luogu P2962 [USACO09NOV]灯Lights 高斯消元

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[TOC]

#题目链接
luogu P2962 [USACO09NOV]灯Lights
#题解
可以折半搜索
map合并
复杂度
2^(n / 2)*logn
高斯消元后得到每个点的翻转状态
爆搜自由元得到最优翻转状态

// luogu-judger-enable-o2
#include<map>
#include<queue> 
#include<cstdio> 
#include<cstring> 
#include<algorithm> 
using namespace std; 
#define gc getchar()
#define pc putchar
inline int read() { 
    int x = 0,f = 1; 
    char c = getchar(); 
    while(c < '0' || c > '9') c = getchar(); 
    while(c <= '9' && c >= '0') x = x * 10 + c - '0',c = getchar(); 
    return x * f; 
} 
void print(int x) { 
    if(x < 0) { 
        putchar('-'); 
        x = -x; 
    } 
    if(x >= 10) print(x / 10); 
    putchar(x % 10 + '0'); 
} 
#define LL long long 
int n,m; 
LL s[200]; 
int cnt,ans = 0x3f3f3f3f; 
std::map<LL,int>b; 
LL ed; 
bool flag; 
void dfs(int x,LL now,int used) { 
    if(x == cnt + 1) { 
        if(now == ed) ans = std::min(used,ans); 
        if(!flag) { 
            int t = b[now]; 
            if(!t || t > used) b[now] = used; 
        } else { 
            int t = b[ed ^ now] ; 
            if(!t) return ; 
            ans = std::min(ans,t + used); 
        } 
        return ; 
    } 	 
    dfs(x + 1,now,used); 
    dfs(x + 1,now ^ s[x],used + 1); 	 
} 
int main() { 
    //freopen("data.cpp","r",stdin); 	
    n = read(), m = read(); 
    for(int i = 1;i <= m;++ i) { 
        int u = read(),v = read(); 
        s[u] |= (1ll << v - 1) ,s[v] |= (1ll << u - 1);   
    } 
    ed = (1ll << n) - 1; 
    //print(ed); pc('\n'); 
    for(int i = 1;i <= n;++ i) s[i] |= (1ll << i - 1); 
    cnt = n / 2; 
    dfs(1,0,0); 
    flag = 1; cnt = n; 
    dfs(n / 2 + 1,0,0); 
    print(ans); 
    pc('\n'); 
}

#include<map>
#include<queue> 
#include<cstdio> 
#include<cstring> 
#include<algorithm> 
using namespace std; 
#define rep(a,b,c) for(int a = b; a <= c;++ a) 
#define per(a,b,c) for(int a = b; a >= c; -- a) 
#define gc getchar()
#define pc putchar
inline int read() { 
	int x = 0,f = 1; 
	char c = getchar(); 
	while(c < '0' || c > '9') c = getchar(); 
	while(c <= '9' && c >= '0') x = x * 10 + c - '0',c = getchar(); 
	return x * f; 
} 
void print(int x) { 
	if(x < 0) { 
		putchar('-'); 
		x = -x; 
	} 
	if(x >= 10) print(x / 10); 
	putchar(x % 10 + '0'); 
} 
#define LL long long 
int n,m; 
const int maxn = 37; 
int f[maxn][maxn]; 
bool flag; 
void guass() { 
	rep(i,1,n) { 
		int j = i; 
		while(!f[j][i] && j <= n) ++ j; 
		if(j == n + 1) continue; 
		if(i != j) swap(f[i],f[j]); 
		rep(j,1,n) 
			if(j != i && f[j][i]) 
				rep(k,1,n + 1) 
					f[j][k] ^= f[i][k]; 
	} 
} 
int zy[maxn],tot = 0,ans = 0x3f3f3f3f; 
void dfs(int now) { 
	if(tot > ans) return; 	
	if(!now) { 
		ans = std::min(ans,tot); 
		return ; 
	} 
	if(f[now][now]) {
		int t = f[now][n + 1]; 
		rep(i,now + 1,n) if(f[now][i]) t ^= zy[i]; 
		zy[now] = t; 
		if(t) tot ++; 
		dfs(now - 1); 
		if(t) tot --; 
	} else { 
		zy[now] = 0; 
		dfs(now - 1); 
		zy[now] = 1; 
		tot ++; 
		dfs(now - 1); 
		tot --; 
	} 
} 
int main() { 
	n = read(),m = read(); 
	rep(i,1,n) f[i][i] = f[i][n + 1] = 1; 
	rep(i,1,m) {
		int x = read(),y = read(); 
		f[x][y] = f[y][x] = 1; 
	} 
	guass(); 
	dfs(n); 
	print(ans); 
	pc('\n'); 
}