Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants
Name
LIPIcs-IPEC-2017-6.pdf
Description
Published version
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501.67 KB
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Adobe PDF
Checksum (MD5)
7da5cf079fca8527f2f8e12d3ceb378f
Author(s)
Williams, Richard Ryan
Björklund, Andreas
Kaski, Petteri
Date Issued
2018
Journal
Leibniz International Proceedings in Informatics, LIPIcs
Citation
Williams, Richard Ryan, Björklund, Andreas and Kaski, Petteri. 2018. "Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants." Leibniz International Proceedings in Informatics, LIPIcs, 89.
Version
Final published version
Abstract
© 2018 Andreas Björklund, Petteri Kaski, and Ryan Williams. 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).
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.4230/LIPIcs.IPEC.2017.6
