最小割树小记

概要

最小割树是解决无向图上任意两点间最小割问题的工具。其核心思想为分治。

现在有一个图 (G=(V,E)) ，可以这样求得它的最小割树：

选取两个点 (u,v) ，求得这两个点之间的最小割。这个最小割将原图分为两部分 (G_s) 和 (G_t) 。任意 (xin G_s) 和 (y in G_y) 之间的最小割 至少 是 (u,v) 之间的最小割。

那么可以对于 (G_s) 和 (G_y) 分别递归。最后两个点之间的最小割就是在这个递归树上路径的最大值。

具体地，对于每一次递归，可以在求得最小割之后在新图上在 (u,v) 之间连一条权值为最小割的边。那么最后两个点之间的最小割就是树上路径上的最大边权。

如果使用 Dinic 求最小割的话，时间复杂度是 (O(n^3m)) 的。但一般认为是 (O(能过)) 。

模板

luogu4897

#include <bits/stdc++.h>
using namespace std;

const int Maxn = 510;
const int Maxm = 1510;
const int INF = 2147483647;
const int MaxLog = 20;
int n, m, Q[ Maxn ], P[ Maxn ], q;
int Start[ Maxn ], Cur[ Maxn ], Next[ Maxm << 1 ], To[ Maxm << 1 ], Flow[ Maxm << 1 ], NowFlow[ Maxm << 1 ], Used;
int Start_[ Maxn ], Next_[ Maxn << 1 ], To_[ Maxn << 1 ], Val_[ Maxn << 1 ], Used_, D[ Maxn ][ MaxLog ][ 2 ], Deep_[ Maxn ];
int Deep[ Maxn ], Dis[ Maxn ], L, R, Queue[ Maxn ];

inline void AddEdge( int x, int y, int z ) {
	++Used;
	Next[ Used ] = Start[ x ];
	To[ Used ] = y;
	Flow[ Used ] = z;
	Start[ x ] = Used;
	return;
}

inline void AddEdge_( int x, int y, int z ) {
	++Used_;
	Next_[ Used_ ] = Start_[ x ];
	To_[ Used_ ] = y;
	Val_[ Used_ ] = z;
	Start_[ x ] = Used_;
	return;
}

bool Bfs( int S, int T ) {
	memset( Deep, 0, sizeof( Deep ) );
	memset( Dis, 0, sizeof( Dis ) );
	Deep[ S ] = 1;
	L = R = 0;
	Queue[ ++R ] = S;
	while( L < R ) {
		int u = Queue[ ++L ];
		for( int t = Start[ u ]; t != -1; t = Next[ t ] ) {
			int v = To[ t ];
			if( Deep[ v ] ) continue;
			if( NowFlow[ t ] <= 0 ) continue;
			Deep[ v ] = Deep[ u ] + 1;
			Queue[ ++R ] = v;
		}
	}
	return Deep[ T ];
}

int Dfs( int u, int Rest, int T ) {
	if( u == T || Rest <= 0 ) return Rest;
	int Ans = 0;
	for( int &t = Cur[ u ]; t != -1; t = Next[ t ] ) {
		int v = To[ t ];
		if( Deep[ v ] != Deep[ u ] + 1 ) continue;
		if( NowFlow[ t ] <= 0 ) continue;
		int d = Dfs( v, min( Rest, NowFlow[ t ] ), T );
		NowFlow[ t ] -= d; NowFlow[ t ^ 1 ] += d;
		Ans += d; Rest -= d;
		if( !Rest ) break;
	}
	return Ans;
}

int Dinic( int S, int T ) {
	int Ans = 0;
	memcpy( NowFlow, Flow, sizeof( Flow ) );
	while( Bfs( S, T ) ) {
		memcpy( Cur, Start, sizeof( Start ) );
		int d = Dfs( S, INF, T );
		while( d ) {
			Ans += d;
			d = Dfs( S, INF, T );
		}
	}
	return Ans;
}

void Build( int Left, int Right ) {
	if( Left >= Right ) return;
	int t = Dinic( Q[ Left ], Q[ Left + 1 ] );
	AddEdge_( Q[ Left ], Q[ Left + 1 ], t );
	Bfs( Q[ Left ], n + 1 );
	t =  Left - 1;
	for( int i = Left; i <= Right; ++i ) 
		if( Deep[ Q[ i ] ] )
			P[ ++t ] = Q[ i ];
	int Cut = t;
	for( int i = Left; i <= Right; ++i ) 
		if( !Deep[ Q[ i ] ] )
			P[ ++t ] = Q[ i ];
	for( int i = Left; i <= Right; ++i ) 
		Q[ i ] = P[ i ];
	Build( Left, Cut );
	Build( Cut + 1, Right );
	return;
}

void Build_( int u, int Fa, int c ) {
	D[ u ][ 0 ][ 0 ] = Fa;
	for( int i = 1; i < MaxLog; ++i ) D[ u ][ i ][ 0 ] = D[ D[ u ][ i - 1 ][ 0 ] ][ i - 1 ][ 0 ];
	D[ u ][ 0 ][ 1 ] = c;
	for( int i = 1; i < MaxLog; ++i ) D[ u ][ i ][ 1 ] = min( D[ u ][ i - 1 ][ 1 ], D[ D[ u ][ i - 1 ][ 0 ] ][ i - 1 ][ 1 ] );
	Deep_[ u ] = Deep_[ Fa ] + 1;
	for( int t = Start_[ u ]; t; t = Next_[ t ] ) {
		int v = To_[ t ];
		if( v == Fa ) continue;
		Build_( v, u, Val_[ t ] );
	}
	return;
}

int Query( int x, int y ) {
	int Ans = INF;
	if( Deep_[ x ] < Deep_[ y ] ) swap( x, y );
	for( int i = MaxLog - 1; i >= 0; --i )
		if( Deep_[ D[ x ][ i ][ 0 ] ] >= Deep_[ y ] ) {
			Ans = min( Ans, D[ x ][ i ][ 1 ] );
			x = D[ x ][ i ][ 0 ];
		}
	if( x == y ) return Ans;
	for( int i = MaxLog - 1; i >= 0; --i ) 
		if( D[ x ][ i ][ 0 ] != D[ y ][ i ][ 0 ] ) {
			Ans = min( Ans, D[ x ][ i ][ 1 ] );
			Ans = min( Ans, D[ y ][ i ][ 1 ] );
			x = D[ x ][ i ][ 0 ];
			y = D[ y ][ i ][ 0 ];
		}
	Ans = min( Ans, D[ x ][ 0 ][ 1 ] );
	Ans = min( Ans, D[ y ][ 0 ][ 1 ] );
	return Ans;
}

int main() {
	Used = -1;
	memset( Start, 255, sizeof( Start ) );
	scanf( "%d%d", &n, &m );
	for( int i = 1; i <= m; ++i ) {
		int x, y, z;
		scanf( "%d%d%d", &x, &y, &z );
		AddEdge( x, y, z );
		AddEdge( y, x, z );
	}
	for( int i = 0; i <= n; ++i ) Q[ i ] = i;
	Build( 0, n );
	Build_( 0, 0, INF );
	scanf( "%d", &q );
	for( int i = 1; i <= q; ++i ) {
		int x, y;
		scanf( "%d%d", &x, &y );
		printf( "%d
", Query( x, y ) );
	}
	return 0;
}