Tree-点分治

原创于 2019-08-17 15:58:37 发布 · 185 阅读

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本文介绍了一种求解树上两点间距离小于特定值k的组数的算法。利用树的重心分解和点分治策略，通过深度优先搜索确定树的重心，再递归求解子树，最后统计满足条件的点对数量。此算法适用于竞赛编程中的图论问题。

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题意：
求树上两点之间距离小于 k 有多少组。

解决必备条件：
树的重心求解
https://blog.youkuaiyun.com/flymoyu/article/details/99692889
点分治
https://blog.youkuaiyun.com/flymoyu/article/details/99693871
代码：


#include <stdio.h>
#include <string.h>
#include <iostream>
#include <algorithm>
#include <vector>
#include <queue>
#include <set>
#include <map>
#include <string>
#include <math.h>
#include <stdlib.h>
#include <time.h>
using namespace std;
const int MAXN = 10010;
const int INF = 0x3f3f3f3f;
struct Edge
{
    int to,next,w;
}edge[MAXN*2];
int head[MAXN],tot;
void init()
{
    tot = 0;
    memset(head,-1,sizeof(head));
}
void addedge(int u,int v,int w)
{
    edge[tot].to = v; edge[tot].w = w;
    edge[tot].next = head[u];head[u] = tot++;
}
bool vis[MAXN];
int size[MAXN],dep[MAXN];
int le,ri;
int dfssize(int u,int pre)
{
    size[u] = 1;
    for(int i = head[u];i != -1;i = edge[i].next)
    {
        int v = edge[i].to;
        if(v == pre || vis[v])continue;
        size[u] += dfssize(v,u);
    }
    return size[u];
}
int minn;
//找重心
void getroot(int u,int pre,int totnum,int &root)
{
    int maxx = totnum - size[u];
    for(int i = head[u];i != -1;i = edge[i].next)
    {
        int v = edge[i].to;
        if(v == pre || vis[v])continue;
        getroot(v,u,totnum,root);
        maxx = max(maxx,size[v]);
    }
    if(maxx < minn){minn = maxx; root = u;}
}
void dfsdepth(int u,int pre,int d)
{
    dep[ri++] = d;
    for(int i = head[u];i != -1;i = edge[i].next)
    {
        int v = edge[i].to;
        if(v == pre || vis[v])continue;
        dfsdepth(v,u,d+edge[i].w);
    }
}
int k;
int getdep(int a,int b)
{
    sort(dep+a,dep+b);
    int ret = 0, e = b-1;
    for(int i = a;i < b;i++)
    {
        if(dep[i] > k)break;
        while(e >= a && dep[e] + dep[i] > k)e--;
        ret += e - a + 1;
        if(e > i)ret--;
    }
    return ret>>1;
}
int solve(int u)
{
    int totnum = dfssize(u,-1);
    int ret = 0;
    minn = INF;
    int root;
    getroot(u,-1,totnum,root);
    vis[root] = true;
    for(int i = head[root];i != -1;i = edge[i].next)
    {
        int v = edge[i].to;
        if(vis[v])continue;
        ret += solve(v);
    }
    le = ri = 0;
    for(int i = head[root];i != -1;i = edge[i].next)
    {
        int v = edge[i].to;
        if(vis[v])continue;
        dfsdepth(v,root,edge[i].w);
        ret -= getdep(le,ri);
        le = ri;
    }
    ret += getdep(0,ri);
    for(int i = 0;i < ri;i++)
    {
        if(dep[i] <= k)ret++;
        else break;
    }
    vis[root] = false;
    return ret;
}

int main()
{
    int n;
    int u,v,w;
    while(scanf("%d%d",&n,&k) == 2)
    {
        if(n == 0 && k == 0)break;
        init();
        for(int i = 1;i < n;i++)
        {
            scanf("%d%d%d",&u,&v,&w);
            addedge(u,v,w);
            addedge(v,u,w);
        }
        memset(vis,false,sizeof(vis));
        printf("%d\n",solve(1));
    }
    return 0;
}