Maximum subarray problem

In computer science, the maximum sum subarray problem, also known as the maximum segment sum problem, is the task of finding a contiguous subarray with the largest sum, within a given one-dimensional array A[1...n] of numbers. It can be solved in ${\displaystyle O(n)}$ time and ${\displaystyle O(1)}$ space.

Formally, the task is to find indices ${\displaystyle i}$ and ${\displaystyle j}$ with ${\displaystyle 1\leq i\leq j\leq n}$, such that the sum

${\displaystyle \sum _{x=i}^{j}A[x]}$

is as large as possible. (Some formulations of the problem also allow the empty subarray to be considered; by convention, the sum of all values of the empty subarray is zero.) Each number in the input array A could be positive, negative, or zero.^[1]

For example, for the array of values [−2, 1, −3, 4, −1, 2, 1, −5, 4], the contiguous subarray with the largest sum is [4, −1, 2, 1], with sum 6.

Some properties of this problem are:

1. If the array contains all non-negative numbers, then the problem is trivial; a maximum subarray is the entire array.
2. If the array contains all non-positive numbers, then a solution is any subarray of size 1 containing the maximal value of the array (or the empty subarray, if it is permitted).
3. Several different sub-arrays may have the same maximum sum.

Although this problem can be solved using several different algorithmic techniques, including brute force,^[2] divide and conquer,^[3] dynamic programming,^[4] and reduction to shortest paths, a simple single-pass algorithm known as Kadane's algorithm solves it efficiently.