The Problem

Given an M x N matrix of integers, find the sub rectangle such that the sum is maximized.


Things to consider

  • Negative values may appear in the matrix
    • If all values were positive the maximum sum sub rectangle would be the entire matrix
  • There is a better solution than checking every sub rectangle

Data representation

  • Nothing overly complicated needed
  • A 2D array will represent the data exactly as needed
  • Usually easier to use a vector<vector<T>> in C++ v.s. T[][] as passing 2D arrays with run time sizes into functions is a little awkward

The Idea

  • We can solve the one dimensional version of this problem in linear time using kadanes algorithm
  • The rows of the 2D array can be flattened into a 1D array by summing them together
  • We can then perform kadanes algorithm on the flattened rows and find the rows bounding the maximum sub sequence

In Action

  • etc.

One Dimensional Algorithm

  • In order to find the row bounds we need to implement the one dimensional maximum sub sequence algorithm
  • Known as Kadane’s algorithm, it goes as follows:
    • Keep track of the sum of the current prefix that we scanned
    • If that prefix ever becomes negative, give up on it and start over
    • After each iteration, compare our current prefix sum with the max seen so far and update accordingly


Using this algorithm

  • To use this algorithm, we simply need to iterate through the matrix fixing the columns
  • We can then flatten the elements in between our fixed left and right columns into a one dimensional array
  • Once we have our flattened columns, we run the result through kadanes algorithm and recieve the maximum sum as well as the bounding rows

Combining the two

The solution