Improved Algorithms for the k Maximum-Sums Problems

Abstract

Given a sequence of n real numbers and an integer k, 1 k 1 2 n(n − 1), the k maximum-sum segments problem is to locate the k segments whose sums are the k largest among all possible segment sums. Recently, Bengtsson and Chen gave an O(min{k + n log 2 n, n √ k})-time algorithm for this problem. Bae and Takaoka later proposed a more efficient algorithm for small k. In this paper, we propose an O(n + k log(min{n, k}))-time algorithm for the same problem, which is superior to both of them when k is o(n log n). We also give the first optimal algorithm for delivering the k maximum-sum segments in non-decreasing order if k n. Then we develop an O(n 2d−1 +k log min{n, k})-time algorithm for the d-dimensional version of the problem, where d > 1 and each dimension, without loss of generality, is of the same size n. This improves the best previously known O(n 2d−1 C)-time algorithm, also by Bengtsson and Chen, where C = min{k + n log 2 n, n √ k}. It should be pointed out that, given a two-dimensional array of size m×n, our algorithm for finding the k maximum-sum subarrays is the first one achieving cubic time provided that k is O(m 2 n/ log n).

Efficient Algorithms for the Sum Selection Problem and K Maximum Sums Problem

“…If k < n, then we can solve the sum selection problem by using the algorithm due to Cheng et al [8]. Let us assume k ≥ n in the following.…”

“…Bengtsson and Chen [5] gave an O(min{k + n log 2 n, nk 1 2 }) time algorithm, or O(n log 2 n + k) time in the worst case. Cheng et al [8] recently gave an O(n + k log(min{n, k})) time algorithm for this problem which is superior to Bengtsson and Chen's when k is o(n log n), but it runs in O(n 2 log n) time in the worst case. Lin and Lee [14] recently gave an expected O(n log n + k) time randomized algorithm based on a randomized algorithm which finds in expected O(n log n) time the segment whose sum is the k-th smallest, for any given pos-…”

Algorithm for K Disjoint Maximum Subarrays

“…Throughout this paper, we assume that K, the number of maximum subarrays we wish to find, is not greater thanK. [7] + a [8] if the index of first element is 1. We denote this by 192 (5,8).…”

“…Recent development by Cheng et al [7] and Bengtsson and Chen [4] established an optimal solution of O(n + K log K) time. For two-dimensions, O(n 3 ) time is possible [2,7]. The goal of the K-disjoint maximum subarray problem is to find K maximum subarrays, which are disjoint from one another.…”

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)…”