Check whether row or column swaps produce maximum size binary sub-matrix with all 1s

5

Given a binary matrix, the task is to find whether row swaps or column swaps give maximum size sub-matrix with all 1’s. In a row swap, we are allowed to swap any two rows. In a column swap we are allowed to swap any two columns. Output “Row Swap” or “Column Swap” and the maximum size.

Examples:

Input : 1 1 1
        1 0 1
Output : Column Swap
         4
By swapping column 1 and column 2(0-based indexing), 
index (0, 0) to (1, 1) makes the largest binary 
sub-matrix.

Input : 0 0 0
        1 1 0
        1 1 0
        0 0 0
        1 1 0 
Output : Row Swap
         6

Input : 1 1 0
        0 0 0
        0 0 0
        1 1 0
        1 1 0
        0 0 0
        1 1 0 
Output : Row Swap
         8

The idea is to find both row swap and column swap maximum size binary submatrix and compare.

To find the maximum sized binary sub-matrix with row swaps allowed, make a 2-D array, say dp[i][j]. Each value of dp[i][j] contains the number of consecutive 1s on right side of (i,j) in i-th row. Now, store each column in the 1-D temporary array one by one, say b[] and sort, and find maximum b[i] * (n – i), since b[i] is indicating the sub-matrix width and (n – i) is sub-matrix height.

Similarly, to find the maximum size binary sub-matrix with column swap allowed, find dp[i][j], where each value contains the number of consecutive 1 below the (i, j) in j-th column. Similarly, store each row in the 1-D temporary array one by one, say b[] and sort. Find maximum b[i] * (m – i), since b[i] is indicating the submatrix height and (n – i) is submatrix width.

Below is C++ implementation of this approach:

// C++ program to find maximum binary sub-matrix
// with row swaps and column swaps.
#include <bits/stdc++.h>
#define R 5
#define C 3
using namespace std;

// Precompute the number of consecutive 1 below the
// (i, j) in j-th column and the number of consecutive 1s
// on right side of (i, j) in i-th row.
void precompute(int mat[R][C], int ryt[][C + 2],
                               int dwn[R + 2][C + 2])
{
    // Travesing the 2d matrix from top-right.
    for (int j=C-1; j>=0; j--)
    {
        for (int i=0; i<R; ++i)
        {
            // If (i,j) contain 0, do nothing
            if (mat[i][j] == 0)
                ryt[i][j] = 0;

            // Counting consecutive 1 on right side
            else
                ryt[i][j] = ryt[i][j + 1] + 1;
        }
    }


    // Travesing the 2d matrix from bottom-left.
    for (int i = R - 1; i >= 0; i--)
    {
        for (int j = 0; j < C; ++j)
        {
            // If (i,j) contain 0, do nothing
            if (mat[i][j] == 0)
                dwn[i][j] = 0;

            // Counting consecutive 1 down to (i,j).
            else
                dwn[i][j] = dwn[i + 1][j] + 1;
        }
    }
}

// Return maximum size submatrix with row swap allowed.
int solveRowSwap(int ryt[R + 2][C + 2])
{
    int b[R] = { 0 }, ans = 0;

    for (int j=0; j<C; j++)
    {
        // Copying the column
        for (int i=0; i<R; i++)
            b[i] = ryt[i][j];

        // Sort the copied array
        sort(b, b + R);

        // Find maximum submatrix size.
        for (int i = 0; i < R; ++i)
            ans = max(ans, b[i] * (R - i));
    }

    return ans;
}

// Return maximum size submatrix with column
// swap allowed.
int solveColumnSwap(int dwn[R + 2][C + 2])
{
    int b[C] = { 0 }, ans = 0;

    for (int i = 0; i < R; ++i)
    {
        // Copying the row.
        for (int j = 0; j < C; ++j)
            b[j] = dwn[i][j];

        // Sort the copied array
        sort(b, b + C);

        // Find maximum submatrix size.
        for (int i = 0; i < C; ++i)
            ans = max(ans, b[i] * (C - i));
    }

    return ans;
}

void findMax1s(int mat[R][C])
{
    int ryt[R + 2][C + 2], dwn[R + 2][C + 2];
    memset(ryt, 0, sizeof ryt);
    memset(dwn, 0, sizeof dwn);

    precompute(mat, ryt, dwn);

    // Solving for row swap and column swap
    int rswap = solveRowSwap(ryt);
    int cswap = solveColumnSwap(dwn);

    // Comparing both.
    (rswap > cswap)? (cout << "Row Swap\n" << rswap << endl):
                     (cout << "Column Swap\n" << cswap << endl);
}

// Driven Program
int main()
{
    int mat[R][C] = {{ 0, 0, 0 },
                     { 1, 1, 0 },
                     { 1, 1, 0 },
                     { 0, 0, 0 },
                     { 1, 1, 0 }};

    findMax1s(mat);
    return 0;
}

Output:

Row Swap
6

This article is contributed by Anuj Chauhan. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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