Improved Algorithms for the K-Maximum Subarray Problem

Abstract

The maximum subarray problem is to find the contiguous array elements having the largest possible sum. We extend this problem to find K maximum subarrays. For general K maximum subarrays where overlapping is allowed, Bengtsson and Chen presented OðminfK + n log 2 n‚ n ffiffiffi ffi K p gÞ time algorithm for one-dimensional case, which finds unsorted subarrays. Our algorithm finds K maximum subarrays in sorted order with improved complexity of O ((n + K ) log K ). For the twodimensional case, we introduce two techniques that establish O(n 3 ) and subcubic time.

Application of the Maximum Convex Sum Algorithm in Determining Environmental Variables that Affect Nigerian Highland Stream Benthic Communities

“…Figure 2 shows an example of a two-dimensional extension of m. The shaded region A r c r c is the region that was found to be the solution existing between r r maximum subarray. Bae and Takoaka modified algorithm to cover the disjoint case by replacing all elements in the first located maximum sub-array with negative infinity [10,15]; therefore, such a region will be voided in the calculation of finding the next maximum sub-array. This process is iterated until the K th K K maximum sub-array is found.…”

Improved Algorithms for the K-Maximum Subarray Problem for Small K

“…As min i is sorted, the produced list of candidates cand i is sorted in non-increasing order, and has the first item cand i [1] being the largest candidate produced from sum [i].…”

“…For the twodimensional case, EREW PRAM solutions achieving O(log n) time with O(n 3 / log n) processors are given in [11,18] and comparable result on interconnection networks is given in [12]. The EREW PRAM version of the subcubic algorithm in [15,17] is given in [1], which also features a VLSI algorithm based on the technique introduced in Bentley's paper. This VLSI algorithm for the maximum subarray problem achieves T = m + n − 2 steps, which is O(n) time using O(n 2 ) sized hardware circuit.…”

On the Lower Bound for the Maximum Consecutive Sub-Sums Problem

“…, n] and a positive number k, consist in locate the k segments whose sum are the k largest among all possible sums. The k Maximum Sum Segments was first presented by Bae and Takaoka (5) and, after different solutions emerged (5,6,7,10,22,50), was optimally solved by Brodal and Jørgensen (15) in O(n + k)…”