[codevs 1922] 骑士共存问题

[codevs 1922] 骑士共存问题


题解:

二分图最大独立集问题。


二分图的最大独立集:

选出一些点,让两两之间没有边相连。


二分图最大独立集问题一般转化为它的对偶问题——最小覆盖集,因为最大独立集要求每条边所连接的两个点最多有一个被选中,而最小覆盖集要求每条边所连接的两个点最少有一个被选中。那么点的个数-最小覆盖集点的个数=最大独立集(显然减掉的越少剩下的越多)。


那么最小覆盖怎么求呢?答案就是建立二分图求最大基数匹配。可以简单证明一下:

1)充分性:如果还有一条边没有被覆盖,那么此时一定可以把这条边作为一条新的匹配,说明此时还不是最大匹配。

2)必要性:如果可以删去一条边,二分图匹配边中是没有交点的,那么删去后这条边一定不会被覆盖。


算法就到此结束。


原题中记得要把总点数减去不能走的格子数。


代码:

dinic算法过了,ISAP死活不对...应该是我写的问题。

Dinic:

总时间耗费: 3172ms  

总内存耗费: 11 kB

#include<cstdio>
#include<iostream>
#include<vector>
#include<cstring>
#include<string>
#include<queue>
#include<algorithm>
using namespace std;

const int maxn = 200 * 200 + 10;
const int INF = 1e9 + 7;

const int dx[] = {-1, -2, -2, -1, 1, 2, 2, 1};
const int dy[] = {-2, -1, 1, 2, 2, 1, -1, -2};

int n, m, s, t;

struct Edge {
	int from, to, cap, flow;
};

vector<Edge> edges;
vector<int> G[maxn];

void AddEdge(int from, int to, int cap) {
	edges.push_back((Edge){from, to, cap, 0});
	edges.push_back((Edge){to, from, 0, 0});
	int sz = edges.size();
	G[from].push_back(sz-2);
	G[to].push_back(sz-1);
}

int id[222][222];
void init() {
	
	cin >> n >> m;
	s = 0; t = n*n - m + 1;
			
	for(int i = 0; i < m; i++) {
		int x, y;
		cin >> x >> y;
		id[x][y] = -1;
	}
	
	int c = 0;
	for(int i = 1; i <= n; i++)
		for(int j = 1; j <= n; j++)
			if(!id[i][j])	id[i][j] = ++c;
			
	for(int x = 1; x <= n; x++)
		for(int y = 1; y <= n; y++) if(id[x][y] > 0) {
			int ID = id[x][y];
			if((x+y)%2 == 0) {
				AddEdge(s, ID, 1);
				for(int i = 0; i < 8; i++) {
					int newx = x + dx[i], newy = y + dy[i];
					int newID = id[newx][newy];
					if(newx <= n && newy <= n && newx >= 1 && newy >= 1 && newID > 0) 
						AddEdge(ID, newID, INF);
				}
			} else {
				AddEdge(ID, t, 1);
			}
		}
}

bool vis[maxn];
int d[maxn];
bool BFS() {
	queue<int> Q;
	Q.push(s);
	memset(vis, 0, sizeof(vis));
	d[s] = 0;
	vis[s] = 1;
	while(!Q.empty()) {
		int u = Q.front(); Q.pop();
		for(int i = 0; i < G[u].size(); i++) {
			Edge& e = edges[G[u][i]];
				if(!vis[e.to] && e.cap > e.flow) {
					vis[e.to] = 1;
					d[e.to] = d[u] + 1;
					Q.push(e.to);
				}
		}
	}
	return vis[t];
}

int cur[maxn];

int DFS(int u, int a) {
	if(a == 0 || u == t) return a;
	int f, flow = 0;
	for(int& i = cur[u]; i < G[u].size(); i++) {
		Edge& e = edges[G[u][i]];
		if(d[u] + 1 == d[e.to] && (f = DFS(e.to, min(a, e.cap-e.flow))) > 0) {
			flow += f;
			e.flow += f;
			a -= f;
			edges[G[u][i]^1].flow -= f;
			if(a == 0) break;
		}
	}
	return flow;
}

void Dinic() {
	int flow = 0;
	while(BFS()) {
		memset(cur, 0, sizeof(cur));
		flow += DFS(s, INF);
	}
	cout << n*n-m-flow << endl;
}

int main() {
	init();
	Dinic();
	return 0;
}


ISAP:

TLE(哪错了呢?)

