### PROBLEM LINK:

**Author:** Vasya Antoniuk

**Testers:** Istvan Nagy

**Editorialist:** Praveen Dhinwa

### DIFFICULTY:

medium

### PREREQUISITES:

partial sums, dequeue, dynamic programming

### PROBLEM:

You are given 2-D matrix A[n][m]. The matrix will be called good if there exists a sub-matrix (continuous rectangular block of matrix) of dimensions a \times b whose all of its elements are equal. In a single operation, you can increase any element of the matrix by 1. Find out minimum number of operations required to make the matrix good.

### QUICK EXPLANATION:

For finding sum of a sub-matrix, you can use partial sums.

For finding maximum element in a sub-matrix, your can use doubly ended queue or deque.

### EXPLANATION:

Let us consider some a \times b dimension sub-matrix. Let x be the largest element in the sub-matrix. Let S be the total sum of elements in the sub-matrix. For making all the elements equal in minimum number of operations, we should aim to make all the elements equal to x. So, the minimum number of operations required will be x * n * m - S.

So, we have to find the sum and maximum element in all the possible sub-matrices of dimension a \times b.

For finding sum of all sub-matrices of sizes a \times b, we can use maintain partial sum sub-matrix. For that, we maintain another array, sum[n][m], where sum[i][j] will denote the sum of elements of the sub-matrix with left top end coordinate being (1, 1) and right bottom coordinate being (i, j). After computing sum matrix, we can find the sum of any sub-matrix.

Now let us learn about how to find maximum element in all sub-matrices of sizes a \times b.

Let maxCol[i][j] be the maximum element of the sub-matrix in the column starting at (i, j - b + 1) and ending at (i, j) (i.e. of column of length b).

Assume that we have computed maxCol array efficiently, let us find how we can use this to calculate maximum of a \times b sub-matrix.

Let max[i][j] denote the maximum value of sub-matrix of A ending at (i, j) (bottom right point), and of dimensions a \times b. Note that max[i][j] is maximum of maxCol[i - a + 1][j], maxCol[i - a + 2][j], \dots, maxCol[i][j].

Note that for calculating max[i][j + 1] from max[i][j], we have to add a new element maxCol[i][j + 1] and remove the element maxCol[i - a + 1][j].

That means, that we have to maintain maximum of a constant size subarray of an array, i.e. at each step, maintain maximum with inserting and deleting one element at each step. Note that we can compute maxCol in similar way.

This can be done either maintaining a multi-set (balanced binary search tree) of K elements, in which insertion/deletion and finding maximum can be done in \mathcal{O}(log(n)) time. Sadly this method is slower for passing largest subtask.

You can find a \mathcal{O}(n) solution for it by using doubly ended queue (aka deque). Please see this link for its very detailed explanation.

### Time Complexity:

\mathcal{O}(n m) for finding both maxCol and max matrices.