eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-12
29:1
29:12
10.4230/LIPIcs.SWAT.2020.29
article
Fast Multi-Subset Transform and Weighted Sums over Acyclic Digraphs
Koivisto, Mikko
1
Röyskö, Antti
1
Department of Computer Science, University of Helsinki, Finland
The zeta and Moebius transforms over the subset lattice of n elements and the so-called subset convolution are examples of unary and binary operations on set functions. While their direct computation requires O(3ⁿ) arithmetic operations, less naive algorithms only use 2ⁿ poly(n) operations, nearly linear in the input size. Here, we investigate a related n-ary operation that takes n set functions as input and maps them to a new set function. This operation, we call multi-subset transform, is the core ingredient in the known inclusion - exclusion recurrence for weighted sums over acyclic digraphs, which extends Robinson’s recurrence for the number of labelled acyclic digraphs. Prior to this work, the best known complexity bound for computing the multi-subset transform was the direct O(3ⁿ). By reducing the task to rectangular matrix multiplication, we improve the complexity to O(2.985ⁿ).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol162-swat2020/LIPIcs.SWAT.2020.29/LIPIcs.SWAT.2020.29.pdf
Bayesian networks
Moebius transform
Rectangular matrix multiplication
Subset convolution
Weighted counting of acyclic digraphs
Zeta transform