POJ 1222 EXTENDED LIGHTS OUT 高斯消元 异或方程 (水

题目链接：点击打开链接

题意：

给定5*6 的灯的目标状态（开始全暗），每按下一个灯就会把这个灯及这灯相邻的4个灯状态改变。

输出一个按灯的方案使得由全暗变成输入的状态。

思路：

每盏灯都作为一个变量，

a1&x1 + a2&x2 + a3&x3 ··· a30&x30 = input[1][1]

 能影响到某盏灯的系数是1，其他是0

实际上是

x4 + x5 + x3 + x10 = input[4]

但其他不相关的变量也要填充进方程，所以前面系数设置为0。

共5*6个变量，系数矩阵30*30,增广矩阵 30*31.

回代时注意是系数和变量之间的符号是&

#include<cstdio>
#include<cstring>
#include<cmath>
#include<vector>
#include<algorithm>
template <class T>
inline bool rd(T &ret) {
	char c; int sgn;
	if (c = getchar(), c == EOF) return 0;
	while (c != '-' && (c<'0' || c>'9')) c = getchar();
	sgn = (c == '-') ? -1 : 1;
	ret = (c == '-') ? 0 : (c - '0');
	while (c = getchar(), c >= '0'&&c <= '9') ret = ret * 10 + (c - '0');
	ret *= sgn;
	return 1;
}
template <class T>
inline void pt(T x) {
	if (x <0) {
		putchar('-');
		x = -x;
	}
	if (x>9) pt(x / 10);
	putchar(x % 10 + '0');
}
const double eps = 1e-8;
typedef long long ll;
using namespace std;
int step[5][2], id[5][6];
int A[30][31], init[30][31];
int ans[30];
int Guass(int mat[][31], int row, int col){
	int r, c, i, j;
	for (r = c = 0; r < row && c < col; r++, c++){
		for (i = r; i < row; i++)if (mat[i][c]>0)break;
		if (i == row){ r--; continue; }
		if (i != r)
			for (j = c; j <= col; j++)	swap(mat[i][j], mat[r][j]);
		for (i = r + 1; i < row; i++)	
			if(mat[i][c]>0)
				for (j = c; j <= col; j++)	mat[i][j] ^= mat[r][j];
	}
	for (i = r; i < row; i++) if (mat[i][col] > 0)return 0;
	for (i = r - 1; i >= 0; i--)
	{
		ans[i] = mat[i][col];
		for (j = i + 1; j < col; j++)
			ans[i] ^= (mat[i][j] && ans[j]);
	}
	return 1;
}
int main() {
	step[0][0] = 0; step[0][1] = 0;
	step[1][0] = 1; step[1][1] = 0;
	step[2][0] = 0; step[2][1] = 1;
	step[3][0] = -1; step[3][1] = 0;
	step[4][0] = 0; step[4][1] = -1;
	int now = 0;
	for (int i = 0; i < 5; i++)for (int j = 0; j < 6; j++)id[i][j] = now++;
	memset(init, 0, sizeof init);
	for (int i = 0; i < 5; i++)for (int j = 0; j < 6; j++){
		for (int k = 0; k < 5; k++)
		{
			int x = i + step[k][0], y = j + step[k][1];
			if (0 <= x&&x<5 && 0 <= y&&y<6)init[id[i][j]][id[x][y]] = 1;
		}
	}
	int T, Cas = 1; rd(T);
	while (T-->0){
		for (int i = 0; i < 30; i++)for (int j = 0; j < 30; j++)A[i][j] = init[i][j];
		for (int i = 0; i < 5; i++)
		for (int j = 0; j < 6; j++)
			rd(A[id[i][j]][now]);
		Guass(A, 30, 30);
		printf("PUZZLE #%d\n", Cas++);
		for (int i = 0, tmp = 0; i < 5; i++)
		for (int j = 0; j < 6; j++){
			pt(ans[tmp++]);
			if (j == 5)puts(""); else putchar(' ');
		}
	}
	return 0;
}

