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Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants

Björklund, Andreas LU ; Kaski, Petteri and Williams, Ryan (2018) 12th International Symposium on Parameterized and Exact Computation, IPEC 2017 In 12th International Symposium on Parameterized and Exact Computation, IPEC 2017 89.
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),... (More)

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).

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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
Besicovitch set, Fermionant, Finite field, Finite vector space, Hamiltonian cycle, Homogeneous polynomial, Kakeya set, Permanent, Polynomial evaluation, Tabulation
in
12th International Symposium on Parameterized and Exact Computation, IPEC 2017
volume
89
publisher
Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
conference name
12th International Symposium on Parameterized and Exact Computation, IPEC 2017
external identifiers
  • scopus:85044722946
ISBN
9783959770514
DOI
10.4230/LIPIcs.IPEC.2017.6
language
English
LU publication?
yes
id
2e3ad8cf-adca-4644-9bf9-bccac635fc83
date added to LUP
2018-04-11 14:32:48
date last changed
2018-05-29 11:08:17
@inproceedings{2e3ad8cf-adca-4644-9bf9-bccac635fc83,
  abstract     = {<p>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)<sup>n+2</sup> tabulated values of P to produce the value of P at any of the qn points using O(nqd<sup>2</sup>) 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)<sup>n+s</sup> tabulated values to produce the value of P at any point using O(nq<sup>s</sup>sq) 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 2<sup>m-Ω(√m/log log m)</sup>, improving an earlier algorithm of Björklund (2016) that runs in time 2<sup>m-Ω(√m/logm)</sup>.</p>},
  author       = {Björklund, Andreas and Kaski, Petteri and Williams, Ryan},
  booktitle    = {12th International Symposium on Parameterized and Exact Computation, IPEC 2017},
  isbn         = {9783959770514},
  keyword      = {Besicovitch set,Fermionant,Finite field,Finite vector space,Hamiltonian cycle,Homogeneous polynomial,Kakeya set,Permanent,Polynomial evaluation,Tabulation},
  language     = {eng},
  month        = {02},
  publisher    = {Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing},
  title        = {Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants},
  url          = {http://dx.doi.org/10.4230/LIPIcs.IPEC.2017.6},
  volume       = {89},
  year         = {2018},
}