Maximum Subarray
There's an interesting problem I recently solved on leetcode based on dynamic programming. My Github repository contains list of all problems that I have solved. I often start with a brute force approach without fretting about time complexity. Later I try to improve my algorithm for a better efficient solution.
Given an integer array nums, find the contiguous subarray (containing at least one number) which has the largest sum and return its sum. A subarray is a contiguous part of an array.1
Brute force algorithm
We can commence with a brute force algorithm. For every element of array, we compare its sum with the rest of the element to calculate maximum sum.
Time complexity is O(n2) and has Time Limit Exceeded on leetcode.
More efficient solution
We can use Kadane algorithm to solve. It scans the given array A[1..n] from left to right. In the jth step, it computes the subarray with the largest sum ending at j; this sum is maintained in variable cSum. It computes the subarray with the largest sum anywhere in A[1..j], maintained in variable oSum;
The time complexity is O(n).
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