图论总结

①DJ

#include<iostream>
#include<cstring>
#include<cstdio>
#include<queue>
using namespace std;
#define e exit(0)
#define re register
#define inf 2147483647
const int N = 10005;
const int M = 500005; 
int n,m,s,cnt,head[N],vis[N],dis[N];
struct bian{int to,next,v;}len[M];
inline int fd(){
    int s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
void add(int from,int to,int v){
    len[++cnt].v = v;
    len[cnt].to = to;
    len[cnt].next = head[from];
    head[from] = cnt;
}
void DJ(){
    for(re int i=1;i<=n;++i)
        vis[i] = 0,dis[i] = inf;
    dis[s] = 0;
    priority_queue<pair<int,int> >Q;
    Q.push(make_pair(0,s));
    while(Q.size()){
        int now = Q.top().second;
        Q.pop();
        if(vis[now]) continue;
        vis[now] = 1;
        for(re int k=head[now];k;k=len[k].next){
            int to = len[k].to,v = len[k].v;
            if(dis[to] > dis[now]+v){
                dis[to] = dis[now]+v;
                Q.push(make_pair(-dis[to],to));
            }
        }
    }
 }
int main()
{
    freopen("P3371.in","r",stdin);
    freopen("P3371.out","w",stdout);
    n = fd(),m = fd(),s = fd();
    for(re int i=1;i<=m;++i){
        int x = fd(),y = fd(),v = fd();
        add(x,y,v);
    }
    DJ();
    for(re int i=1;i<=n;++i)
        printf("%d ",dis[i]);
    return 0;
}

 ②spfa

#include<iostream>
#include<cstring>
#include<cstdio>
#include<queue>
using namespace std;
#define e exit(0)
#define re register
#define inf 2147483647
const int N = 10005;
const int M = 500005;
int n,m,s,cnt,head[N],vis[N],dis[N];
struct bian{int to,next,v;}len[M];
inline int fd(){
    int s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
void add(int from,int to,int v){
    len[++cnt].v = v;
    len[cnt].to = to;
    len[cnt].next = head[from];
    head[from] = cnt;
}
void spfa(){
    for(re int i=1;i<=n;++i)
        vis[i] = 0,dis[i] = inf;
    queue<int> Q;
    vis[s] = 1,dis[s] = 0;
    Q.push(s);
    while(Q.size()){
        int now = Q.front();
        Q.pop();vis[now] = 0;
        for(re int k=head[now];k;k=len[k].next){
            int to = len[k].to,v = len[k].v;
            if(dis[to] > dis[now]+v){
                dis[to] = dis[now]+v;
                if(!vis[to]){
                    vis[to] = 1;
                    Q.push(to);
                }
            }
        }
    }
}
int main()
{
    n = fd(),m = fd(),s = fd();
    for(re int i=1;i<=m;++i){
        int x = fd(),y = fd(),v = fd();
        add(x,y,v);
    }
    spfa();
    for(re int i=1;i<=n;++i)
        printf("%d ",dis[i]);
    return 0;
}

③Floyd

#include<iostream>
#include<cstring>
#include<cstdio>
using namespace std;
#define e exit(0)
#define re register
#define inf 2147483647
int n,m,s,f[5010][5010];
inline int fd(){
    int s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
int main()
{
    n = fd(),m = fd(),s = fd();
    for(re int i=0;i<5010;++i)
        for(re int j=0;j<5010;++j)
            if(i == j) f[i][j] = 0;
            else f[i][j] = inf;
    for(re int i=1;i<=m;++i){
        int x = fd(),y = fd(),v = fd();
        f[x][y] = min(f[x][y],v);
    }
    for(re int k=1;k<=n;++k)
        for(re int i=1;i<=n;++i)
            for(re int j=1;j<=n;++j){
                if(f[i][k] == inf||f[k][j] == inf) continue;
                f[i][j] = min(f[i][j],f[i][k]+f[k][j]);
            }
    for(re int i=1;i<=n;++i)
        printf("%d ",f[s][i]);
    return 0;
}

