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) log n) 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. © 2012 Springer-Verlag.
CITATION STYLE
Kaski, P., Koivisto, M., & Korhonen, J. H. (2012). Fast monotone summation over disjoint sets. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7535 LNCS, pp. 159–170). https://doi.org/10.1007/978-3-642-33293-7_16
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