Fast Monotone Summation over Disjoint Sets

  • Petteri Kaski
  • Mikko Koivisto
  • Janne H. Korhonen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7535)

Abstract

We study the problem of computing an ensemble of multiple sums where the summands in each sum are indexed by subsets of size p of an n-element ground set. More precisely, the task is to compute, for each subset of size q of the ground set, the sum over the values of all subsets of size p that are disjoint from the subset of size q. We present an arithmetic circuit that, without subtraction, solves the problem using O((np + nq)logn) arithmetic gates, all monotone; for constant p, q this is within the factor logn of the optimal. The circuit design is based on viewing the summation as a “set nucleation” task and using a tree-projection approach to implement the nucleation. Applications include improved algorithms for counting heaviest k-paths in a weighted graph, computing permanents of rectangular matrices, and dynamic feature selection in machine learning.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Petteri Kaski
    • 1
  • Mikko Koivisto
    • 2
  • Janne H. Korhonen
    • 2
  1. 1.Helsinki Institute for Information Technology HIIT & Department of Information and Computer ScienceAalto UniversityFinland
  2. 2.Helsinki Institute for Information Technology HIIT & Department of Computer ScienceUniversity of HelsinkiFinland

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