披荆斩棘，乘风破浪：动态规划引领下的矩阵区域和巅峰之旅

2024-01-01 08:52:13

激荡江湖路：动态规划の真谛

动态规划，一种解决问题的范式，以其简便灵活之姿，驰骋于计算机科学的辽阔疆域。其核心精髓在于将复杂问题分解为一系列子问题，逐一征服，最终收复问题之全境。

以矩阵区域和问题为例，我们并非一味蛮干，而是将矩阵分割成一个个矩形区域，逐一计算其和。此举，大大降低了算法的复杂度，令其得以在有限的时间内，求得问题的最佳解。

乘风踏浪，步步为营

def matrix_region_sum(matrix, k):
    """
    Calculate the sum of all submatrices with a sum less than or equal to k.
    
    Args:
    matrix: A 2D array of integers.
    k: The maximum sum allowed for the submatrices.
    
    Returns:
    The sum of all submatrices with a sum less than or equal to k.
    """
    # Initialize a 2D array to store the cumulative sums of the submatrices.
    cumulative_sums = [[0 for _ in range(len(matrix[0]) + 1)] for _ in range(len(matrix) + 1)]

    # Calculate the cumulative sums of the submatrices.
    for i in range(1, len(matrix) + 1):
        for j in range(1, len(matrix[0]) + 1):
            cumulative_sums[i][j] = cumulative_sums[i - 1][j] + cumulative_sums[i][j - 1] - cumulative_sums[i - 1][j - 1] + matrix[i - 1][j - 1]

    # Initialize the count of submatrices with a sum less than or equal to k.
    count = 0

    # Iterate over all possible submatrices and check if their sum is less than or equal to k.
    for i in range(1, len(matrix) + 1):
        for j in range(1, len(matrix[0]) + 1):
            for p in range(i):
                for q in range(j):
                    submatrix_sum = cumulative_sums[i][j] - cumulative_sums[p][j] - cumulative_sums[i][q] + cumulative_sums[p][q]
                    if submatrix_sum <= k:
                        count += 1

    # Return the count of submatrices with a sum less than or equal to k.
    return count