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Fast multiplication of matrices over a finitely generated semiring
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
2008 (English)In: Information Processing Letters, ISSN 0020-0190, Vol. 107, no 6, 230-234 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we show that n×n matrices with entries from a semiring R which is generated additively by q generators can be multiplied in time O(q2nω), where nω is the complexity for matrix multiplication over a ring (Strassen: ω<2.807, Coppersmith and Winograd: ω<2.376).

We first present a combinatorial matrix multiplication algorithm for the case of semirings with q elements, with complexity O(n3/log2qn), matching the best known methods in this class.

Next we show how the ideas used can be combined with those of the fastest known boolean matrix multiplication algorithms to give an O(q2nω) algorithm for matrices of, not necessarily finite, semirings with q additive generators.

For finite semirings our combinatorial algorithm is simple enough to be a practical algorithm and is expected to be faster than the O(q2nω) algorithm for matrices of practically relevant sizes.

Place, publisher, year, edition, pages
2008. Vol. 107, no 6, 230-234 p.
Keyword [en]
Matrix multiplication, Semirings
URN: urn:nbn:se:umu:diva-2348DOI: doi:10.1016/j.ipl.2008.03.004OAI: diva2:140303
Available from: 2007-05-10 Created: 2007-05-10 Last updated: 2009-08-24Bibliographically approved
In thesis
1. On the Ising problem and some matrix operations
Open this publication in new window or tab >>On the Ising problem and some matrix operations
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The first part of the dissertation concerns the Ising problem proposed to Ernst Ising by his supervisor Wilhelm Lenz in the early 20s. The Ising model, or perhaps more correctly the Lenz-Ising model, tries to capture the behaviour of phase transitions, i.e. how local rules of engagement can produce large scale behaviour.

Two decades later Lars Onsager solved the Ising problem for the quadratic lattice without an outer field. Using his ideas solutions for other lattices in two dimensions have been constructed. We describe a method for calculating the Ising partition function for immense square grids, up to linear order 320 (i.e. 102400 vertices).

In three dimensions however only a few results are known. One of the most important unanswered questions is at which temperature the Ising model has its phase transition. In this dissertation it is shown that an upper bound for the critical coupling Kc, the inverse absolute temperature, is 0.29 for the tree dimensional cubic lattice.

To be able to get more information one has to use different statistical methods. We describe one sampling method that can use simple state generation like the Metropolis algorithm for large lattices. We also discuss how to reconstruct the entropy from the model, in order to obtain parameters as the free energy.

The Ising model gives a partition function associated with all finite graphs. In this dissertation we show that a number of interesting graph invariants can be calculated from the coefficients of the Ising partition function. We also give some interesting observations about the partition function in general and show that there are, for any N, N non-isomorphic graphs with the same Ising partition function.

The second part of the dissertation is about matrix operations. We consider the problem of multiplying them when the entries are elements in a finite semiring or in an additively finitely generated semiring. We describe a method that uses O(n3 / log n) arithmetic operations.

We also consider the problem of reducing n x n matrices over a finite field of size q using O(n2 / logq n) row operations in the worst case.

Place, publisher, year, edition, pages
Umeå: Matematik och matematisk statistik, 2007. 7 p.
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 37
Ising problem, phase tansition, matrix multiplicatoin, matrix inversion
National Category
Discrete Mathematics
urn:nbn:se:umu:diva-1129 (URN)987-91-7264-323-9 (ISBN)
Public defence
2007-05-31, MA 121, MIT, Umeå, 13:15 (English)
Available from: 2007-05-10 Created: 2007-05-10 Last updated: 2011-04-21Bibliographically approved

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