【bzoj4390】[Usaco2015 dec]Max Flow LCA

题目描述

Farmer John has installed a new system of N−1 pipes to transport milk between the N stalls in his barn (2≤N≤50,000), conveniently numbered 1…N. Each pipe connects a pair of stalls, and all stalls are connected to each-other via paths of pipes.
FJ is pumping milk between KK pairs of stalls (1≤K≤100,000). For the iith such pair, you are told two stalls sisi and titi, endpoints of a path along which milk is being pumped at a unit rate. FJ is concerned that some stalls might end up overwhelmed with all the milk being pumped through them, since a stall can serve as a waypoint along many of the KK paths along which milk is being pumped. Please help him determine the maximum amount of milk being pumped through any stall. If milk is being pumped along a path from sisi to titi, then it counts as being pumped through the endpoint stalls sisi and titi, as well as through every stall along the path between them.

给定一棵有N个点的树，所有节点的权值都为0。

有K次操作，每次指定两个点s,t，将s到t路径上所有点的权值都加一。

请输出K次操作完毕后权值最大的那个点的权值。

输入

The first line of the input contains NN and KK.
The next N−1 lines each contain two integers x and y (x≠y，x≠y) describing a pipe between stalls x and y.
The next K lines each contain two integers ss and t describing the endpoint stalls of a path through which milk is being pumped.

输出

An integer specifying the maximum amount of milk pumped through any stall in the barn.

样例输入

5 10
3 4
1 5
4 2
5 4
5 4
5 4
3 5
4 3
4 3
1 3
3 5
5 4
1 5
3 4

样例输出

9

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题解

LCA

在x到y的路径上加1，差分一下就是：在x和y上加1，在lca和fa[lca]上减1。

于是使用看得过去一点的求LCA方法求出LCA（代码写了树上倍增），打上差分标记，然后自下而上统计一遍答案即可。

#include <cstdio>
#include <cstring>
#include <algorithm>
#define N 50010
using namespace std;
int head[N] , to[N << 1] , next[N << 1] , cnt , fa[N][20] , deep[N] , log[N] , sum[N];
void add(int x , int y)
{
	to[++cnt] = y , next[cnt] = head[x] , head[x] = cnt;
}
void dfs(int x)
{
	int i;
	for(i = 1 ; (1 << i) <= deep[x] ; i ++ ) fa[x][i] = fa[fa[x][i - 1]][i - 1];
	for(i = head[x] ; i ; i = next[i])
		if(to[i] != fa[x][0])
			fa[to[i]][0] = x , deep[to[i]] = deep[x] + 1 , dfs(to[i]);
}
int lca(int x , int y)
{
	int i;
	if(deep[x] < deep[y]) swap(x , y);
	for(i = log[deep[x] - deep[y]] ; ~i ; i -- )
		if((1 << i) <= deep[x] - deep[y])
			x = fa[x][i];
	if(x == y) return x;
	for(i = log[deep[x]] ; ~i ; i -- )
		if((1 << i) <= deep[x] && fa[x][i] != fa[y][i])
			x = fa[x][i] , y = fa[y][i];
	return fa[x][0];
}
int solve(int x)
{
	int i , ans = 0;
	for(i = head[x] ; i ; i = next[i])
		if(to[i] != fa[x][0])
			ans = max(ans , solve(to[i])) , sum[x] += sum[to[i]];
	return max(ans , sum[x]);
}
int main()
{
	int n , m , i , x , y , z;
	scanf("%d%d" , &n , &m);
	for(i = 2 ; i <= n ; i ++ ) scanf("%d%d" , &x , &y) , add(x , y) , add(y , x) , log[i] = log[i >> 1] + 1;
	dfs(1);
	for(i = 1 ; i <= m ; i ++ ) scanf("%d%d" , &x , &y) , z = lca(x , y) , sum[x] ++ , sum[y] ++ , sum[z] -- , sum[fa[z][0]] -- ;
	printf("%d\n" , solve(1));
	return 0;
}