④最小生成树

#include<iostream>
#include<algorithm>
#include<cstring>
#include<cstdio>
using namespace std;
#define e exit(0)
#define re register
const int N = 5005;
const int M = 200005;
int n,m,fa[N];
long long ans;
struct bian{int x,y,v;}len[M];
inline int fd(){
    int s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
int find(int x){
    if(x == fa[x]) return x;
    else return fa[x] = find(fa[x]);
}
bool cmp(bian a,bian b){return a.v < b.v;}
int main()
{
    freopen("P3366.in","r",stdin);
    freopen("P3366.out","w",stdout);
    n = fd(),m = fd();
    for(re int i=1;i<=n;++i)
        fa[i] = i;
    for(re int i=1;i<=m;++i)
        len[i].x = fd(),len[i].y = fd(),len[i].v = fd();
    sort(len+1,len+1+m,cmp);
    for(re int i=1;i<=m;++i){
        int fa1 = find(len[i].x),fa2 = find(len[i].y);
        if(fa1 != fa2){
            ans += len[i].v;
            fa[fa1] = fa2;
        }
    }
    printf("%lld",ans);
    return 0;
}

⑤树的直径

#include<iostream>
#include<cstring>
#include<cstdio>
using namespace std;
#define e exit(0)
#define re register
const int M = 50005;
int n,m,cnt,ans,d[M],vis[M],head[M];
struct bian{int to,next,v;}len[M<<1];
inline int fd(){
    int s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
void add(int from,int to,int v){
    len[++cnt].v = v;
    len[cnt].to = to;
    len[cnt].next = head[from];
    head[from] = cnt;
}
void dfs(int x){
    vis[x] = 1;
    for(re int k=head[x];k;k=len[k].next){
        int to = len[k].to,v = len[k].v;
        if(vis[to]) continue;
        dfs(to);
        ans = max(ans,d[x]+d[to]+v);
        d[x] = max(d[x],d[to]+v);
    }
}
void work1(){
    dfs(1);
    printf("%d",ans);
}
int main()
{
    freopen("P5021.in","r",stdin);
    freopen("P5021.out","w",stdout);
    n = fd(),m = fd();
    for(re int i=1;i<=n-1;++i){
        int x = fd(),y = fd(),v = fd();
        add(x,y,v),add(y,x,v);
    }
    if(m == 1){work1();return 0;}
    return 0;
}

 ⑥Lca

#include<iostream>
#include<cstring>
#include<cstdio>
using namespace std;
#define e exit(0)
#define re register
const int N = 5e5+5;
int n,m,s,cnt,head[N],vis[N],dis[N],h[N],fa[N],bz[N][24];
struct bian{int to,next;}len[N<<1];
inline int fd(){
    int s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
void add(int from,int to){
    len[++cnt].to = to;
    len[cnt].next = head[from];
    head[from] = cnt;
}
void dfs(int x){
    for(re int k=head[x];k;k=len[k].next){
        int to = len[k].to;
        if(!h[to]){
            h[to] = h[x]+1;
            fa[to] = x;
            dfs(to);
        }
    }
}
void makebz(){
    for(re int i=1;i<=n;++i)
        bz[i][0] = fa[i];
    for(re int j=1;j<=20;++j)
        for(re int i=1;i<=n;++i)
            bz[i][j] = bz[bz[i][j-1]][j-1];
}
int Lca(int x,int y){
    if(h[x] <= h[y])
        swap(x,y);
    for(re int j=19;j>=0;--j)
        if(h[bz[x][j]] >= h[y])
            x = bz[x][j];
    if(x == y)
        return x;
    for(re int j=19;j>=0;--j)
        if(bz[x][j] != bz[y][j])
            x = bz[x][j],y = bz[y][j];
    return fa[x];
}
int main()
{
    freopen("P3379.in","r",stdin);
    freopen("P3379.out","w",stdout);
    n = fd(),m = fd(),s = fd();
    for(re int i=1;i<=n-1;++i){
        int x = fd(),y = fd();
        add(x,y),add(y,x);
    }
    h[s] = s,fa[s] = s;
    dfs(s);makebz();
    while(m--){
        int a = fd(),b = fd();
        printf("%d\n",Lca(a,b));
    }
    return 0;
}

⑦树的重心 NO1.

