Question

A “generalized diagonal” in a NXN matrix is a selection of N cells, such that:

1. Exactly one cell is selected from each row and from each column
2. Every selected cell contains a non-zero value

I am looking for an algorithm to find a generalized diagonal in O(n^3). It would seem to me that the following dynamic programming algorithm is “good enough”, but I’m not sure how to analyze its complexity.

Set<Set<Integer>> failedCache = new HashSet<Set<Integer>>();

List<Integer> find(int[][] matrix, Set<Integer> used, int row) {
    int N = matrix.length;
    if (failedCache.contains(used))
        return null;

    if (row == N) return new ArrayList<Integer>();

    for (int col = 0; col < N; ++col) {
        if (matrix[row][col] == 0)
            continue;

        if (used.contains(col))
            continue;

        Set<Integer> newUsed = new HashSet<Integer>(used);
        newUsed.add(col);
        List<Integer> answer = find(matrix, newUsed, row + 1);
        if (answer != null) {
            answer.add(col);
            return answer;
        }
    }

    failedCache.add(used);
    return null;
}

Answer 1

The algorithm runs in worst-case exponential time because on the following matrix

 11111
 11111
 11111
 11111
 00000

it will try about n! possible combinations.

For polynomial time solution, create a bipartite graph using the matrix, and find perfect matching.

For example, with matrix

 011
 101
 001

you create the graph

 A    X
 B    Y
 C    Z

with edges A->Y, A->Z, B->X, B->Z, C->Z.