GESP C++ Level 4 Real Exam Questions (2D Arrays) [202503] Second Order Matrix (luogu-B4264)

GESP C++ Level 4 real exam questions from March 2025. This question mainly tests the application of 2D arrays. It is considered a simple question among level 4 questions. Difficulty ⭐⭐☆☆☆. This question is rated as <span>beginner</span> by Luogu.

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luogu-B4264 [GESP202503 Level 4] Second Order Matrix

Problem Requirements

Problem Description

Little A has a row column matrix .

Little A considers a matrix to be good if and only if . Here, represents the element in the row and column of the matrix.

Little A wants to know how many good submatrices are in .

Input Format

The first line contains two positive integers .

Next, lines, each containing integers .

Output Format

One line, an integer representing the number of good submatrices in .

Input Output Example #1

Input #1

3 4
1 2 1 0
2 4 2 1
0 3 3 0

Output #1

2

Explanation/Hint

Example Explanation

The good submatrices in the example are as follows:

Data Range

For all test points, it is guaranteed that ,,

Problem Analysis

The core of this problem is to find all submatrices that meet the conditions in a matrix.

1. Data Structure Design

• Use a 2D array <span>num_ary[505][505]</span> to store the input matrix
• The array size is set to 505 to meet the maximum range requirement of the problem ()

2. Core Algorithm Idea

• Traverse all possible top-left positions of 2×2 submatrices
• Row traversal range:<span>[0, n-2]</span>, ensuring enough space to accommodate a 2×2 matrix
• Column traversal range:<span>[0, m-2]</span>, similarly
• Check:
• Main diagonal:<span>num_ary[i][j]</span> and <span>num_ary[i+1][j+1]</span>
• Anti-diagonal:<span>num_ary[i][j+1]</span> and <span>num_ary[i+1][j]</span>
• Condition satisfied:<span>num_ary[i][j] * num_ary[i+1][j+1] == num_ary[i][j+1] * num_ary[i+1][j]</span>

3. Algorithm Complexity Analysis

1. Time Complexity

• Traverse all possible top-left positions of submatrices:
• Judgment operation for each position:
• Overall time complexity:

Space Complexity

• Store the input matrix:
• Other variables:
• Overall space complexity:

Example Code

#include <iostream>

// Define a 2D array to store matrix elements, supporting a maximum of 500x500 matrix
int num_ary[505][505];

int main() {
    // Define the number of rows n and columns m of the matrix
    int n, m;
    // Read the number of rows and columns of the matrix
    std::cin >> n >> m;

    // Read all elements of the matrix
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < m; j++) {
            std::cin >> num_ary[i][j];
        }
    }

    // Counter to count the number of 2x2 submatrices that meet the conditions
    int count = 0;
    
    // Traverse all possible top-left positions of 2x2 submatrices
    for (int i = 0; i < n - 1; i++) {
        for (int j = 0; j < m - 1; j++) {
            // Check if the current 2x2 submatrix meets the condition: product of main diagonal elements equals product of anti-diagonal elements
            if (num_ary[i][j] * num_ary[i+1][j+1] == num_ary[i][j+1] * num_ary[i+1][j]) {
                count++;
            }
        }
    }

    // Output the result
    std::cout << count;
    return 0;
}

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