#include<iostream>
#include<cstring>
#include<cstdio>
using namespace std;
#define e exit(0)
#define re register
const int M = 5e5+5;
int n,cnt,ans,pos,head[M],size[M],vis[M];
struct bian{int to,next;}len[M];
inline int fd(){
    int s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
void add(int from,int to){
    len[++cnt].to = to;
    len[cnt].next = head[from];
    head[from] = cnt;
}
void dfs(int x){
    size[x] = vis[x] = 1;
    int max_part = 0;
    for(re int k=head[x];k;k=len[k].next){
        int to = len[k].to;
        if(vis[to]) continue;
        dfs(to);
        size[x] += size[to];
        max_part = max(max_part,size[to]);
    }
    max_part = max(max_part,n-size[x]);
    if(max_part < ans){
        ans = max_part;
        pos = x;
    }
}
int main()
{
    freopen("kkk.in","r",stdin);
    freopen("kkk.out","w",stdout);
    n = fd();
    for(re int i=1;i<=n-1;++i){
        int x = fd(),y = fd();
        add(x,y);
    }
    dfs(1);
    printf("%d",pos);
    return 0;
}

 ⑧树的重心 NO2.

#include<iostream>
#include<cstring>
#include<cstdio>
using namespace std;
#define e exit(0)
#define re register
const int M = 5e5+5;
int n,cnt,vis[M],head[M],maxx[M],fa[M],bz[M][24],ans[M],size[M];
struct bian{int to,next;}len[M<<1];
inline int fd(){
    int s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
void add(int from,int to){
    len[++cnt].to = to;
    len[cnt].next = head[from];
    head[from] = cnt;
}
void dfs(int x){
    size[x] = vis[x] = 1;
    for(re int k=head[x];k;k=len[k].next){
        int to = len[k].to;
        if(vis[to]) continue;
        dfs(to);
        size[x] += size[to];
        maxx[x] = max(maxx[x],size[to]);
    }
}
void makebz(){
    for(re int j=1;j<=20;++j)
        for(re int i=1;i<=n;++i)
            bz[i][j] = bz[bz[i][j-1]][j-1];
}
void work(int x){
    int pos = x;
    for(re int j=19;j>=0;--j)
        if(bz[pos][j]&&maxx[x]*2-size[bz[pos][j]]>0)
            pos = bz[pos][j];
    if(pos != x) pos = fa[pos];
    while(pos&&size[x]*2-size[pos]>=0){
        if(maxx[x]*2-size[pos]<=0) ans[pos] = x;
        pos = fa[pos];
    }
    for(re int k=head[x];k;k=len[k].next){
        int to = len[k].to;
        work(to);
    }
}
int main()
{
    freopen("kkk.in","r",stdin);
    freopen("kkk.out","w",stdout);
    n = fd();
    for(re int i=2;i<=n;++i){
        fa[i] = fd();
        bz[i][0] = fa[i];
        add(fa[i],i);
    }
    dfs(1);makebz();work(1);
    return 0;
}

⑨二分图最大匹配

#include<iostream>
#include<cstring>
#include<cstdio>
using namespace std; 
#define re register
const int M = 2001;
int n,m,e,ans,Max,vis[M],match[M],check[M][M];
inline int fd(){
    int s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
bool Dfs(int x){
    for(re int j=n+1;j<=m+n;++j){
        if(!check[x][j]||vis[j]) continue;
        vis[j] = 1;
        if(!match[j]||Dfs(match[j])){
            match[j] = x;
            match[x] = j;
            return true;
        }
    }
    return false;
}
int main()
{    
    freopen("P3386.in","r",stdin);
    freopen("P3386.out","w",stdout);
    n = fd(),m = fd(),e = fd();
    for(re int i=1;i<=e;++i){
        int x = fd(),y = fd();
        if(x > n||y > m) continue;
        y += n;
        check[x][y] = 1;
    }
    memset(match,0,sizeof(match));
    for(re int i=1;i<=n;++i){
        memset(vis,0,sizeof(vis));
        if(match[i]) continue;
        if(Dfs(i)) ++ans;
    }
    printf("%d",ans);
    return 0;
}

二分图最佳匹配(最小费用最大流)

