luogu P2962 [USACO09NOV]灯Lights 高斯消元

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目录

• [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');
}