Randomized Algorithm for the Sum Selection Problem

Abstract

Given a sequence of n real numbers A = a ₁, a ₂,..., a _n and a positive integer k, the Sum Selection Problem is to find the segment A( i,j) = a _i , a _i + 1,..., a _j such that the rank of the sum $s(i, j) = \sum_{t = i}^{j}{a_{t}}$ is k over all ${n(n-1)} \over {2}$ segments. We will give a randomized algorithm for this problem that runs in expected O(n log n) time. Applying this algorithm we can obtain an algorithm for the k Maximum Sums Problem, i.e., the problem of enumerating the k largest sum segments, that runs in expected O(n log n + k) time. The previously best known algorithm for the k Maximum Sums Problem runs in O(n log² n + k) time in the worst case.

Research supported in part by the National Science Council under the Grants NSC-92-3112-B-001-018-Y, NSC-92-3112-B-001-021-Y, NSC-92-2218-E-001-001, NSC 93-2422-H-001-0001, NSC 93-2213-E-001-013 and NSC 93-2752-E-002-005-PAE, and by the Taiwan Information Security Center (TWISC), National Science Council under the Grants NSC 94-3114-P-001-001-Y and NSC 94-3114-P-011-001.