#include<cstdio>
#include<iostream>
#include<vector>
#include<queue>
#include<algorithm>
using namespace std;

const int maxn = 200 * 200 + 10;
const int INF = 1e9 + 7;

const int dx[] = {-1, -2, -2, -1, 1, 2, 2, 1};
const int dy[] = {-2, -1, 1, 2, 2, 1, -1, -2};

int n, m, s, t;

struct Edge {
	int from, to, cap, flow;
};

vector<Edge> edges;
vector<int> G[maxn];

void AddEdge(int from, int to, int cap) {
	edges.push_back((Edge){from, to, cap, 0});
	edges.push_back((Edge){to, from, 0, 0});
	int sz = edges.size();
	G[from].push_back(sz-2);
	G[to].push_back(sz-1);
}

int id[222][222];
void init() {
	cin >> n >> m;
	s = 0; t = n*n - m + 1;
			
	for(int i = 0; i < m; i++) {
		int x, y;
		cin >> x >> y;
		id[x][y] = -1;
	}
	
	int c = 0;
	for(int i = 1; i <= n; i++)
		for(int j = 1; j <= n; j++)
			if(!id[i][j])	id[i][j] = ++c;
			
	for(int x = 1; x <= n; x++)
		for(int y = 1; y <= n; y++) if(id[x][y] > 0) {
			int ID = id[x][y];
			if((x+y)%2 == 0) {
				AddEdge(s, ID, 1);
				for(int i = 0; i < 8; i++) {
					int newx = x + dx[i], newy = y + dy[i];
					int newID = id[newx][newy];
					if(newx <= n && newy <= n && newx >= 1 && newy >= 1 && newID > 0) 
						AddEdge(ID, newID, INF);
				}
			} else {
				AddEdge(ID, t, 1);
			}
		}
}

bool vis[maxn];
int d[maxn], p[maxn], cur[maxn], num[maxn];

int Augment() {
	int x = t, a = INF;
	while(x != 1) {
		Edge& e = edges[p[x]];
		a = min(a, e.cap-e.flow);
		x = e.from;
	}
	x = t;
	while(x != 1) {
		edges[p[x]].flow += a;
		edges[p[x]^1].flow -= a;
		x = edges[p[x]].from;
	}
	return a;
}

void BFS() {
	queue<int> Q;
	bool vis[maxn];
	memset(vis, 0, sizeof(vis));
	d[t] = 0;
	Q.push(t);
	while(!Q.empty()) {
		int x = Q.front(); Q.pop();
		for(int i = 0; i < G[x].size(); i++) {
			Edge& e = edges[G[x][i]];
			if(e.cap == 0 && !vis[e.to]) {
				d[e.to] = d[x] + 1;
				vis[e.to] = 1;
				Q.push(e.to);
			}
		}
	}
} 

void Maxflow() {
  int flow = 0;
  BFS();
  memset(num, 0, sizeof(num));
  for(int i = 0; i <= t; i++) num[d[i]]++;
  int x = s;
  memset(cur, 0, sizeof(cur));
  while(d[s] < t) {
  	if(x == t) {
  		flow += Augment();
  		x = s;
  	}
	int ok = 0;
  	for(int i = cur[x]; i < G[x].size(); i++) {
  		Edge& e = edges[G[x][i]];
	  	if(e.cap > e.flow && d[x] == d[e.to] + 1) {
  			ok = 1;
  			p[e.to] = G[x][i];
	  		cur[x] = i;
  			x = e.to;
  			break;
	  	}
	  }
  	if(!ok) {
  		int _m = t - 1;
	  	for(int i = 0; i < G[x].size(); i++) {
  			Edge& e = edges[G[x][i]];
  			if(e.cap > e.flow) _m = min(_m, d[e.to]);
	  	}
  		if(--num[d[x]] == 0) break;
  		num[d[x] = _m+1]++;
	  	cur[x] = 0;
  		if(x != s) x = edges[p[x]].from;
	  }
	}
	cout << n*n-m-flow << endl;
}

int main() {
	init();
	Maxflow();
	return 0;
}
我的ISAP连草地排水都WA一个点。。。


dinic算法过了,ISAP死活不对...应该是我写的问题。
TLE
posted @ 2015-02-01 13:51  wfwbz  阅读(160)  评论(0编辑  收藏  举报