#### Modified Binary Search

```/**
* @file
* @brief [Modified binary search algorithm](https://arxiv.org/abs/1406.1677)
*/
#include <stdio.h>
#include <stdlib.h>

/** This function does Binary search for `x` in `i`-th row from `j_low` to
* `j_high`.
* @param mat 2D matrix to search within
* @param i row to search in
* @param j_low start column index
* @param j_high end column index
* @param x value to search for
* @return column where `x` was found
* @return -1 if value not found
*/
int binarySearch(const int **mat, int i, int j_low, int j_high, int x)
{
while (j_low <= j_high)
{
int j_mid = (j_low + j_high) / 2;

// Element found
if (mat[i][j_mid] == x)
{
printf("Found at (%d,%d)\n", i, j_mid);
return j_mid;
}
else if (mat[i][j_mid] > x)
j_high = j_mid - 1;
else
j_low = j_mid + 1;
}

return -1;
}

/** Function to perform binary search on the mid values of row to get the
* desired pair of rows where the element can be found
* @param [in] mat matrix to search for the value in
* @param n number of rows in the matrix
* @param m number of columns in the matrix
* @param x value to search for
*/
void modifiedBinarySearch(const int **mat, int n, int m, int x)
{  // If Single row matrix
if (n == 1)
{
binarySearch(mat, 0, 0, m - 1, x);
return;
}

// Do binary search in middle column.
// Condition to terminate the loop when the 2 desired rows are found.
int i_low = 0, i_high = n - 1, j_mid = m / 2;
while ((i_low + 1) < i_high)
{
int i_mid = (i_low + i_high) / 2;
// element found
if (mat[i_mid][j_mid] == x)
{
printf("Found at (%d,%d)\n", i_mid, j_mid);
return;
}
else if (mat[i_mid][j_mid] > x)
i_high = i_mid;
else
i_low = i_mid;
}
// If element is present on the mid of the two rows
if (mat[i_low][j_mid] == x)
printf("Found at (%d,%d)\n", i_low, j_mid);
else if (mat[i_low + 1][j_mid] == x)
printf("Found at (%d,%d)\n", i_low + 1, j_mid);

// Search element on 1st half of 1st row
else if (x <= mat[i_low][j_mid - 1])
binarySearch(mat, i_low, 0, j_mid - 1, x);

// Search element on 2nd half of 1st row
else if (x >= mat[i_low][j_mid + 1] && x <= mat[i_low][m - 1])
binarySearch(mat, i_low, j_mid + 1, m - 1, x);

// Search element on 1st half of 2nd row
else if (x <= mat[i_low + 1][j_mid - 1])
binarySearch(mat, i_low + 1, 0, j_mid - 1, x);

// search element on 2nd half of 2nd row
else
binarySearch(mat, i_low + 1, j_mid + 1, m - 1, x);
}

/** Main function */
int main()
{
int x;     // element to be searched
int m, n;  // m = columns, n = rows

scanf("%d %d %d\n", &n, &m, &x);

int **mat = (int **)malloc(n * sizeof(int *));
for (int i = 0; i < m; i++) mat[i] = (int *)malloc(m * sizeof(int));

for (int i = 0; i < n; i++)
{
for (int j = 0; j < m; j++)
{
scanf("%d", &mat[i][j]);
}
}

modifiedBinarySearch(mat, n, m, x);

for (int i = 0; i < n; i++) free(mat[i]);
free(mat);
return 0;
}
```