import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.math.BigInteger;
import java.text.DecimalFormat;
import java.util.ArrayDeque;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collection;
import java.util.Collections;
import java.util.Comparator;
import java.util.Deque;
import java.util.HashMap;
import java.util.Iterator;
import java.util.LinkedList;
import java.util.Map;
import java.util.PriorityQueue;
import java.util.Scanner;
import java.util.Stack;
import java.util.StringTokenizer;
import java.util.TreeMap;
import java.util.TreeSet;
import java.util.Queue;
import java.io.File;
import java.io.FileInputStream;
import java.io.FileNotFoundException;
import java.io.FileOutputStream;

public class Main {
	int[][] step = new int[5][2], id = new int[N][N];
	int[][] A = new int[N][N], init = new int[N][N];
	int[] ans = new int[N];
	int Guass(int[][] mat, int row, int col){
		int r, c, i, j;
		for (r = c = 0; r < row && c < col; r++, c++){
			for (i = r; i < row; i++)if (mat[i][c]>0)break;
			if (i == row){ r--; continue; }
			if (i != r)
				for (j = c; j <= col; j++)
				{int tmp = mat[i][j]; mat[i][j] = mat[r][j]; mat[r][j] = tmp;}
			for (i = r + 1; i < row; i++)	
				if(mat[i][c]>0)
					for (j = c; j <= col; j++)	mat[i][j] ^= mat[r][j];
		}
		for (i = r; i < row; i++) if (mat[i][col] > 0)return 0;
		for (i = r - 1; i >= 0; i--)
		{
			ans[i] = mat[i][col];
			for (j = i + 1; j < col; j++)
				ans[i] ^= (mat[i][j] & ans[j]);
		}
		return 1;
	}
	void work() throws Exception {
		step[0][0] = 0; step[0][1] = 0;
		step[1][0] = 1; step[1][1] = 0;
		step[2][0] = 0; step[2][1] = 1;
		step[3][0] = -1; step[3][1] = 0;
		step[4][0] = 0; step[4][1] = -1;
		int now = 0;
		for(int i = 0; i < 5; i++)for(int j = 0; j < 6; j++)id[i][j] = now++;
		for(int i = 0; i < N; i++)for(int j = 0; j < N; j++)init[i][j] = 0;
		for(int i = 0; i < 5; i++)for(int j = 0; j < 6; j++){
			for(int k = 0; k < 5; k++)
			{
				int x = i+step[k][0], y = j+step[k][1];
				if(0<=x&&x<5&&0<=y&&y<6)init[id[i][j]][id[x][y]] = 1;
			}
		}
		int T = Int(), Cas = 1;
		while (T-->0){
			for(int i = 0; i < N; i++)for(int j = 0; j < N; j++)A[i][j] = init[i][j];
			for(int i = 0; i < 5; i++)
				for(int j = 0; j < 6; j++)
					A[id[i][j]][now] = Int();
			Guass(A, now, now);
			out.println("PUZZLE #" + (Cas++));
			for(int i = 0, tmp = 0; i < 5; i++)
				for(int j = 0; j < 6; j++){
					out.print(ans[tmp++]);
					if(j==5)out.println(); else out.print(" ");
				}
		}
	}
	public static void main(String[] args) throws Exception {
		Main wo = new Main();
        in = new BufferedReader(new InputStreamReader(System.in));  
        out = new PrintWriter(System.out);  
		// in = new BufferedReader(new InputStreamReader(new FileInputStream(new File("input.txt"))));
		// out = new PrintWriter(new File("output.txt"));
		wo.work();
		out.close();
	}

	static int N = 56;
	static int M = N * 2;
	DecimalFormat df = new DecimalFormat("0.0000");
	static int inf = (int) 1e9;
	static long inf64 = (long) 1e18;
	static double eps = 1e-8;
	static double Pi = Math.PI;
	static int mod = (int) 1e9 + 7;

	private String Next() throws Exception {
		while (str == null || !str.hasMoreElements())
			str = new StringTokenizer(in.readLine());
		return str.nextToken();
	}

	private int Int() throws Exception {
		return Integer.parseInt(Next());
	}

	private long Long() throws Exception {
		return Long.parseLong(Next());
	}

	private double Double() throws Exception {
		return Double.parseDouble(Next());
	}

