HDU2586 How far away? —— 倍增LCA

题目链接：http://acm.hdu.edu.cn/showproblem.php?pid=2586

How far away ？

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 16425    Accepted Submission(s): 6252

Problem Description

There are n houses in the village and some bidirectional roads connecting them. Every day peole always like to ask like this "How far is it if I want to go from house A to house B"? Usually it hard to answer. But luckily int this village the answer is always unique, since the roads are built in the way that there is a unique simple path("simple" means you can't visit a place twice) between every two houses. Yout task is to answer all these curious people.

 

Input

First line is a single integer T(T<=10), indicating the number of test cases.
  For each test case,in the first line there are two numbers n(2<=n<=40000) and m (1<=m<=200),the number of houses and the number of queries. The following n-1 lines each consisting three numbers i,j,k, separated bu a single space, meaning that there is a road connecting house i and house j,with length k(0<k<=40000).The houses are labeled from 1 to n.
  Next m lines each has distinct integers i and j, you areato answer the distance between house i and house j.

 

Output

For each test case,output m lines. Each line represents the answer of the query. Output a bland line after each test case.

 

Sample Input

2
3 2
1 2 10
3 1 15
1 2
2 3

2 2
1 2 100
1 2
2 1

 

Sample Output

10
25
100
100

题解：

1.可知这是一棵无根树，那么把它转化为有根树，再用倍增LCA求出每个结点到根节点的距离。

2.两点的距离：dist = dis[u] + dis[v] - 2 * dis[ LCA(u,v) ]。

3.复杂度O（nlogn）。

对倍增LCA的理解：

对于每一个结点，由于在倍增的时候，每个祖先以及每条边只会被扫过一次，不会出现重复，所以可以用倍增LCA求距离。

代码如下：

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <vector>
#include <cmath>
#include <queue>
#include <stack>
#include <map>
#include <string>
#include <set>
#define ms(a,b) memset((a),(b),sizeof((a)))
using namespace std;
typedef long long LL;
const int INF = 2e9;
const LL LNF = 9e18;
const int mod = 1e9+7;
const int maxn = 4e4+10;
const int DEG = 20;

int n, m;

struct edge
{
    int to, w, next;
}edge[maxn*2];
int head[maxn], tot;
int fa[maxn][DEG], deg[maxn], dis[maxn];

void add(int u, int v, int w)
{
    edge[tot].to = v;
    edge[tot].w = w;
    edge[tot].next = head[u];
    head[u] = tot++;
}

void bfs(int root)  //一边建树，一边求出每个节点到根节点的距离，以及深度
{
    queue<int>que;
    deg[root] = 0;
    dis[root] = 0;
    fa[root][0] = root;
    que.push(root);
    while(!que.empty())
    {
        int tmp = que.front();
        que.pop();
        for(int i = 1; i<DEG; i++)
            fa[tmp][i] = fa[fa[tmp][i-1]][i-1];
        for(int i = head[tmp]; i!=-1; i = edge[i].next)
        {
            int v = edge[i].to, w = edge[i].w;
            if(v==fa[tmp][0]) continue;
            deg[v] = deg[tmp]+1;
            dis[v] = dis[tmp]+w;
            fa[v][0] = tmp;
            que.push(v);
        }
    }
}

int LCA(int u, int v)
{
    if(deg[u]>deg[v]) swap(u,v);
    int hu = deg[u], hv = deg[v];
    int tu = u, tv = v;
    for(int det = hv-hu, i = 0; det; det>>=1, i++)
        if(det&1)
            tv = fa[tv][i];
    if(tv==tu) return tu;
    for(int i = DEG-1; i>=0; i--)
    {
        if(fa[tu][i]==fa[tv][i]) continue;
        tu = fa[tu][i];
        tv = fa[tv][i];
    }
    return fa[tu][0];
}

int main()
{
    int T;
    cin>>T;
    while(T--)
    {
        tot = 0;
        ms(head, -1);
        scanf("%d%d",&n,&m);
        for(int i = 1; i<n; i++)
        {
            int u, v, w;
            scanf("%d%d%d",&u,&v,&w);
            add(u,v,w);
            add(v,u,w);
        }

        bfs(1);
        for(int i = 0; i<m; i++)
        {
            int u, v;
            scanf("%d%d",&u,&v);
            printf("%d
", dis[u]+dis[v]-2*dis[LCA(u,v)]);
        }
    }
}