Skip to content
Related Articles

Related Articles

Improve Article
Save Article
Like Article

Summed Area Table – Submatrix Summation

  • Difficulty Level : Easy
  • Last Updated : 11 Feb, 2018

Given a matrix of size M x N, there are large number of queries to find submatrix sums. Inputs to queries are left top and right bottom indexes of submatrix whose sum is to find out.

How to preprocess the matrix so that submatrix sum queries can be performed in O(1) time.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.  To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

In case you wish to attend live classes with experts, please refer DSA Live Classes for Working Professionals and Competitive Programming Live for Students.


tli :  Row number of top left of query submatrix
tlj :  Column number of top left of query submatrix
rbi :  Row number of bottom right of query submatrix
rbj :  Column number of bottom right of query submatrix

Input: mat[M][N] = {{1, 2, 3, 4, 6},
                    {5, 3, 8, 1, 2},
                    {4, 6, 7, 5, 5},
                    {2, 4, 8, 9, 4} };
Query1: tli = 0, tlj = 0, rbi = 1, rbj = 1
Query2: tli = 2, tlj = 2, rbi = 3, rbj = 4
Query3: tli = 1, tlj = 2, rbi = 3, rbj = 3;

Query1: 11  // Sum between (0, 0) and (1, 1)
Query2: 38  // Sum between (2, 2) and (3, 4)
Query3: 38  // Sum between (1, 2) and (3, 3)

Naive Algorithm:
We can loop all the queries and calculate each query in O (q*(N*M)) worst case which is too large for a large range of numbers.

// Pseudo code of Naive algorithm.
Arr[][] = input_matrix
For each query:
    Input tli, tlj, rbi, rbj
    sum = 0
    for i from tli to tbi (inclusive):
        for j  from tlj to rbj(inclusive):
            sum += Arr[i][j]

Optimized Solution :
Summed Area Table can reduce this type of query into preprocessing time of O(M*N) and each query will execute in O(1).
Summed Area Table is a data structure and algorithm for quickly and efficiently generating the sum of values in a rectangular subset of a grid.
The value at any point (x, y) in the summed area table is just the sum of all the values above and to the left of (x, y), inclusive :

The optimized solution is implemented in below post.
Implementation of optimized approach

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

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

My Personal Notes arrow_drop_up
Recommended Articles
Page :