	StringTokenizer str;
	static Scanner cin = new Scanner(System.in);
	static BufferedReader in;
	static PrintWriter out;

	
	class Edge{
		int from, to, dis, nex;
		Edge(){} 
		Edge(int from, int to, int	 dis, int nex)
		{
		this.from = from; this.to = to; this.dis = dis; this.nex =	nex; 
		}
	} 
	Edge[] edge = new Edge[M<<1]; 
	int[] head = new int[N]; int edgenum; 
	void init_edge(){ for(int i = 0; i < N; i++)head[i] = -1; edgenum = 0;} 
	void add(int u, int v, int dis){ 
		edge[edgenum] = new Edge(u, v, dis, head[u]); 
		head[u] = edgenum++; 
	}
	 /*
	 */
	int upper_bound(int[] A, int l, int r, int val) {// upper_bound(A+l,A+r,val)-A;
		int pos = r;
		r--;
		while (l <= r) {
			int mid = (l + r) >> 1;
			if (A[mid] <= val) {
				l = mid + 1;
			} else {
				pos = mid;
				r = mid - 1;
			}
		}
		return pos;
	}

	int lower_bound(int[] A, int l, int r, int val) {// upper_bound(A+l,A+r,val)-A;
		int pos = r;
		r--;
		while (l <= r) {
			int mid = (l + r) >> 1;
			if (A[mid] < val) {
				l = mid + 1;
			} else {
				pos = mid;
				r = mid - 1;
			}
		}
		return pos;
	}

	int Pow(int x, int y) {
		int ans = 1;
		while (y > 0) {
			if ((y & 1) > 0)
				ans *= x;
			y >>= 1;
			x = x * x;
		}
		return ans;
	}

	double Pow(double x, int y) {
		double ans = 1;
		while (y > 0) {
			if ((y & 1) > 0)
				ans *= x;
			y >>= 1;
			x = x * x;
		}
		return ans;
	}

	int Pow_Mod(int x, int y, int mod) {
		int ans = 1;
		while (y > 0) {
			if ((y & 1) > 0)
				ans *= x;
			ans %= mod;
			y >>= 1;
			x = x * x;
			x %= mod;
		}
		return ans;
	}

	long Pow(long x, long y) {
		long ans = 1;
		while (y > 0) {
			if ((y & 1) > 0)
				ans *= x;
			y >>= 1;
			x = x * x;
		}
		return ans;
	}

	long Pow_Mod(long x, long y, long mod) {
		long ans = 1;
		while (y > 0) {
			if ((y & 1) > 0)
				ans *= x;
			ans %= mod;
			y >>= 1;
			x = x * x;
			x %= mod;
		}
		return ans;
	}

	int gcd(int x, int y) {
		if (x > y) {
			int tmp = x;
			x = y;
			y = tmp;
		}
		while (x > 0) {
			y %= x;
			int tmp = x;
			x = y;
			y = tmp;
		}
		return y;
	}

	int max(int x, int y) {
		return x > y ? x : y;
	}

	int min(int x, int y) {
		return x < y ? x : y;
	}

	double max(double x, double y) {
		return x > y ? x : y;
	}

	double min(double x, double y) {
		return x < y ? x : y;
	}

	long max(long x, long y) {
		return x > y ? x : y;
	}

	long min(long x, long y) {
		return x < y ? x : y;
	}

	int abs(int x) {
		return x > 0 ? x : -x;
	}

	double abs(double x) {
		return x > 0 ? x : -x;
	}

	long abs(long x) {
		return x > 0 ? x : -x;
	}

	boolean zero(double x) {
		return abs(x) < eps;
	}

	double sin(double x) {
		return Math.sin(x);
	}

	double cos(double x) {
		return Math.cos(x);
	}

	double tan(double x) {
		return Math.tan(x);
	}

	double sqrt(double x) {
		return Math.sqrt(x);
	}
	double fabs(double x){return x>0?x:-x;}
}