c][i],G[i][id]=G[i][c];
}
rep(i,1,n+1)if(belong[c][i])belong[id][i]=c;
}
recalc(id);
}
void expand(int b){
for(auto& c:flower[b])set(c,c);
int x=belong[b][G[b][par[b]].u];
isS[x]=2,par[x]=par[b];
int pos=findeven(b,x);
for(int i=0;i<pos;i+=2){
int T=flower[b][i],S=flower[b][i+1];
isS[S]=1,isS[T]=2;
par[T]=G[S][T].u;
slack[S]=slack[T]=0;
push(S);
}
rep(i,pos+1,flower[b].size()){
isS[flower[b][i]]=0;
recalc(flower[b][i]);
}
flower[b].clear();
root[b]=0;
}
bool path(const E& e){
int u=root[e.u],v=root[e.v];
if(!isS[v]){
par[v]=e.u;
isS[v]=2;
int nu=root[mate[v]];
slack[v]=slack[nu]=0;
isS[nu]=1;
push(nu);
}
else if(isS[v]==1){
int lca=0,bu=u,bv=v;
in++;
while(bu){
used[bu]=in;
bu=root[mate[bu]];
if(bu)bu=root[par[bu]];
}
while(bv){
if(used[bv]==in){
lca=bv;
break;
}
bv=root[mate[bv]];
if(bv)bv=root[par[bv]];
}
if(lca)make(u,v,lca);
else{
link(u,v),link(v,u);
return true;
}
}
return false;
}
bool augment(){
isS=slack=par=vector<int>(n*2);
que=queue<int>();
rep(i,1,m+1)if(root[i]==i and !mate[i]){
isS[i]=1;
push(i);
}
if(que.empty())return false;
for(;;){
while(!que.empty()){
int v=que.front();
que.pop();
rep(i,1,n+1)if(G[v][i].w and root[i]!=root[v]){
if(!dist(G[v][i])){
if(path(G[v][i]))return true;
}
else if(isS[root[i]]!=2)update(v,root[i]);
}
}
ll delta=INF;
rep(i,n+1,m+1)if(root[i]==i and isS[i]==2)chmin(delta,dual[i]/2);
rep(i,1,m+1)if(root[i]==i and slack[i] and isS[i]!=2){
if(!isS[i])chmin(delta,dist(G[slack[i]][i]));
else chmin(delta,dist(G[slack[i]][i])/2);
}
rep(i,1,n+1){
if(isS[root[i]]==1){
dual[i]-=delta;
if(dual[i]<=0)return false;
}
else if(isS[root[i]]==2)dual[i]+=delta;
}
rep(i,n+1,m+1)if(root[i]==i and isS[i]){
if(isS[i]==1)dual[i]+=delta*2;
else dual[i]-=delta*2;
}
rep(i,1,m+1)if(root[i]==i and slack[i] and root[slack[i]]!=i){
if(dist(G[slack[i]][i])==0 and path(G[slack[i]][i]))return true;
}
rep(i,n+1,m+1)if(root[i]==i and isS[i]==2 and dual[i]==0)expand(i);
}
}
public:
GeneralWeightedMatching(int _n):n(_n),m(n),in(0),G(n*2,vector<E>(n*2)),
mate(n*2),root(n*2),used(n*2),flower(n*2),belong(n*2,vector<int>(n*2)),dual(n*2){
rep(i,0,n+1){
root[i]=i;
belong[i][i]=i;
if(i)dual[i]=INF;
rep(j,0,n+1)G[i][j]=E{i,j,0};
}
}
void add_edge(int u,int v,ll w){
u++,v++;
chmax(G[u][v].w,w*2);
chmax(G[v][u].w,w*2);
}
vector<int> run(){
while(augment());
vector<int> res(n,-1);
rep(i,1,n+1)if(mate[i])res[i-1]=mate[i]-1;
return res;
}
};
/**
* @brief General Weighted Matching
*/#line 2 "Graph/generalweightedmatching.hpp"
// reference: http://www.cs.kent.edu/~dragan/GraphAn/p23-galil.pdf
class GeneralWeightedMatching{
struct E{int u,v; ll w;};
int n,m,in;
vector<vector<E>> G;
vector<int> mate,slack,root,par,isS,used;
vector<vector<int>> flower,belong;
vector<ll> dual;
queue<int> que;
ll dist(const E& e){return dual[e.u]+dual[e.v]-e.w;}
void update(int u,int v){
if(!slack[v] or dist(G[u][v])<dist(G[slack[v]][v]))slack[v]=u;
}
void recalc(int v){
slack[v]=0;
rep(i,1,n+1)if(G[i][v].w and root[i]!=v and isS[root[i]]==1)update(i,v);
}
void push(int v){
if(v<=n)que.push(v);
else for(auto& nxt:flower[v])push(nxt);
}
void set(int v,int rt){
root[v]=rt;
if(v>n)for(auto& nxt:flower[v])set(nxt,rt);
}
int findeven(int b,int v){
int pos=find(ALL(flower[b]),v)-flower[b].begin();
if(pos&1){
reverse(flower[b].begin()+1,flower[b].end());
pos=flower[b].size()-pos;
}
return pos;
}
void match(int u,int v){