#include<iostream>
#include<cstring>
#include<cstdio>
#include<queue>
using namespace std;
#define e exit(0)
#define re register
#define inf 2147483647
#define LL int
const LL N = 1001;
const LL M = 30005;
LL n,m,cnt=1,s,t,sum,ans,head[N],dis[N],vis[N],pre[N];
struct bian{LL from,to,v,flow,next;}len[M<<1];
inline LL fd(){
    LL s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
void add(LL from,LL to,LL flow,LL v){
    len[++cnt].v = v;
    len[cnt].to = to;
    len[cnt].flow = flow;
    len[cnt].from = from;
    len[cnt].next = head[from];
    head[from] = cnt;
}
bool spfa(){
    for(re LL i=0;i<=n;++i)
        pre[i] = -1,dis[i] = inf,vis[i] = 0;
    vis[1] = 1,dis[1] = 0;
    queue<LL> Q;
    Q.push(1);
    while(Q.size()){
        LL now = Q.front();
        Q.pop();
        vis[now] = 0;
        for(re LL k=head[now];k;k=len[k].next){
            if(len[k].flow <= 0) continue;
            LL to = len[k].to,v = len[k].v;
            if(dis[to] > dis[now]+v){
                dis[to] = dis[now]+v;
                pre[to] = k;
                if(!vis[to]){
                    vis[to] = 1;
                    Q.push(to);
                }
            }
        }
    }
    if(dis[n] == inf)
        return false;
    return true;
}
void costflow(){
    while(spfa()){
        LL Min = inf;
        for(re LL k=pre[n];k!=-1;k = pre[len[k].from])
            Min = min(Min,len[k].flow);
        sum += Min;
        for(re LL k=pre[n];k!=-1;k = pre[len[k].from]){
            ans += Min*len[k].v;
            len[k].flow -= Min;
            len[k^1].flow += Min;
        }
    }
    printf("%lld %lld",sum,ans);
}
int main()
{
    freopen("LOJ102.in","r",stdin);
    freopen("LOJ102.out","w",stdout);
    n = fd(),m = fd();
    for(re LL i=1;i<=m;++i){
        LL x = fd(),y = fd(),flow = fd(),v = fd();
        add(x,y,flow,v),add(y,x,0,-v);
    }
    costflow();
    return 0;
}

tarjan.缩点

#include<iostream>
#include<cstring>
#include<cstdio>
#include<queue>
#include<cmath>
using namespace std;
#define re register
const int N = 1e4+5;
const int M = 1e5+5;
int n,m,c,cnt,lx,deep,ans,top,head[N],col[N],H[N],in[N];
int Q[N],dfn[N],low[N],vis[N],val[N],sum[N],f[N];
struct bian{int from,to,next;}len[M];
struct Eg{int to,next,v;}e[M];
inline int fd(){
    int s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
void add(int from,int to){
    len[++cnt].to = to;
    len[cnt].from = from;
    len[cnt].next = head[from];
    head[from] = cnt;
}
void con(int from,int to,int v){
    e[++lx].v = v;
    e[lx].to = to;
    e[lx].next = H[from];
    H[from] = lx;
}
void tarjan(int x){
    low[x] = dfn[x] = ++deep;
    Q[++top] = x;
    vis[x] = 1;
    for(re int k=head[x];k;k=len[k].next){
        int to = len[k].to;
        if(!dfn[to]){
            tarjan(to);
            low[x] = min(low[x],low[to]);
        }
        else if(vis[to])
            low[x] = min(low[x],dfn[to]);//防止连到兄弟边.
    }
    if(low[x] == dfn[x]){
        col[x] = ++c;
        sum[c] += val[x];
        vis[x] = 0;
        while(top&&Q[top] != x){
            vis[Q[top]] = 0;
            sum[c] += val[Q[top]];
            col[Q[top--]] = c;
        }
        --top;
    }
}
int main()
{
    freopen("P3387.in","r",stdin);
    freopen("P3387.out","w",stdout);
    n = fd(),m = fd();
    for(re int i=1;i<=n;++i)
        val[i] = fd();
    for(re int i=1;i<=m;++i){
        int x = fd(),y = fd();
        add(x,y);
    }
    for(re int i=1;i<=n;++i)
        if(!dfn[i]) tarjan(i);
    for(re int i=1;i<=cnt;++i){
        int x = len[i].from,y = len[i].to;
        if(col[x] == col[y]) continue;
        int v = sum[col[y]];
        con(col[x],col[y],v);
        ++in[col[y]];
    }
    queue<int> Q;
    for(re int i=1;i<=c;++i){
        if(!in[i]) Q.push(i);
        f[i] = sum[i];
    }
    while(Q.size()){
        int now = Q.front();Q.pop();
        for(re int k=H[now];k;k=e[k].next){
            int to = e[k].to,v = e[k].v;
            f[to] = max(f[to],f[now]+v);
            --in[to];
            if(in[to] == 0) Q.push(to);
        }
    }
    for(re int i=1;i<=c;++i)
        ans = max(ans,f[i]);
    printf("%d",ans);
    return 0;
}

