图论模板

单源最短路（dij+堆优化）

#include<bits/stdc++.h>
#define ll long long int
using namespace std;
inline ll gcd(ll a,ll b){return b?gcd(b,a%b):a;}
inline ll lcm(ll a,ll b){return a/gcd(a,b)*b;}
int moth[13]={0,31,28,31,30,31,30,31,31,30,31,30,31};
int dir[4][2]={1,0 ,0,1 ,-1,0 ,0,-1};
int dirs[8][2]={1,0 ,0,1 ,-1,0 ,0,-1, -1,-1 ,-1,1 ,1,-1 ,1,1};
const int inf=0x3f3f3f3f;
const ll mod=1e9+7;
const int N=500007;
priority_queue<pair<int,int> > q;
struct edge{
    int next,to,w;
};
edge e[N<<1];
int head[N],cnt;
int d[N],vis[N];
void init(){
    cnt=0;
    memset(head,0,sizeof(head));
    memset(vis,0,sizeof(vis));
    memset(d,inf,sizeof(d));
}
void add(int u, int v, int w){
    e[++cnt]={head[u],v,w};
    head[u] = cnt;
}
void dij(int s){
    d[s]=0;
    q.push(make_pair(0,s));
    while(!q.empty()){
        int u=q.top().second;
        q.pop();
        if(vis[u]) continue;
        vis[u]=1;
        for(int i=head[u];i;i=e[i].next){
            int v=e[i].to; int w=e[i].w;
            if(d[v]>d[u]+w){
                d[v]=d[u]+w;
                q.push(make_pair(-d[v],v));
            }
        }
    }
}
int main(){
    //ios::sync_with_stdio(false);
    init();
    int n,m,s;
    scanf("%d%d%d",&n,&m,&s);
    for(int i=1;i<=m;i++){
        int u,v,w; scanf("%d%d%d",&u,&v,&w);
        add(u,v,w);
    }
    dij(s);
    for(int i=1;i<=n;i++){
        if(i==1) printf("%d",d[i]);
        else printf(" %d",d[i]);
    }
    printf("
");
    return 0;
}

 LCA(倍增)(在线算法) nlogn预处理 logn回答

#include<bits/stdc++.h>
#define ll long long int
using namespace std;
inline ll gcd(ll a,ll b){return b?gcd(b,a%b):a;}
inline ll lcm(ll a,ll b){return a/gcd(a,b)*b;}
int moth[13]={0,31,28,31,30,31,30,31,31,30,31,30,31};
int dir[4][2]={1,0 ,0,1 ,-1,0 ,0,-1};
int dirs[8][2]={1,0 ,0,1 ,-1,0 ,0,-1, -1,-1 ,-1,1 ,1,-1 ,1,1};
const int inf=0x3f3f3f3f;
const ll mod=1e9+7;
const int N=500007;
struct edge{
    int next,v;
};
edge e[N<<1];
int head[N],cnt,f[N][30],d[N];
void add(int u,int v){
    e[++cnt]=(edge){head[u],v};
    head[u]=cnt;
}
int T;
void dfs(int u,int fa){
    d[u]=d[fa]+1; 
    for(int i=head[u];i;i=e[i].next){
        int v=e[i].v;
        if(v==fa) continue;
        f[v][0]=u;
        for(int j=1;j<=T;j++)
            f[v][j]=f[f[v][j-1]][j-1];
        dfs(v,u);
    }
}
int lca(int x,int y){
    if(d[x]>d[y]) swap(x,y);
    for(int i=T;i>=0;i--)
        if(d[f[y][i]]>=d[x]) y=f[y][i];
    if(x==y) return x;
    for(int i=T;i>=0;i--)
        if(f[x][i]!=f[y][i]) x=f[x][i],y=f[y][i];
    return f[x][0];
}
int main(){
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n,m,s; cin>>n>>m>>s; 
    T=log2(n)+1;
    for(int i=1;i<n;i++){
        int u,v; cin>>u>>v;
        add(u,v); add(v,u);
    }
    dfs(s,0);
    for(int i=1;i<=m;i++){
        int x,y; cin>>x>>y;
        cout<<lca(x,y)<<"
";
    }    
    return 0;
}

