A Linear Time Algorithm for the k Maximal Sums Problem

  • Gerth Stølting Brodal
  • Allan Grønlund Jørgensen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)


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(m 2·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(n 2d − 1 + k) time. The space usage of all the algorithms can be reduced to O(n d − 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.


Binary Tree Time Algorithm Large Element Linear Time Algorithm Space Usage 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© 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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