The Maximum Subarray ProblemDefining problem , its brute force solution, divide and conquer solution

Presented by: Kamran Ashraf

Formal Problem Definition

•Given a sequence of numbers <a1,a2,…..an> we work to find a subsequence of A that is contiguous and whose values have the maximum sum.

•Maximum Subarray problem is only challenging when having both positive and negative numbers.

Example…

Here, the subarray A[8…11] is the Maximum Subarray, with sum 43, has the greatest sum of any contiguous subarray of array A.

Brute Force SolutionTo solve the Maximum Subarray Problem we can use brute force solution however in brute force we will have to compare all possible continuous subarray which will increase running time. Brute force will result in Ѳ(n2), and the best we can hope for is to evaluate each pair in constant time which will result in Ὠ(n2).

Divide and Conquer is a more efficient method for solving large number of problems resulting in Ѳ(nlogn).

Divide and Conquer Solution

•First we Divide the array A [low … high] into two subarrays A [low … mid] (left) and A [mid +1 … high] (right).

•We recursively find the maximum subarray of A [low … mid] (left), and the maximum of A [mid +1 … high] (right).

•We also find maximum crossing subarray that crosses the midpoint.

•Finally we take a subarray with the largest sum out of three.

Example…

Time Analysis

•Find-Max-Cross-Subarray takes: Ѳ (n) time

•Two recursive calls on input size n/2 takes: 2T(n/2) time

•Hence:T(n) = 2T(n/2) + Ѳ (n)T(n) = Ѳ (n log n)

Pseudo CodeMax-subarray(A, Left, Right)if (Right == Left)

return (left, right, A[left])else mid= [(left+right)/2]L1=Find-Maximum-Subarray(A,left,mid)R1=Find-Maximum-Subarray(A,mid+1,right)M1=Find-Max-Crossing-Subarray(A,left,mid,right)If sum(L1) > sum(R1) and sum(L1) > sum(M1)Return L1elseif sum(R1) > sum(L1) and sum(R1) > sum(M1)Return R1Else return M1