Fast zeta transforms for lattices with few irreducibles

We investigate fast algorithms for changing between the standard basis and an orthogonal basis of idempotents for M¨obius 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 M¨obius 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.

• Type: Journal
• Year: 2012
• Where: SODA
• Authors: Andreas Björklund, Mikko Koivisto, Thore Husfeldt, Jesper Nederlof, Petteri Kaski, Pekka Parviainen
• Keywords: Algorithms, Arithmetic Circuits, Obius, SODA 2012, Vector Space