 LCA(tarjan并查集优化)(离线算法) n+m回答

#include<bits/stdc++.h>
#define ll long long int
using namespace std;
inline ll gcd(ll a,ll b){return b?gcd(b,a%b):a;}
inline ll lcm(ll a,ll b){return a/gcd(a,b)*b;}
int moth[13]={0,31,28,31,30,31,30,31,31,30,31,30,31};
int dir[4][2]={1,0 ,0,1 ,-1,0 ,0,-1};
int dirs[8][2]={1,0 ,0,1 ,-1,0 ,0,-1, -1,-1 ,-1,1 ,1,-1 ,1,1};
const int inf=0x3f3f3f3f;
const ll mod=1e9+7;
const int N=500007;
struct edge{
    int next,v;
};
edge e[N<<1];
int head[N],cnt,f[N],vis[N];
vector<pair<int,int> > qu[N];
int ans[N];
void add(int u,int v){
    e[++cnt]=(edge){head[u],v};
    head[u]=cnt;
}
int find(int x){
    if(x!=f[x])
        f[x]=find(f[x]);
    return f[x];
}
void join(int x,int y){
    int xx=find(x); int yy=find(y);
    if(xx!=yy){
        f[yy]=xx;
    }
}
void tarjan(int u){
    vis[u]=1;
    for(int i=head[u];i;i=e[i].next){
        int v=e[i].v;
        if(vis[v]) continue;
        tarjan(v);
        join(u,v);
    }
    vis[u]=2;
    for(int i=0;i<qu[u].size();i++){
        if(vis[qu[u][i].first]==2){
            int lca=find(qu[u][i].first);
            ans[qu[u][i].second]=lca;
        }
    }
}
int main(){
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n,m,s; cin>>n>>m>>s;
    for(int i=0;i<=n;i++) f[i]=i;
    for(int i=1;i<n;i++){
        int u,v; cin>>u>>v;
        add(u,v); add(v,u);
    }
    for(int i=1;i<=m;i++){
        int x,y; cin>>x>>y;
        qu[x].push_back(make_pair(y,i));
        qu[y].push_back(make_pair(x,i));
    }
    tarjan(s);
    for(int i=1;i<=m;i++){
        cout<<ans[i]<<"
";
    }
    return 0;
}

tarjan算法(无向图割边)

#include<bits/stdc++.h>
#define ll long long int
using namespace std;
inline ll gcd(ll a,ll b){return b?gcd(b,a%b):a;}
inline ll lcm(ll a,ll b){return a/gcd(a,b)*b;}
int moth[13]={0,31,28,31,30,31,30,31,31,30,31,30,31};
int dir[4][2]={1,0 ,0,1 ,-1,0 ,0,-1};
int dirs[8][2]={1,0 ,0,1 ,-1,0 ,0,-1, -1,-1 ,-1,1 ,1,-1 ,1,1};
const int inf=0x3f3f3f3f;
const int N=1e6+7;
const ll mod=1e9+7;
struct edge{
    int v,next,w;
};
edge e[N<<1];
int head[N],cnt = 1,dfn[N],low[N],num = 0,bridge[N];
void init(){
    memset(head,0,sizeof(head));
    memset(dfn,0,sizeof(dfn));
    memset(low,0,sizeof(low));
    memset(bridge,0,sizeof(bridge));
    cnt=1; num=0;
}
void add(int u,int v,int w){
    e[++cnt]=(edge){v,head[u],w};
    head[u]=cnt;
}
void tarjan(int u,int edge){
    dfn[u]=low[u]=++num;
    for(int i=head[u];i;i=e[i].next){
        int v=e[i].v; int w=e[i].w;
        if(!dfn[v]){
            tarjan(v,i);
            low[u]=min(low[u],low[v]);
            if(low[v]>dfn[u])
                bridge[i]=bridge[i^1]=1;
        }else if(i!=(edge^1))
            low[u]=min(low[u],low[v]);
    }
}
int main(){
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n,m;
    cin>>n>>m; 
    init();
    for(int i=1;i<=m;i++){
        int u,v,w; cin>>u>>v>>w;
        add(u,v,w); add(v,u,w);
    }
    for(int i=1;i<=n;i++){
        if(dfn[i]) continue;
        tarjan(i,0);
    }
    int ans=inf;
    for(int i=2;i<cnt;i+=2){
        if(bridge[i])
            cout<<e[i^1].v<<" "<<e[i].v<<endl;
    }
    return 0;
}

 tarjan算法(无向图割点)

