A Linear Time Algorithm for the k Maximal Sums Problem

  • Gerth Stølting Brodal
  • Allan Grønlund Jørgensen
Conference paper

DOI: 10.1007/978-3-540-74456-6_40

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)
Cite this paper as:
Brodal G.S., Jørgensen A.G. (2007) A Linear Time Algorithm for the k Maximal Sums Problem. In: Kučera L., Kučera A. (eds) Mathematical Foundations of Computer Science 2007. MFCS 2007. Lecture Notes in Computer Science, vol 4708. Springer, Berlin, Heidelberg

Abstract

Finding the sub-vector with the largest sum in a sequence of n numbers is known as the maximum sum problem. Finding the k sub-vectors with the largest sums is a natural extension of this, and is known as the k maximal sums problem. In this paper we design an optimal O(n + k) time algorithm for the k maximal sums problem. We use this algorithm to obtain algorithms solving the two-dimensional k maximal sums problem in O(m2·n + k) time, where the input is an m ×n matrix with m ≤ n. We generalize this algorithm to solve the d-dimensional problem in O(n2d − 1 + k) time. The space usage of all the algorithms can be reduced to O(nd − 1 + k). This leads to the first algorithm for the k maximal sums problem in one dimension using O(n + k) time and O(k) space.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Gerth Stølting Brodal
    • 1
  • Allan Grønlund Jørgensen
    • 1
  1. 1.BRICS, MADALGO, Department of Computer Science, University of AarhusDenmark

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