Unusual Product

最新推荐文章于 2019-04-02 22:31:32 发布

原创最新推荐文章于 2019-04-02 22:31:32 发布 · 522 阅读

CC 4.0 BY-SA版权

本文介绍了一种特殊的矩阵运算——“不寻常”的平方，并提供了一个解决该问题的算法实现。该算法通过跟踪对角线元素的变化来高效地计算经过行翻转或列翻转操作后的矩阵的“不寻常”平方。

Little Chris is a huge fan of linear algebra. This time he has been given a homework about the unusual square of a square matrix.

The dot product of two integer number vectors x and y of size n is the sum of the products of the corresponding components of the vectors. The unusual square of an n × n square matrix A is defined as the sum of n dot products. The i-th of them is the dot product of the i-th row vector and the i-th column vector in the matrix A.

Fortunately for Chris, he has to work only in GF(2)! This means that all operations (addition, multiplication) are calculated modulo 2. In fact, the matrix A is binary: each element of A is either 0 or 1. For example, consider the following matrix A:

$\text{[math]}$

The unusual square of A is equal to (1·1 + 1·0 + 1·1) + (0·1 + 1·1 + 1·0) + (1·1 + 0·1 + 0·0) = 0 + 1 + 1 = 0.

However, there is much more to the homework. Chris has to process q queries; each query can be one of the following:

given a row index i, flip all the values in the i-th row in A;
given a column index i, flip all the values in the i-th column in A;
find the unusual square of A.

To flip a bit value w means to change it to 1 - w, i.e., 1 changes to 0 and 0 changes to 1.

Given the initial matrix A, output the answers for each query of the third type! Can you solve Chris's homework?

Input

The first line of input contains an integer n (1 ≤ n ≤ 1000), the number of rows and the number of columns in the matrix A. The next n lines describe the matrix: the i-th line contains n space-separated bits and describes the i-th row of A. The j-th number of the i-th line a_ij (0 ≤ a_ij ≤ 1) is the element on the intersection of the i-th row and the j-th column of A.

The next line of input contains an integer q (1 ≤ q ≤ 10⁶), the number of queries. Each of the next q lines describes a single query, which can be one of the following:

1 i — flip the values of the i-th row;
2 i — flip the values of the i-th column;
3 — output the unusual square of A.

Note: since the size of the input and output could be very large, don't use slow output techniques in your language. For example, do not use input and output streams (cin, cout) in C++.

Output

Let the number of the 3rd type queries in the input be m. Output a single string s of length m, where the i-th symbol of s is the value of the unusual square of A for the i-th query of the 3rd type as it appears in the input.

Sample test(s)

Input

Output
01001

分析：分析规律之后其实只有对角线上的对结果产生影响，所以只需要每变化一次对结果取反就可以了。

#include <stdio.h>
#include <string.h>
#include <stdlib.h>

int map[1010][1010];

int main()
{
    int n, i, j, q, k;
    int value = 0;
    scanf("%d",&n);
    for(i = 1; i <= n; i++)
        for(j = 1; j <= n; j++)
            scanf("%d",&map[i][j]);
    scanf("%d", &q);
    for(i = 1; i <= n; i++)
    {
        value += map[i][i]*map[i][i];
        value %= 2;
    }
    for(i = 0;i < q; i++)
    {
        scanf("%d", &k);
        if(k == 1 || k == 2)
        {
            scanf("%d", &k);
            value = !value;
        }
        else printf("%d",value);
    }

    return 0;
}