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

原创 于 2015-03-26 21:52:23 发布 · 1.3k 阅读 · 0 ·
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本文详细阐述了解决特定矩阵灯状态转换问题的算法过程，包括问题背景、数学建模、求解步骤及代码实现，旨在提供一种有效的方法来实现从初始全暗状态到指定目标状态的灯操作方案。

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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;}
}


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