#include<bits/stdc++.h>
#define ll long long int
using namespace std;
inline ll gcd(ll a,ll b){return b?gcd(b,a%b):a;}
inline ll lcm(ll a,ll b){return a/gcd(a,b)*b;}
int moth[13]={0,31,28,31,30,31,30,31,31,30,31,30,31};
int dir[4][2]={1,0 ,0,1 ,-1,0 ,0,-1};
int dirs[8][2]={1,0 ,0,1 ,-1,0 ,0,-1, -1,-1 ,-1,1 ,1,-1 ,1,1};
const int inf=0x3f3f3f3f;
const int N=1e5+7;
const ll mod=1e9+7;
struct edge{
    int v,next;
};
edge e[N<<1];
int head[N],cnt=0,dfn[N],low[N],num=0,root,cut[N];
void add(int u,int v){
    e[++cnt]=(edge){v,head[u]};
    head[u]=cnt;
}
void tarjan(int u){
    dfn[u]=low[u]=++num;
    int flag=0;
    for(int i=head[u];i;i=e[i].next){
        int v=e[i].v;
        if(!dfn[v]){
            tarjan(v);
            low[u]=min(low[u],low[v]);
            if(dfn[u]<=low[v]){
                flag++;
                if(u!=root||flag>1) cut[u]=1;
            }
        }else{
            low[u]=min(low[u],dfn[v]);
        }
    }
}
int main(){
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n,m; cin>>n>>m;
    for(int i=1;i<=m;i++){
        int u,v; cin>>u>>v;
        add(u,v); add(v,u);
    }
    for(int i=1;i<=n;i++){
        if(!dfn[i]){
            root=i;
            tarjan(i);
        }
    }
    int c=0;
    for(int i=1;i<=n;i++)
        if(cut[i]) c++;
    cout<<c<<"
";
    for(int i=1;i<=n;i++)
        if(cut[i]) cout<<i<<" ";
    return 0;
}

 e-DCC（边双联通分量的求法）

把图中的桥删去后剩下的联通分量就是边双连通分量

#include<bits/stdc++.h>
#define ll long long int
using namespace std;
inline ll gcd(ll a,ll b){return b?gcd(b,a%b):a;}
inline ll lcm(ll a,ll b){return a/gcd(a,b)*b;}
int moth[13]={0,31,28,31,30,31,30,31,31,30,31,30,31};
int dir[4][2]={1,0 ,0,1 ,-1,0 ,0,-1};
int dirs[8][2]={1,0 ,0,1 ,-1,0 ,0,-1, -1,-1 ,-1,1 ,1,-1 ,1,1};
const int inf=0x3f3f3f3f;
const int N=1e6+7;
const ll mod=1e9+7;
struct edge{
    int v,next,w;
};
edge e[N<<1];
int head[N],cnt = 1,dfn[N],low[N],num = 0,bridge[N];
void init(){
    memset(head,0,sizeof(head));
    memset(dfn,0,sizeof(dfn));
    memset(low,0,sizeof(low));
    memset(bridge,0,sizeof(bridge));
    cnt=1; num=0;
}
void add(int u,int v,int w){
    e[++cnt]=(edge){v,head[u],w};
    head[u]=cnt;
}
void tarjan(int u,int edge){
    dfn[u]=low[u]=++num;
    for(int i=head[u];i;i=e[i].next){
        int v=e[i].v; int w=e[i].w;
        if(!dfn[v]){
            tarjan(v,i);
            low[u]=min(low[u],low[v]);
            if(low[v]>dfn[u])
                bridge[i]=bridge[i^1]=1;
        }else if(i!=(edge^1))
            low[u]=min(low[u],low[v]);
    }
}
int c[N],dcc=0;
void dfs(int u){
  c[u]=dcc;
  for(int i=head[u];i;i=e[i].next){
      int v=e[i].v;
      if(c[v]||bridge[v]) continue;
      dfs(v);
  }
}
int main(){
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n,m;
    cin>>n>>m; 
    init();
    for(int i=1;i<=m;i++){
        int u,v,w; cin>>u>>v>>w;
        add(u,v,w); add(v,u,w);
    }
    for(int i=1;i<=n;i++){
        if(dfn[i]) continue;
        tarjan(i,0);
    }
    for(int i=1;i<=n;i++){
        if(!c[i]){
            ++dcc;
            dfs(i);
        }
    }
    return 0;
}

