Finding the rank of a boolean matrix

Question

I am doing a project in which I need to check whether a bool array 'vector' is linearly independent of the columns of the 'matrix'. In MATLAB it can be done by finding the rank of the augmented matrix [matrix vector] using the command rank(gf([matrix vector])). 'gf' because the matrix is Boolean. But how to do it in C++. This is what I have tried:

#include <stdlib.h>
#include <stdio.h>
#include <string.h>
#include "engine.h"
#define  BUFSIZE 256

int main()
{
Engine *ep;
mxArray *M = NULL, *V = NULL, *result = NULL;
bool matrix[4][4]={1,1,1,0,0,1,1,0,0,1,0,0}, vector[4][1]={1,1,1,1};
double *rank;

if (!(ep = engOpen("\0"))) {
    fprintf(stderr, "\nCan't start MATLAB engine\n");
    return EXIT_FAILURE;
}

V = mxCreateDoubleMatrix(4, 1, mxREAL);
M = mxCreateDoubleMatrix(4, 4, mxREAL);
memcpy((void *)mxGetPr(V), (void *)vector, sizeof(vector));
memcpy((void *)mxGetPr(M), (void *)matrix, sizeof(matrix));
engPutVariable(ep, "V", V);
engPutVariable(ep, "M", M);

engEvalString(ep, "R = rank(gf([M V]));");
result = engGetVariable(ep, "R");
engClose(ep);
rank = mxGetPr(result);
printf("%f", *rank);

printf("Done with LI\n");
mxDestroyArray(M);
mxDestroyArray(V);
mxDestroyArray(result);
engEvalString(ep, "close;");
}

The above code works and I am getting the desired results. But it runs very slow. Can anyone suggest me a way to make it fast? Or suggest some other way to find the rank of a boolean matrix. Some libraries are out there, but they seem to have functions only for int or double matrices.

Answer 1

You can find the rank of the Boolean Matrix by finding rank in the Galois Field of 2 (as you are doing in your Matlab code), which is essentially mod 2 arithmetic.

The code below finds the rank of the Boolean Matrix using the same idea by using Gauss Elimination with partial pivoting.

#include <iostream>
#include <vector>

using namespace std;

class BooleanMatrix{
    vector< vector<bool> > mat; //boolean matrix
    int n, m;           //size of matrix nxm
    int rank;           //rank of the matrix

    public:

    /*Constructor
     * Required Parameters:
     * M ==> boolean matrix
     * n ==> number of rows
     * m ==> number of columns
     */
    template <size_t size_m>
    BooleanMatrix(bool M[][size_m], int n, int m){
        this -> n = n;
        this -> m = m;
        for (int i = 0; i < n; i++){
            vector<bool> row(m);
            for (int j = 0; j < m; j++) row[j] = M[i][j];
            mat.push_back(row);         
        }
        gaussElimination();
    }

    /* Does Gauss Elimination with partial pivoting on the matrix */
     void gaussElimination(){
        rank = n;
        for (int i = 0; i < n; i++){
            if (!mat[i][i]){
                int j;
                for (j = i+1; j < n && !mat[j][i]; j++);
                if (j == n){
                       rank--;
                       continue;
                }
                else
                    for (int k = i; k < m; k++){
                        bool t = mat[i][k];
                        mat[i][k] = mat[j][k];
                        mat[j][k] = t;
                    }
            }
            for (int j = i+1; j < n; j++){
                if (mat[j][i]){
                    for (int k = i; k < m; k++)
                        mat[j][k] = mat[j][k] - mat[i][k];
                }
            }
        }
    }

    /* Get the row rank of the boolean matrix
     * If you require the rank of the matrix, make sure that n > m.
     * i.e. if n < m, call the constructor over the transpose.
     */
    int getRank(){
        return rank;
    }
};

int main(){
    bool M1[3][3] = {   {1, 0, 1},
                {0, 1, 1}, 
                {1, 1, 0}   };
    BooleanMatrix booleanMatrix1(M1, 3, 3);
    cout << booleanMatrix1.getRank() << endl;   

    bool M2[4][4] = {   {1,1,1,0},
                {0,1,1,0},
                {0,1,0,0},
                {1,1,1,1}   };
    BooleanMatrix booleanMatrix2(M2, 4, 4);
    cout << booleanMatrix2.getRank() << endl;   
}

This gives result as expected for both the case. The algorithm should work well for all practical purposes. Trivial improvements & application specific changes could be made to suit as per your requirements.

I haven't tested it thoroughly though. If anybody finds any bug, please edit the answer to correct it.

Hope this helps.

Answer 2

A simple solution is to solve the least square problem where operators are defined in the boolean sense:

min_x |matrix * x - vector|^2

Then, if vector is in the span of column vectors of the matrix, the solutions's residual error should be very small.