Traffic Real Time Query System 圆方树+LCA

题目描述

City C is really a nightmare of all drivers for its traffic jams. To solve the traffic problem, the mayor plans to build a RTQS (Real Time Query System) to monitor all traffic situations. City C is made up of N crossings and M roads, and each road connects two crossings. All roads are bidirectional. One of the important tasks of RTQS is to answer some queries about route-choice problem. Specifically, the task is to find the crossings which a driver MUST pass when he is driving from one given road to another given road.

Input

There are multiple test cases.
For each test case:
The first line contains two integers N and M, representing the number of the crossings and roads.
The next M lines describe the roads. In those M lines, the i th line (i starts from 1)contains two integers X i and Y i, representing that road i connects crossing X i and Y i (X i≠Y i).
The following line contains a single integer Q, representing the number of RTQs.
Then Q lines follows, each describing a RTQ by two integers S and T(S≠T) meaning that a driver is now driving on the roads and he wants to reach roadt . It will be always at least one way from roads to roadt.
The input ends with a line of “0 0”.
Please note that: 0<N<=10000, 0<M<=100000, 0<Q<=10000, 0<X i,Y i<=N, 0<S,T<=M

Output

For each RTQ prints a line containing a single integer representing the number of crossings which the driver MUST pass.

样例

Sample Input

5 6
1 2
1 3
2 3
3 4
4 5
3 5
2
2 3
2 4
0 0

Sample Output

0
1

分析

题意：一个城市有n个路口，m条无向公路。现在要求从第S条路到第T条路必须经过的点有几个。

这道题和压力那一道题比较像，但是比那一道题要简单一些。

题目中要求从一条路到另一条路的必经点，我们先不去考虑边，我们先只去考虑点

如果是点的话那就很好说了，我们用Tarjan对点双进行缩点，使之成为一棵圆方树

缩点之后的树一定是圆点与方点交替分布的，我们算出两点之间经过的圆点个数就可以了

要求经过的圆点个数，我们要先求两点之间的距离

最后再把这个距离除以2，再向下取整减去1就是我们想要的结果

(ans(xx,yy)=(dep[xx]+dep[yy]-dep[LCA(xx,yy)] imes2)/2-1)

这个结果是怎么来的呢，我们简单地推导一下

很显然的是，原先的的边所连接的两个点在新的圆方树中都是圆点

而且这两个点的的最近公共祖先不是方点就是圆点

我们设这两个点分别为xx、yy

1、如果它们的最近公共祖先是圆点

xx和它的最近公共祖先间圆点的个数=((dep[xx]-dep[LCA(xx,yy)])/2)

yy和它的最近公共祖先间圆点的个数=((dep[yy]-dep[LCA(xx,yy)])/2)

因为它们的最近公共祖先作为圆点被计算了两次，所以两式相加后还要减去1

(ans（xx,yy）=(dep[xx]-dep[LCA(xx,yy)])/2+(dep[yy]-dep[LCA(xx,yy)])/2-1)

化简得到(ans(xx,yy)=(dep[xx]+dep[yy]-dep[LCA(xx,yy)] imes2)/2-1)

2、如果它们的最近公共祖先是方点

xx和它的最近公共祖先间圆点的个数=((dep[xx]-dep[LCA(xx,yy)]-1)/2)

yy和它的最近公共祖先间圆点的个数=((dep[yy]-dep[LCA(xx,yy)]- 1)/2)

两式相加

(ans（xx,yy）=(dep[xx]-dep[LCA(xx,yy)]-1)/2+(dep[yy]-dep[LCA(xx,yy)]-1)/2)

化简得到(ans(xx,yy)=(dep[xx]+dep[yy]-dep[LCA(xx,yy)] imes2)/2-1)

因此最终的结果为(ans(xx,yy)=(dep[xx]+dep[yy]-dep[LCA(xx,yy)] imes2)/2-1)

