Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants

Andreas Björklund, Petteri Kaski, Ryan Williams

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review

2 Citations (Scopus)
82 Downloads (Pure)

Abstract

We present two new data structures for computing values of an n-variate polynomial P of degree at most d over a finite field of q elements. Assuming that d divides q-1, our first data structure relies on (d+1)n+2 tabulated values of P to produce the value of P at any of the qn points using O(nqd2) arithmetic operations in the finite field. Assuming that s divides d and d/s divides q-1, our second data structure assumes that P satisfies a degree-separability condition and relies on (d/s + 1)n+s tabulated values to produce the value of P at any point using O(nqssq) arithmetic operations. Our data structures are based on generalizing upper-bound constructions due to Mockenhaupt and Tao (2004), Saraf and Sudan (2008), and Dvir (2009) for Kakeya sets in finite vector spaces from linear to higher-degree polynomial curves. As an application we show that the new data structures enable a faster algorithm for computing integer-valued fermionants, a family of self-reducible polynomial functions introduced by Chandrasekharan and Wiese (2011) that captures numerous fundamental algebraic and combinatorial invariants such as the determinant, the permanent, the number of Hamiltonian cycles in a directed multigraph, as well as certain partition functions of strongly correlated electron systems in statistical physics. In particular, a corollary of our main theorem for fermionants is that the permanent of an m × m integer matrix with entries bounded in absolute value by a constant can be computed in time 2m-Ω(√m/log log m), improving an earlier algorithm of Björklund (2016) that runs in time 2m-Ω(√m/logm).

Original languageEnglish
Title of host publication12th International Symposium on Parameterized and Exact Computation, IPEC 2017
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages1-13
ISBN (Electronic)9783959770514
DOIs
Publication statusPublished - 1 Feb 2018
MoE publication typeA4 Article in a conference publication
EventInternational Symposium on Parameterized and Exact Computation - Vienna, Austria
Duration: 6 Sep 20178 Sep 2017
Conference number: 12

Publication series

NameLeibniz International Proceedings in Informatics
PublisherSchloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH
Volume89
ISSN (Electronic)1868-8969

Conference

ConferenceInternational Symposium on Parameterized and Exact Computation
Abbreviated titleIPEC
CountryAustria
CityVienna
Period06/09/201708/09/2017

Keywords

  • Besicovitch set
  • Fermionant
  • Finite field
  • Finite vector space
  • Hamiltonian cycle
  • Homogeneous polynomial
  • Kakeya set
  • Permanent
  • Polynomial evaluation
  • Tabulation

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