mate[u]=G[u][v].v;
if(u>n){
int x=belong[u][G[u][v].u];
int pos=findeven(u,x);
rep(i,0,pos)match(flower[u][i],flower[u][i^1]);
match(x,v);
rotate(flower[u].begin(),flower[u].begin()+pos,flower[u].end());
}
}
void link(int u,int v){
for(;;){
int nv=root[mate[u]];
match(u,v);
if(!nv)break;
v=nv,u=root[par[nv]];
match(v,u);
}
}
void make(int u,int v,int lca){
int id=n+1;
while(id<=m and root[id])id++;
if(id>m)m++;
flower[id].clear();
rep(i,1,m+1)G[id][i].w=G[i][id].w=0;
rep(i,1,n+1)belong[id][i]=0;
isS[id]=1,dual[id]=0,mate[id]=mate[lca];
while(u!=lca){
flower[id].push_back(u);
u=root[mate[u]];
push(u);
flower[id].push_back(u);
u=root[par[u]];
}
flower[id].push_back(lca);
reverse(ALL(flower[id]));
while(v!=lca){
flower[id].push_back(v);
v=root[mate[v]];
push(v);
flower[id].push_back(v);
v=root[par[v]];
}
set(id,id);
for(auto& c:flower[id]){
rep(i,1,m+1)if(!G[id][i].w or dist(G[c][i])<dist(G[id][i])){
G[id][i]=G[c][i],G[i][id]=G[i][c];
}
rep(i,1,n+1)if(belong[c][i])belong[id][i]=c;
}
recalc(id);
}
void expand(int b){
for(auto& c:flower[b])set(c,c);
int x=belong[b][G[b][par[b]].u];
isS[x]=2,par[x]=par[b];
int pos=findeven(b,x);
for(int i=0;i<pos;i+=2){
int T=flower[b][i],S=flower[b][i+1];
isS[S]=1,isS[T]=2;
par[T]=G[S][T].u;
slack[S]=slack[T]=0;
push(S);
}
rep(i,pos+1,flower[b].size()){
isS[flower[b][i]]=0;
recalc(flower[b][i]);
}
flower[b].clear();
root[b]=0;
}
bool path(const E& e){
int u=root[e.u],v=root[e.v];
if(!isS[v]){
par[v]=e.u;
isS[v]=2;
int nu=root[mate[v]];
slack[v]=slack[nu]=0;
isS[nu]=1;
push(nu);
}
else if(isS[v]==1){
int lca=0,bu=u,bv=v;
in++;
while(bu){
used[bu]=in;
bu=root[mate[bu]];
if(bu)bu=root[par[bu]];
}
while(bv){
if(used[bv]==in){
lca=bv;
break;
}
bv=root[mate[bv]];
if(bv)bv=root[par[bv]];
}
if(lca)make(u,v,lca);
else{
link(u,v),link(v,u);
return true;
}
}
return false;
}
bool augment(){
isS=slack=par=vector<int>(n*2);
que=queue<int>();
rep(i,1,m+1)if(root[i]==i and !mate[i]){
isS[i]=1;
push(i);
}
if(que.empty())return false;
for(;;){
while(!que.empty()){
int v=que.front();
que.pop();
rep(i,1,n+1)if(G[v][i].w and root[i]!=root[v]){
if(!dist(G[v][i])){
if(path(G[v][i]))return true;
}
else if(isS[root[i]]!=2)update(v,root[i]);
}
}
ll delta=INF;
rep(i,n+1,m+1)if(root[i]==i and isS[i]==2)chmin(delta,dual[i]/2);
rep(i,1,m+1)if(root[i]==i and slack[i] and isS[i]!=2){
if(!isS[i])chmin(delta,dist(G[slack[i]][i]));
else chmin(delta,dist(G[slack[i]][i])/2);
}
rep(i,1,n+1){
if(isS[root[i]]==1){
dual[i]-=delta;
if(dual[i]<=0)return false;
}
else if(isS[root[i]]==2)dual[i]+=delta;
}
rep(i,n+1,m+1)if(root[i]==i and isS[i]){
if(isS[i]==1)dual[i]+=delta*2;
else dual[i]-=delta*2;
}
rep(i,1,m+1)if(root[i]==i and slack[i] and root[slack[i]]!=i){
if(dist(G[slack[i]][i])==0 and path(G[slack[i]][i]))return true;
}
rep(i,n+1,m+1)if(root[i]==i and isS[i]==2 and dual[i]==0)expand(i);
}
}
public:
GeneralWeightedMatching(int _n):n(_n),m(n),in(0),G(n*2,vector<E>(n*2)),
mate(n*2),root(n*2),used(n*2),flower(n*2),belong(n*2,vector<int>(n*2)),dual(n*2){
rep(i,0,n+1){
root[i]=i;
belong[i][i]=i;
if(i)dual[i]=INF;
rep(j,0,n+1)G[i][j]=E{i,j,0};
}
}
void add_edge(int u,int v,ll w){
u++,v++;
chmax(G[u][v].w,w*2);
chmax(G[v][u].w,w*2);
}
vector<int> run(){
while(augment());
vector<int> res(n,-1);
rep(i,1,n+1)if(mate[i])res[i-1]=mate[i]-1;
return res;
}
};
/**
* @brief General Weighted Matching
*/