A linear time algorithm for the k maximal sums problem

Authors:
Gerth Stølting Brodal;Allan Grønlund Jørgensen
Affiliations:
BRICS, MADALGO, Department of Computer Science, University of Aarhus, Denmark;BRICS, MADALGO, Department of Computer Science, University of Aarhus, Denmark
Venue:
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
Year:
2007

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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 ċ 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.