Rank Transform of a Matrix

Batch equal values before committing

Sort all cells by value. For one equal-value group, temporarily connect cells sharing a row or column, then for each component compute the best prior rank from its rows and columns. Give every cell in that component best + 1. Only after all cells of the value group have been computed should you update the persistent row and column rank maps. This prevents equal values from incorrectly forcing one another higher. The invariant is that persistent ranks describe strictly smaller values only. A practical implementation uses DSU per group plus row/column maps. Sorting is O(mn log(mn)) for an m x n matrix; DSU work is near-linear and storage is O(mn). Test duplicate values in one row, disconnected equal cells, negative values, one row, and ties that share both a row and column.