 最大流（EK）

#include <bits/stdc++.h>
using namespace std;
const double pi = acos(-1.0);
const int maxn = 1e5+7;
const int inf = 0x3f3f3f3f;
const double eps = 1e-6;
typedef long long ll;
const ll mod = 1e7+9;

struct Edge {
  int from, to, cap, flow;
  Edge(int u, int v, int c, int f) : from(u), to(v), cap(c), flow(f) {}
};

struct EK {
  int n, m;
  vector<Edge> edges;
  vector<int> G[maxn];
  int a[maxn], p[maxn];

  void init(int n) {
    for (int i = 0; i < n; i++) G[i].clear();
    edges.clear();
  }

  void AddEdge(int from, int to, int cap) {
    edges.push_back(Edge(from, to, cap, 0));
    edges.push_back(Edge(to, from, 0, 0));
    m = edges.size();
    G[from].push_back(m - 2);
    G[to].push_back(m - 1);
  }

  int Maxflow(int s, int t) {
    int flow = 0;
    while(1){
      memset(a, 0, sizeof(a));
      queue<int> Q;
      Q.push(s);
      a[s] = inf;
      while (!Q.empty()) {
        int x = Q.front();
        Q.pop();
        for (int i = 0; i < G[x].size(); i++) {
          Edge& e = edges[G[x][i]];
          if (!a[e.to] && e.cap > e.flow) {
            p[e.to] = G[x][i];
            a[e.to] = min(a[x], e.cap - e.flow);
            Q.push(e.to);
          }
        }
        if (a[t]) break;
      }
      if (!a[t]) break;
      for (int u = t; u != s; u = edges[p[u]].from) {
        edges[p[u]].flow += a[t];
        edges[p[u] ^ 1].flow -= a[t];
      }
      flow += a[t];
    }
    return flow;
  }
};

 最大流（dinic）

#include <bits/stdc++.h>
using namespace std;
const double pi = acos(-1.0);
const int maxn = 1e4+7;
const int inf = 0x3f3f3f3f;
const double eps = 1e-6;
typedef long long ll;
const ll mod = 1e7+9;
struct Edge {
  int from, to, cap, flow;
  Edge(int u, int v, int c, int f) : from(u), to(v), cap(c), flow(f) {}
};

struct Dinic {
  int n, m, s, t;
  vector<Edge> edges;
  vector<int> G[maxn];
  int d[maxn], cur[maxn];
  bool vis[maxn];

  void init(int n) {
    for (int i = 0; i < n; i++) G[i].clear();
    edges.clear();
  }

  void AddEdge(int from, int to, int cap) {
    edges.push_back(Edge(from, to, cap, 0));
    edges.push_back(Edge(to, from, 0, 0));
    m = edges.size();
    G[from].push_back(m - 2);
    G[to].push_back(m - 1);
  }

  bool BFS() {
    memset(vis, 0, sizeof(vis));
    queue<int> Q;
    Q.push(s);
    d[s] = 0;
    vis[s] = 1;
    while (!Q.empty()) {
      int x = Q.front();
      Q.pop();
      for (int i = 0; i < G[x].size(); i++) {
        Edge& e = edges[G[x][i]];
        if (!vis[e.to] && e.cap > e.flow) {
          vis[e.to] = 1;
          d[e.to] = d[x] + 1;
          Q.push(e.to);
        }
      }
    }
    return vis[t];
  }

  int DFS(int x, int a) {
    if (x == t || a == 0) return a;
    int flow = 0, f;
    for (int& i = cur[x]; i < G[x].size(); i++) {
      Edge& e = edges[G[x][i]];
      if (d[x] + 1 == d[e.to] && (f = DFS(e.to, min(a, e.cap - e.flow))) > 0) {
        e.flow += f;
        edges[G[x][i] ^ 1].flow -= f;
        flow += f;
        a -= f;
        if (a == 0) break;
      }
    }
    return flow;
  }

  int Maxflow(int s, int t) {
    this->s = s;
    this->t = t;
    int flow = 0;
    while (BFS()) {
      memset(cur, 0, sizeof(cur));
      flow += DFS(s, inf);
    }
    return flow;
  }
};