Fast Multi-Subset Transform and Weighted Sums over Acyclic Digraphs
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).
Bayesian networks
Moebius transform
Rectangular matrix multiplication
Subset convolution
Weighted counting of acyclic digraphs
Zeta transform
Theory of computation~Design and analysis of algorithms
29:1-29:12
Regular Paper
This work was partially supported by the Academy of Finland, Grant 316771.
We thank Petteri Kaski for valuable discussions about the topic of the paper.
Mikko
Koivisto
Mikko Koivisto
Department of Computer Science, University of Helsinki, Finland
Antti
Röyskö
Antti Röyskö
Department of Computer Science, University of Helsinki, Finland
10.4230/LIPIcs.SWAT.2020.29
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Mikko Koivisto and Antti Röyskö
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