Fast zeta transforms for lattices with few irreducibles

A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, J. Nederlof, P. Parviainen

Research output: Contribution to journalArticleAcademicpeer-review

6 Citations (Scopus)

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) 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.

Original languageEnglish
Article number4
JournalACM Transactions on Algorithms
Volume12
Issue number1
DOIs
Publication statusPublished - 1 Dec 2015

Keywords

  • Arithmetic circuit
  • Fast multiplication
  • Lattice
  • Möbius inversion
  • Möbius transform
  • Semigroup algebra
  • Zeta transform

Fingerprint Dive into the research topics of 'Fast zeta transforms for lattices with few irreducibles'. Together they form a unique fingerprint.

Cite this