Fast zeta transforms for lattices with few irreducibles

Authors:
Andreas Björklund;Mikko Koivisto;Thore Husfeldt;Jesper Nederlof;Petteri Kaski;Pekka Parviainen
Affiliations:
Lund University, Sweden;IT University of Copenhagen, Denmark;Aalto University, Finland;University of Helsinki, Finland;University of Bergen, Norway;University of Helsinki, Finland
Venue:
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Year:
2012

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Abstract

We investigate fast algorithms for changing between the standard basis and an orthogonal basis of idempotents for Möbius algebras of finite lattices. We show that every lattice with v elements, n of which are nonzero and join-irreducible (or, by a dual result, nonzero and meet-irreducible), has arithmetic circuits of size O(vn) for computing the zeta transform and its inverse, thus enabling fast multiplication in the Möbius algebra. Furthermore, the circuit construction in fact gives optimal (up to constants) circuits for a number of lattices of combinatorial and algebraic relevance, such as the lattice of subsets of a finite set, the lattice of set partitions of a finite set, the lattice of vector subspaces of a finite vector space, and the lattice of positive divisors of a positive integer.