BZOJ 2965 保护古迹 平面图转对偶图+最小割

标签：bzoj   平面图转对偶图   计算几何   网络流   最小割   

题目大意：给出一个平面图，这个平面图中分布着一些点，可以用平面图中的边将一些点围住，问围住k个点的最小花费是多少。

思路：这题重点是平面图转对偶图。做法不难理解。先将所有的边拆成两条，枚举所有的边，若这个边没有被标记过，那么就对这条边进行搜索，弄出来以这个边为一边的平面区域，可以顺时针或者逆时针。将所有边挂在这条边的起点上，在所有点上按照每条边的极角排序，每次找的时候找大于(或小于)当前边的反向边的第一条边作为搜索的下一条边。直到回到最开始的点。找边的过程中记录面积，判断面积的正负来判断这个平面区域是有限区域还是无限区域，顺便记录所有的点在那个平面区域内。

第二部分是最小割部分。注意到由于点只有10个，我们可以O(2^n)枚举那些点被保护，源->这些被保护的点，f:INF保证不被隔断。平面图中的边所连接的两个平面区域之间连边，边权是平面图上的边的边权。跑最小割然后不断更新答案即可。

CODE：

#define _CRT_SECURE_NO_WARNINGS

#include <queue>
#include <cmath>
#include <vector>
#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
#define MAX 1100
#define MAXE 50100
#define S 0
#define T (MAX - 1)
#define INF 0x3f3f3f3f
using namespace std;

struct MaxFlow{
	int head[MAX],total;
	int next[MAXE],aim[MAXE],flow[MAXE];

	int deep[MAX];

	void Reset() {
		total = 1;
		memset(head,0,sizeof(head));
	}
	void Add(int x,int y,int f) {
		next[++total] = head[x];
		aim[total] = y;
		flow[total] = f;
		head[x] = total;
	}
	void Insert(int x,int y,int f) {
		Add(x,y,f);
		Add(y,x,f);
	}
	bool BFS() {
		static queue<int> q;
		while(!q.empty())	q.pop();
		memset(deep,0,sizeof(deep));
		deep[S] = 1;
		q.push(S);
		while(!q.empty()) {
			int x = q.front(); q.pop();
			for(int i = head[x]; i; i = next[i])
				if(flow[i] && !deep[aim[i]]) {
					deep[aim[i]] = deep[x] + 1;
					q.push(aim[i]);
					if(aim[i] == T)	return true;
				}
		}
		return false;
	}
	int Dinic(int x,int f) {
		if(x == T)	return f;
		int temp = f;
		for(int i = head[x]; i; i = next[i])
			if(flow[i] && temp && deep[aim[i]] == deep[x] + 1) {
				int away = Dinic(aim[i],min(temp,flow[i]));
				if(!away)	deep[aim[i]] = 0;
				flow[i] -= away;
				flow[i^1] += away;
				temp -= away;
			}
		return f - temp;
	}
}solver;

struct Line;

struct Point{
	int x,y;

	Point(int _,int __):x(_),y(__) {}
	Point() {}
	Point operator +(const Point &a)const {
		return Point(x + a.x,y + a.y);
	}
	Point operator -(const Point &a)const {
		return Point(x - a.x,y - a.y);
	}
	void Read() {
		scanf("%d%d",&x,&y);
	}
}point[MAX],arch[MAX];
vector<Line *> e[MAX];

inline long long Cross(const Point &p1,const Point &p2)
{
	return (long long)p1.x * p2.y - (long long)p1.y * p2.x;
}

struct Line{
	Point p,v;
	double alpha;
	int flag,x,y,len;

	Line *another;

	Line(Point _,Point __,int ___,int ____,int _____):p(_),v(__),x(___),y(____),len(_____) {
		alpha = atan2(v.y,v.x);
		flag = 0;
	}
	Line() {}
}line[MAX * MAX];

bool out[MAX];

struct Edge{
	int x,y,len;

	Edge(int _,int __,int ___):x(_),y(__),len(___) {
		if(out[y])	y = T;
		if(out[x])	x = T;
	}
	Edge() {}
}edge[MAX * MAX];

inline bool cmp(Line *l1,Line *l2)
{
	return l1->alpha < l2->alpha;
}

inline bool OnLeft(Line *l,const Point &p)
{
	return Cross(l->v,p - l->p) > 0;
}

int archs,points,lines,edges;
int belong[MAX];

inline void Find(Line *l,int p)
{
	static bool v[MAX];
	memset(v,true,sizeof(v));
	int start = l->x,now = l->y;
	long long area = Cross(point[l->x],point[l->y]);
	for(int i = 1; i <= archs; ++i)
		if(v[i])
			v[i] = !OnLeft(l,arch[i]);
	l->flag = p;
	do {
		vector<Line *>::iterator it = upper_bound(e[now].begin(),e[now].end(),l->another,cmp);
		if(it == e[now].end())	it = e[now].begin();
		l = *it;
		now = l->y;
		l->flag = p;
		area += Cross(point[l->x],point[l->y]);
		for(int i = 1; i <= archs; ++i)
			if(v[i])
				v[i] = !OnLeft(l,arch[i]);
	}while(now != start);
	for(int i = 1; i <= archs; ++i)
		if(v[i])
			belong[i] = p;
	if(area > 0)	out[p] = true;
}

inline Line MakeLine(int x,int y,int len)
{
	Line re(point[x],point[y] - point[x],x,y,len);
	return re;
}

inline int MakeGraph(int status)
{
	solver.Reset();
	int re = 0;
	for(int i = 1; i <= archs; ++i)
		if((status >> (i - 1))&1) {
			solver.Insert(S,belong[i],INF);
			++re;
		}
	for(int i = 1; i <= edges; ++i)
		solver.Insert(edge[i].x,edge[i].y,edge[i].len);
	return re;
}

int ans[MAX];

int main()
{
	cin >> archs >> points >> lines;
	for(int i = 1; i <= archs; ++i)
		arch[i].Read();
	for(int i = 1; i <= points; ++i)
		point[i].Read();
	for(int x,y,z,i = 1; i <= lines; ++i) {
		scanf("%d%d%d",&x,&y,&z);
		line[i << 1] = MakeLine(x,y,z);
		line[i << 1|1] = MakeLine(y,x,z);
		line[i << 1].another = &line[i << 1|1];
		line[i << 1|1].another = &line[i << 1];
		e[x].push_back(&line[i << 1]);
		e[y].push_back(&line[i << 1|1]);
	}
	for(int i = 1; i <= points; ++i)
		sort(e[i].begin(),e[i].end(),cmp);
	int blocks = 0;
	for(int i = 2; i <= (lines << 1|1); ++i)
		if(!line[i].flag)
			Find(&line[i],++blocks);
	for(int i = 2; i <= (lines << 1|1); i += 2)
		edge[++edges] = Edge(line[i].flag,line[i^1].flag,line[i].len);
	memset(ans,0x3f,sizeof(ans));
	for(int i = 1; i < (1 << archs); ++i) {
		int cnt = MakeGraph(i),max_flow = 0;
		while(solver.BFS())
			max_flow += solver.Dinic(S,INF);
		ans[cnt] = min(ans[cnt],max_flow);
	}
	for(int i = 1; i <= archs; ++i)
		printf("%d\n",ans[i]);
	return 0;
}

BZOJ 2965 保护古迹 平面图转对偶图+最小割

标签：bzoj   平面图转对偶图   计算几何   网络流   最小割   

原文地址：http://blog.csdn.net/jiangyuze831/article/details/43201335