Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants

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 q n points usi...

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Vydáno v:Algorithmica Ročník 81; číslo 10; s. 4010 - 4028
Hlavní autoři: Björklund, Andreas, Kaski, Petteri, Williams, Ryan
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.10.2019
Springer Nature B.V
Témata:
Algorithm Analysis and Problem Complexity
Algorithms
Arithmetic
Combinatorial analysis
Computation
Computer and Information Sciences
Computer Science
Computer Sciences
Computer Systems Organization and Communication Networks
Data structures
Data Structures and Information Theory
Data- och informationsvetenskap (Datateknik)
Datavetenskap (Datalogi)
Fields (mathematics)
Functions (mathematics)
Integers
Mathematical analysis
Mathematics of Computing
Natural Sciences
Naturvetenskap
Partitions
Partitions (mathematics)
Polynomials
Special Issue: Parameterized and Exact Computation
Theorems
Theory of Computation
Upper bounds
Vector spaces
Finite field
Finite vector space
Hamiltonian cycle
Permanent
Besicovitch set
Tabulation
Fermionant
Homogeneous polynomial
Kakeya set
Polynomial evaluation
ISSN:0178-4617, 1432-0541, 1432-0541
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Popis
Shrnutí: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 q n points using O ( n q d 2 ) 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 n q s s q arithmetic operations. Our data structures are based on generalizing upper-bound constructions due to Mockenhaupt and Tao (Duke Math J 121(1):35–74, 2004 ), Saraf and Sudan (Anal PDE 1(3):375–379, 2008 ) and Dvir (Incidence theorems and their applications, 2012 . arXiv:1208.5073 ) 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 (Partition functions of strongly correlated electron systems as fermionants, 2011 . arXiv:1108.2461v1 ) that captures numerous fundamental algebraic and combinatorial functions 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 m - Ω m / log log m , improving an earlier algorithm of Björklund (in: Proceedings of the 15th SWAT, vol 17, pp 1–11, 2016 ) that runs in time 2 m - Ω m / log m .
Bibliografie:ObjectType-Article-1
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ISSN:0178-4617
1432-0541
1432-0541
DOI:10.1007/s00453-018-0513-7