[POJ1753]Flip Game（异或方程组，高斯消元，枚举自由变量）

题目链接：http://poj.org/problem?id=1753

题意：同上。

这回翻来翻去要考虑自由变元了，假设返回了自由变元数量，则需要枚举自由变元。

/*
━━━━━┒ギリギリ♂ eye！
┓┏┓┏┓┃キリキリ♂ mind！
┛┗┛┗┛┃＼○／
┓┏┓┏┓┃ /
┛┗┛┗┛┃ノ)
┓┏┓┏┓┃
┛┗┛┗┛┃
┓┏┓┏┓┃
┛┗┛┗┛┃
┓┏┓┏┓┃
┛┗┛┗┛┃
┓┏┓┏┓┃
┃┃┃┃┃┃
┻┻┻┻┻┻
*/
#include <algorithm>
#include <iostream>
#include <iomanip>
#include <cstring>
#include <climits>
#include <complex>
#include <cassert>
#include <cstdio>
#include <bitset>
#include <vector>
#include <deque>
#include <queue>
#include <stack>
#include <ctime>
#include <set>
#include <map>
#include <cmath>
using namespace std;

const int maxn = 230;
int equ, var;
int a[maxn][maxn];
int x[maxn];
int free_x[maxn];
int free_num;
int ret1, ret2;

int gauss() {
    int max_r, col, k;
    free_num = 0;
    for(k = 0, col = 0; k < equ && col < var; k++, col++) {
        max_r = k;
        for(int i = k + 1; i < equ; i++) {
            if(abs(a[i][col]) > abs(a[max_r][col]))
                max_r = i;
        }
        if(a[max_r][col] == 0) {
            k--;
            free_x[free_num++] = col;
            continue;
        }
        if(max_r != k) {
            for(int j = col; j < var + 1; j++)
                swap(a[k][j], a[max_r][j]);
        }
        for(int i = k + 1; i < equ; i++) {
            if(a[i][col] != 0) {
                for(int j = col; j < var + 1; j++) {
                    a[i][j] ^= a[k][j];
                }
            }
        }
    }
    for(int i = k; i < equ; i++) {
        if(a[i][col] != 0)
            return -1;
    }
    if(k < var) return var - k;
    for(int i = var - 1; i >= 0; i--) {
        x[i] = a[i][var];
        for(int j = i + 1; j < var; j++) {
            x[i] ^= (a[i][j] & x[j]);
        }
    }
    return 0;
}

char G[maxn][maxn];
int n;

int main() {
    // freopen("in", "r", stdin);
    n = 4;
    var = equ = 16;
    ret1 = ret2 = 0;
    memset(a, 0, sizeof(a));
    for(int i = 0; i < n; i++) scanf("%s", G[i]);
    for(int i = 0; i < n; i++) {
        for(int j = 0; j < n; j++) {
            if(G[i][j] == 'b') a[i*n+j][var] = 1;
            else a[i*n+j][var] = 0;
        }
    }
    for(int i = 0; i < n; i++) {
        for(int j = 0; j < n; j++) {
            int q = i * n + j;
            a[q][q] = 1;
            if(i > 0) a[(i-1)*n+j][q] = 1;
            if(i < n - 1) a[(i+1)*n+j][q] = 1;
            if(j > 0) a[i*n+j-1][q] = 1;
            if(j < n - 1) a[i*n+j+1][q] = 1;
        }
    }
    int v1 = gauss();
    if(v1 == -1) ret1 = -1;
    else if(v1 != 0) {
        ret1 = maxn;
        int tot = 1 << v1;
        for(int i = 0; i < tot; i++) {
            int cnt = 0;
            for(int j = 0; j < v1; j++) {
                if(i&(1<<j)) {
                    x[free_x[j]] = 1;
                    cnt++;
                }
                else x[free_x[j]] = 0;
            }
            for(int j = var - v1 - 1; j >= 0; j--) {
                int idx;
                for(idx = j; idx < var; idx++) {
                    if(a[j][idx]) break;
                }
                x[idx] = a[j][var];
                for(int l = idx + 1; l < var; l++) {
                    if(a[j][l]) x[idx] ^= x[l];
                }
                cnt += x[idx];
            }
            ret1 = min(ret1, cnt);
        }
    }
    else for(int i = 0; i < var; i++) ret1 += x[i];

    memset(a, 0, sizeof(a));
    for(int i = 0; i < n; i++) {
        for(int j = 0; j < n; j++) {
            if(G[i][j] == 'w') a[i*n+j][var] = 1;
            else a[i*n+j][var] = 0;
        }
    }
    for(int i = 0; i < n; i++) {
        for(int j = 0; j < n; j++) {
            int q = i * n + j;
            a[q][q] = 1;
            if(i > 0) a[(i-1)*n+j][q] = 1;
            if(i < n - 1) a[(i+1)*n+j][q] = 1;
            if(j > 0) a[i*n+j-1][q] = 1;
            if(j < n - 1) a[i*n+j+1][q] = 1;
        }
    }

    int v2 = gauss();
    if(v2 == -1) ret2 = -1;
    else if(v2 != 0) {
        ret2 = maxn;
        int tot = 1 << v2;
        for(int i = 0; i < tot; i++) {
            int cnt = 0;
            for(int j = 0; j < v2; j++) {
                if(i&(1<<j)) {
                    x[free_x[j]] = 1;
                    cnt++;
                }
                else x[free_x[j]] = 0;
            }
            for(int j = var - v2 - 1; j >= 0; j--) {
                int idx;
                for(idx = j; idx < var; idx++) {
                    if(a[j][idx]) break;
                }
                x[idx] = a[j][var];
                for(int l = idx + 1; l < var; l++) {
                    if(a[j][l]) x[idx] ^= x[l];
                }
                cnt += x[idx];
            }
            ret2 = min(ret2, cnt);
        }
    }
    else for(int i = 0; i < var; i++) ret2 += x[i];
    if(ret1==-1&&ret2==-1) puts("Impossible");
    else if(v1==-1) printf("%d
", ret2);
    else if(ret2==-1) printf("%d
", ret1);
    else printf("%d
", min(ret1, ret2));
    return 0;
}