但是我们要求的是边到边之间经过的圆点的个数，这怎么办呢

其实我们只要把每条边的两个顶点分别枚举求最大值就可以了

代码

#include<cstdio>
#include<iostream>
#include<cstring>
#include<algorithm>
#include<vector>
using namespace std;
const int maxd=40005,maxb=400005;
struct asd{
    int from,to,next;
}b[maxb],b2[maxb];
int tot=1,t2=1,head[maxd],h2[maxd],nn;
void ad(int aa,int bb){
    b[tot].from=aa;
    b[tot].to=bb;
    b[tot].next=head[aa];
    head[aa]=tot++;
}
void ad2(int aa,int bb){
    b2[t2].from=aa;
    b2[t2].to=bb;
    b2[t2].next=h2[aa];
    h2[aa]=t2++;
}
int dfn[maxd],low[maxd],dfnc;
int n,m,sta[maxd],top;
void tarjan(int now,int fa){
    dfn[now]=low[now]=++dfnc;
    sta[++top]=now;
    for(int i=head[now];i!=-1;i=b[i].next){
        int u=b[i].to;
        if(!dfn[u]){
            tarjan(u,now);
            low[now]=min(low[now],low[u]);
            if(dfn[now]<=low[u]){
                nn++;
                ad2(nn,now),ad2(now,nn);
                while(1){
                    int xx=sta[top--];
                    ad2(xx,nn),ad2(nn,xx);
                    if(xx==u) break;
                }
            }
        } else if(u!=fa){
            low[now]=min(low[now],dfn[u]);
        }
    }
}
int f[maxd][22],dep[maxd];
bool vis[maxd];
void dfs(int now,int fa){
    vis[now]=1;
            dep[now]=dep[fa]+1;
            f[now][0]=fa;
    for(int i=1;(1<<i)<=dep[now];i++){
        f[now][i]=f[f[now][i-1]][i-1];
    }
    for(int i=h2[now];i!=-1;i=b2[i].next){
        int u=b2[i].to;
        if(u!=fa && !vis[u]){
            dfs(u,now);
        }
    }
}
int LCA(int xx,int yy){
    if(dep[xx]>dep[yy]) swap(xx,yy);
    int len=dep[yy]-dep[xx],k=0;
    while(len){
        if(len&1){
            yy=f[yy][k];
        }
        k++,len>>=1;
    }
    if(xx==yy) return xx;
    for(int i=20;i>=0;i--){
        if(f[xx][i]==f[yy][i]) continue;
        xx=f[xx][i],yy=f[yy][i];
    }
    return f[xx][0];
}
int solve(int xx,int yy){
    return (dep[xx]+dep[yy]-2*dep[LCA(xx,yy)])/2-1; 
}
int main(){
    while(scanf("%d%d",&n,&m)!=EOF && (n|m)){
        memset(head,-1,sizeof(head));
        memset(h2,-1,sizeof(h2));
        memset(&b,0,sizeof(struct asd));
        memset(&b2,0,sizeof(struct asd));
        memset(dfn,0,sizeof(dfn));
        memset(low,0,sizeof(low));
        memset(f,0,sizeof(f));
        memset(dep,0,sizeof(dep));
        memset(sta,0,sizeof(sta));
        memset(vis,0,sizeof(vis));
        tot=1,t2=1,dfnc=0,top=0,nn=n;
        for(int i=1;i<=m;i++){
            int aa,bb;
            scanf("%d%d",&aa,&bb);
            ad(aa,bb);
            ad(bb,aa);
        }
        for(int i=1;i<=n;i++){
            if(!dfn[i]) tarjan(i,-1);
        }
        for(int i=1;i<=nn;i++){
            if(!vis[i]) dfs(i,0);
        }
        int q;
        scanf("%d",&q);
        while(q--){
            int aa,bb;
            scanf("%d%d",&aa,&bb);
            int ans1=solve(b[aa*2].from,b[bb*2].from);
            int ans2=solve(b[aa*2].to,b[bb*2].from);
            int ans3=solve(b[aa*2].from,b[bb*2].to);
            int ans4=solve(b[aa*2].to,b[bb*2].to);
            int ans=max(max(ans1,ans2),max(ans3,ans4));
            printf("%d
",ans);
        }
    }
    return 0;
}