Problem Description
You are given an m x n binary matrix matrix and an integer numSelect. Your goal is to select exactly numSelect distinct columns from matrix such that you cover as many rows as possible. A row is considered covered if all the 1's in that row are also part of a column that you have selected. If a row does not have any 1's, it is also considered covered.
Key Insights
- A row is covered if all its 1's are in the selected columns or if it contains no 1's.
- The problem can be approached using combinations of columns since the constraints are small (m, n <= 12).
- We can use a bitmask to represent selected columns and determine covered rows efficiently.
Space and Time Complexity
Time Complexity: O(2^n * m) — where n is the number of columns, and m is the number of rows. We generate all combinations of columns and check each row against the selected columns. Space Complexity: O(1) — we use a constant amount of space for counting covered rows.
Solution
The solution involves generating all possible combinations of columns that can be selected using backtracking or bit manipulation. For each combination, we check how many rows can be covered based on the selected columns. The maximum count of covered rows across all combinations is returned as the result.