tarjan.缩点

#include<iostream>
#include<cstring>
#include<cstdio>
using namespace std;
#define e exit(0)
#define re register
const int N = 20005;
const int M = 100005;
int n,m,cnt,deep,ans,head[N],cut[N],dfn[N],low[N];
struct bian{int to,next;}len[M<<1];
inline int fd(){
    int s=1,t=0;char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
void add(int from,int to){
    len[++cnt].to = to;
    len[cnt].next = head[from];
    head[from] = cnt;
}
void tarjan(int x,int rt,int fa){
    int lx = 0;
    dfn[x] = low[x] = ++deep;
    for(re int k=head[x];k;k=len[k].next){
        int to = len[k].to;
        if(to == fa) continue;
        if(!dfn[to]){ 
            if(x == rt) ++lx;
            tarjan(to,rt,x);
            low[x] = min(low[x],low[to]);
            if(low[to] >= dfn[x]&&x != rt)
                cut[x] = 1;
        }
        else low[x] = min(low[x],dfn[to]);
    }
    if(x == rt&&lx >= 2)
        cut[x] = 1;
}
int main()
{
    freopen("P3388.in","r",stdin);
    freopen("P3388.out","w",stdout);
    n = fd(),m = fd();
    for(re int i=1;i<=m;++i){
        int x = fd(),y = fd();
        add(x,y),add(y,x);
    }
    for(re int i=1;i<=n;++i)
        if(!dfn[i]) tarjan(i,i,i);
    for(re int i=1;i<=n;++i)
        if(cut[i] == 1) ++ans;
    printf("%d\n",ans);
    for(re int i=1;i<=n;++i)
        if(cut[i] == 1) printf("%d ",i);
    return 0;
}

tarjan.桥

#include<iostream>
#include<cstring>
#include<cstdio>
using namespace std;
#define e exit(0)
#define re register
const int N = 5e5+5;
const int M = 5e5+5;
int n,m,cnt,tot,deep,head[N],dfn[N],low[N],E[N][2];
struct bian{int to,next;}len[M];
inline int fd(){
    int s=1,t=0;
    char c=getchar();
    while(c<'0'||c>'9'){if(c=='-')s=-1;c=getchar();}
    while(c>='0'&&c<='9'){t=t*10+c-'0';c=getchar();}
    return s*t;
}
void add(int from,int to){
    len[++cnt].to = to;
    len[cnt].next = head[from];
    head[from] = cnt;
}
void tarjan(int x,int fa){
    low[x] = dfn[x] = ++deep;
    for(re int k=head[x];k;k=len[k].next){
        int to = len[k].to;
        if(to == fa) continue;
        if(!dfn[to]){
            tarjan(to,x);
            low[x] = min(low[x],low[to]);
            if(low[to] > dfn[x])
                E[++tot][0] = x,E[tot][1] = to;
        }
        else low[x] = min(low[x],dfn[to]);
    }
}
int main()
{
    freopen("kkk.in","r",stdin);
    freopen("kkk.out","w",stdout);
    n = fd(),m = fd();
    for(re int i=1;i<=m;++i){
        int x = fd(),y = fd();
        add(x,y),add(y,x);
    }
    for(re int i=1;i<=n;++i)
        if(dfn[i]) tarjan(i,i);
    return 0;
}