Solution: Minimize Maximum Value in a Grid
Explore a method to replace integers in a matrix to minimize the maximum value while keeping the relative order in rows and columns intact. Understand how to sort grid elements, track row and column maxima, and assign new values to meet constraints. Develop skills to solve matrix transformation problems efficiently with careful ordering and complexity considerations.
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Statement
You are given an m x n integer matrix, grid, containing distinct positive integers.
Your task is to replace each integer in the matrix with a positive integer such that the following conditions are satisfied:
1. Preserve relative order: The relative order of every two elements that are in the same row or column should stay the same after the replacements.
2. Minimize maximum value: The maximum number in the matrix after the replacement should be as small as possible.
The relative order is preserved if, for all pairs of elements in the original matrix, the following condition holds:
If grid[r1][c1] > grid[r2][c2] and either r1 == r2 or c1 == c2, then the corresponding replacement values must also satisfy grid[r1][c1] > grid[r2][c2].
For example, if grid = [[2, 4, 5], [6, 3, 8]], valid replacements could be:
[[1, 2, 3], [2, 1, 4]][[1, 2, 3], [3, 1, 4]]
Return the resulting matrix after the replacement. If there are multiple valid solutions, return any of them.
Constraints:
mgrid.lengthn...