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Fast Monotone Summation over Disjoint Sets

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Parameterized and Exact Computation (IPEC 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7535))

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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((n p + n q)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.

This research was supported in part by the Academy of Finland, Grants 252083 (P.K.), 256287 (P.K.), and 125637 (M.K.), and by the Helsinki Doctoral Programme in Computer Science - Advanced Computing and Intelligent Systems (J.K.).

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Author information

Authors and Affiliations

  1. Helsinki Institute for Information Technology HIIT & Department of Information and Computer Science, Aalto University, Finland

    Petteri Kaski

  2. Helsinki Institute for Information Technology HIIT & Department of Computer Science, University of Helsinki, Finland

    Mikko Koivisto & Janne H. Korhonen

Authors
  1. Petteri Kaski
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  2. Mikko Koivisto
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  3. Janne H. Korhonen
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Editor information

Editors and Affiliations

  1. Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, 15784, Athens, Greece

    Dimitrios M. Thilikos

  2. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Gerhard J. Woeginger

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© 2012 Springer-Verlag Berlin Heidelberg

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Kaski, P., Koivisto, M., Korhonen, J.H. (2012). Fast Monotone Summation over Disjoint Sets. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_16

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  • DOI: https://doi.org/10.1007/978-3-642-33293-7_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33292-0

  • Online ISBN: 978-3-642-33293-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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