Fast monotone summation over disjoint sets

  • Authors:

  • Petteri Kaski;Mikko Koivisto;Janne H. Korhonen

  • Affiliations:

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

  • Venue:
  • IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
  • Year:
  • 2012

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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((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.