Solution -「洛谷 P6158」封锁

\(\mathcal{Description}\)

  Link.

  给定一个 \(n\times n\) 的格点图，横纵相邻的两格点有一条边权为二元组 \((w,e)\) 的边。求对于 \(S=(1,1)\) 和 \(T=(n,n)\) 的一个割，使得 \((\sum w)(\sum c)\) 最小。

  \(n\le400\)。

\(\mathcal{Solution}\)

  套路题，P5540 + P4001。所以我把这两题题解合二为一。

  假设边权都是普通的数字，考虑怎么快速求出这个格点图的最小割。可以发现，这个图一定是一个平面图，那么把它形象化为图形，脑补一下得出一个割肯定是一条完整的曲线，从图的左下方贯穿到图的右上方。所以建立左下方和右上方的超级源汇，每条边相当于连接其左右两个面，最后求源汇之间的最短路即为原图最小割。

  回到本题，对于任意一个割，设其 \(\sum w=w_0\)，\(\sum e=e_0\)，将它体现为一个坐标 \((w_0,e_0)\)，问题就转化为：\(\mathbb R^2\) 的一象限有若干个点，求出其中 \(xy\) 最小的点。

  记 \(A(x_1,y_1)\) 为这些点中 \(x\) 最小的，\(B(x_2,y_2)\) 为 \(y\) 最小的（都能直接求出），考虑在 \(AB\) 的下方取出一个特殊的点 \(C(x_3,y_3)\)，最大化 \(S_{\triangle ABC}\)。推一下式子：

\[\begin{aligned} S_{\triangle ABC}&=\frac{-\vec{AB}\times \vec{AC}}{2}\\ &=-\frac{1}{2}[(x_2-x_1)(y_3-y_1)-(x_3-x_1)(y_2-y_1)]\\ &=-\frac{1}2[(y_1-y_2)x_3+(x_2-x_1)y_3-x_2y_1+x_1y_2] \end{aligned} \]

  把 \((y_1-y_2)w+(x_2-x_1)e\) 作为边 \((w,e)\) 的权，跑最小割即得 \(C\)。用 \(C\) 更新答案最后，分治处理 \((A,C)\) 和 \((C,B)\) 直到 \(C\) 不存在终止，答案就求到啦。

\(\mathcal{Code}\)

/* Clearink */

#include <queue>
#include <cstdio>
#include <assert.h>

typedef long long LL;

inline int rint () {
	int x = 0, f = 1; char s = getchar ();
	for ( ; s < '0' || '9' < s; s = getchar () ) f = s == '-' ? -f : f;
	for ( ; '0' <= s && s <= '9'; s = getchar () ) x = x * 10 + ( s ^ '0' );
	return x * f;
}

inline void chkmin ( LL& a, const LL b ) { b < a && ( a = b, 0 ); }

const int MAXN = 400, INF = 0x3f3f3f3f;
int n, S, T;
LL coeW, coeE, ans = 1ll << 60;

struct Value {
	LL w, e;
	Value ( const LL v = 0 ): w ( v ), e ( v ) {}
	Value ( const LL a, const LL b ): w ( a ), e ( b ) {}
	inline operator LL () const { return w + e; }
	inline LL operator * ( const Value& v ) const { return w * v.e - e * v.w; }
	inline Value operator + ( const Value& v ) const { return { w + v.w, e + v.e }; }
	inline Value operator - ( const Value& v ) const { return { w - v.w, e - v.e }; }
	inline bool operator < ( const Value& v ) const {
		return coeW * w + coeE * e < coeW * v.w + coeE * v.e;
	}
};

typedef std::pair<Value, int> PVI;

struct Graph {
	static const int MAXND = MAXN * MAXN + 2, MAXEG = 4 * MAXN * ( MAXN + 1 );
	int bound, ecnt, head[MAXND + 5], to[MAXEG + 5], nxt[MAXEG + 5];
	Value cst[MAXEG + 5], dist[MAXND + 5];

	inline void operator () ( const int s, const int t, const Value& c ) {
		#ifdef RYBY
			printf ( "%d %d (%lld,%lld)\n", s, t, c.w, c.e );
		#endif
		to[++ecnt] = t, cst[ecnt] = c, nxt[ecnt] = head[s];
		head[s] = ecnt;
		to[++ecnt] = s, cst[ecnt] = c, nxt[ecnt] = head[t];
		head[t] = ecnt;
	}

	inline Value dijkstra ( const int s, const int t ) {
		static bool vis[MAXND + 5];
		static std::priority_queue<PVI, std::vector<PVI>, std::greater<PVI> > heap;
		for ( int i = 1; i <= bound; ++i ) vis[i] = false, dist[i] = INF;
		heap.push ( { dist[s] = 0, s } );
		while ( !heap.empty () ) {
			PVI p ( heap.top () ); heap.pop ();
			if ( vis[p.second] ) continue;
			vis[p.second] = true;
			for ( int i = head[p.second], v; i; i = nxt[i] ) {
				if ( Value d ( p.first + cst[i] ); d < dist[v = to[i]] ) {
					heap.push ( { dist[v] = d, v } );
				}
			}
		}
		return dist[t];
	}
} graph;

inline int id ( const int i, const int j ) {
	if ( !i || j == n ) return S;
	if ( i == n || !j ) return T;
	return ( i - 1 ) * ( n - 1 ) + j;
}

inline Value calc ( const LL a, const LL b ) {
	coeW = a, coeE = b;
	Value ret ( graph.dijkstra ( S, T ) );
	#ifdef RYBY
		printf ( "calc(%lld,%lld) = (%lld,%lld)\n", a, b, ret.w, ret.e );
	#endif
	return graph.dijkstra ( S, T );
}

inline void solve ( const Value& A, const Value& B ) {
	LL a = A.e - B.e, b = B.w - A.w;
	Value C ( calc ( a, b ) );
	chkmin ( ans, C.w * C.e );
	#ifdef RYBY
		printf ( "(%lld,%lld), (%lld,%lld), (%lld,%lld)\n", A.w, A.e, C.w, C.e, B.w, B.e );
		printf ( "%lld...%lld\n", ( B - A ) * ( C - A ), a * C.w + b * C.e + A.w * B.e - A.e * B.w );
	#endif
	if ( ( B - A ) * ( C - A ) >= 0 ) return ;
	solve ( A, C ), solve ( C, B );
}

int main () {
	n = rint ();
	S = ( n - 1 ) * ( n - 1 ) + 1, T = S + 1;
	graph.bound = ( n - 1 ) * ( n - 1 ) + 2;
	for ( int i = 1; i < n; ++i ) {
		for ( int j = 1; j <= n; ++j ) {
			int cw = rint (), ce = rint ();
			graph ( id ( i, j - 1 ), id ( i, j ), { cw, ce } );
		}
	}
	for ( int i = 1; i <= n; ++i ) {
		for ( int j = 1; j < n; ++j ) {
			int rw = rint (), re = rint ();
			graph ( id ( i - 1, j ), id ( i, j ), { rw, re } );
		}
	}
	Value A ( calc ( 1, 0 ) ), B ( calc ( 0, 1 ) );
	chkmin ( ans, A.w * A.e ), chkmin ( ans, B.w * B.e );
	solve ( A, B );
	printf ( "%lld\n", ans );
	return 0;
}

\(\mathcal{Details}\)

  虽然套路但毕竟是黑的，一眼秒掉好开心 owo。