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Interview Question

Machine Learning Software Engineer Interview Palo Alto, CA

Suppose you have a matrix of numbers. How can you easily

  compute the sum of any rectangle (i.e. a range [row_start, row_end, col_start, col_end]) of those numbers? How would you code this?
Tags:
algorithm, code
Answer

Interview Answer

8 Answers

2

It can be done in constant time by precalculating sums of some basic rectangles (extending all the way to the border of the matrix). That precalculation times time O(n) by simple dynamic programming.

Interview Candidate on Jan 21, 2010
1

Please elaborate, which "basic rectangles"? Are you recursively dividing each rectangle into 4 smaller rectangles? Precalc time for doing that is not O(n)?!?

maxzed2k on Feb 10, 2010
15

Compute the sum of the rectangles, for all i,j, bounded by (i,j), (i,m), (n,j), (n,m), where (n,m) is the size of the matrix M. Call that sum s(i,j). You can calculate s(i,j) by dynamic programming: s(i,j) = M(i,j) + s(i+1,j) + s(i,j+1) - s(i+1,j+1). And the sum of any rectangle can be computed from s(i,j).

Anonymous on Feb 10, 2010
3

Awesome!!

Anonymous on Feb 21, 2010
3

The answer is already popular in computer vision fields!! It is called integral imaging.
See this page http://en.wikipedia.org/wiki/Haar-like_features

seungwooYoo on Oct 13, 2012
5

It wasn't 100% clear to me, then I found the Wiki page

http://en.wikipedia.org/wiki/Summed_area_table

Anonymous on Oct 23, 2014
0

Let a[][] be the 2d array, int i=0; for( j = row_start; j <= row_end; j++) for( k = col_start; k <= col_end; k++) i+=a[j][k];

Anonymous on Mar 4, 2019
0

Iterate over matrix as an array storing (new sums array) in each position the cumulative sum up to that point. For each row in the desired submatrix we can compute its sum as a difference between its end and start positions. Repeat for other rows. Add up all the row sums.

Edi on Mar 8, 2019

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