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Fast Zeta Transforms for Lattices with Few Irreducibles
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2016 (English)In: ACM Transactions on Algorithms, ISSN 1549-6325, E-ISSN 1549-6333, Vol. 12, no 1, 4Article in journal (Refereed) PublishedText
Abstract [en]

We investigate fast algorithms for changing between the standard basis and an orthogonal basis of idempotents for Mobius 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 Mobius algebra. Furthermore, the circuit construction in fact gives optimal (up to constants) monotone circuits for several 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.

Place, publisher, year, edition, pages
Association for Computing Machinery (ACM), 2016. Vol. 12, no 1, 4
Keyword [en]
Arithmetic circuit, fast multiplication, lattice, zeta transform, Mobius transform, Mobius inversion, semigroup algebra
National Category
Computer Science
URN: urn:nbn:se:kth:diva-184971DOI: 10.1145/2629429ISI: 000371364600004ScopusID: 2-s2.0-84954312180OAI: diva2:917799
23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), JAN 17-19, 2012, Kyoto, JAPAN
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Available from: 2016-04-07 Created: 2016-04-07 Last updated: 2016-04-07Bibliographically approved

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Parviainen, Pekka
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