Solution: Minimize Maximum Value in a Grid
Explore how to transform an integer matrix by replacing its values to minimize the maximum number while preserving the relative order in rows and columns. Understand the use of sorting and auxiliary arrays to assign new values efficiently, ensuring order is maintained and the maximum value is minimized. This lesson covers the algorithmic approach, time and space complexity, and practical implementation details in solving grid-based matrix problems.
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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...