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Generalized Kakeya Sets for Polynomial Evaluation and Faster Computation of Fermionants

Authors Andreas Björklund, Petteri Kaski, Ryan Williams

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Andreas Björklund
Petteri Kaski
Ryan Williams

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Andreas Björklund, Petteri Kaski, and Ryan Williams. Generalized Kakeya Sets for Polynomial Evaluation and Faster Computation of Fermionants. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 6:1-6:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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(nqd^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(nq^ssq) 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-by-m integer matrix with entries bounded in absolute value by a constant can be computed in time 2^{m-Omega(sqrt(m/log log m))}, improving an earlier algorithm of Bjorklund (2016) that runs in time 2^{m-Omega(sqrt(m/log m))}.
  • Besicovitch set
  • fermionant
  • finite field
  • finite vector space
  • Hamiltonian cycle
  • homogeneous polynomial
  • Kakeya set
  • permanent
  • polynomial evaluatio


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  1. Eric T. Bax and Joel Franklin. A finite-difference sieve to count paths and cycles by length. Inf. Process. Lett., 60(4):171-176, 1996. URL:
  2. A. S. Besicovitch. On Kakeya’s problem and a similar one. Math. Z., 27(1):312-320, 1928. URL:
  3. Andreas Björklund. Below all subsets for some permutational counting problems. In Rasmus Pagh, editor, 15th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2016, June 22-24, 2016, Reykjavik, Iceland, volume 53 of LIPIcs, pages 17:1-17:11. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL:
  4. Andreas Björklund, Thore Husfeldt, and Isak Lyckberg. Computing the permanent modulo a prime power. Inf. Process. Lett., 125:20-25, 2017. URL:
  5. Andreas Björklund and Petteri Kaski. How proofs are prepared at camelot: Extended abstract. In George Giakkoupis, editor, Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25-28, 2016, pages 391-400. ACM, 2016. URL:
  6. Andreas Björklund, Petteri Kaski, and Ioannis Koutis. Directed hamiltonicity and out-branchings via generalized laplacians. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 91:1-91:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL:
  7. Shailesh Chandrasekharan and Uwe-Jens Wiese. Partition functions of strongly correlated electron systems as "fermionants". cond-mat.str-el, abs/1108.2461, 2011. URL:
  8. Marek Cygan and Marcin Pilipczuk. Faster exponential-time algorithms in graphs of bounded average degree. In Fedor V. Fomin, Rusins Freivalds, Marta Z. Kwiatkowska, and David Peleg, editors, Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, volume 7965 of Lecture Notes in Computer Science, pages 364-375. Springer, 2013. URL:
  9. Zeev Dvir. From randomness extraction to rotating needles. Electronic Colloquium on Computational Complexity (ECCC), 16:77, 2009. URL:
  10. Zeev Dvir. Incidence theorems and their applications. CoRR, abs/1208.5073, 2012. URL:
  11. Kiran S. Kedlaya and Christopher Umans. Fast polynomial factorization and modular composition. SIAM J. Comput., 40(6):1767-1802, 2011. URL:
  12. Donald E. Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms. Addison-Wesley, 3rd edition, 1998. Google Scholar
  13. Swastik Kopparty, Vsevolod F. Lev, Shubhangi Saraf, and Madhu Sudan. Kakeya-type sets in finite vector spaces. J. Algebraic Combin., 34(3):337-355, 2011. URL:
  14. Gohar Kyureghyan, Peter Müller, and Qi Wang. On the size of Kakeya sets in finite vector spaces. Electron. J. Combin., 20(3):#P36, 2013. URL:
  15. J. M. Landsberg. An introduction to geometric complexity theory. Eur. Math. Soc. Newsl., 99:10-18, 2016. Google Scholar
  16. Carsten Lund, Lance Fortnow, Howard J. Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. J. ACM, 39(4):859-868, 1992. URL:
  17. Stephan Mertens and Cristopher Moore. The complexity of the fermionant, and immanants of constant width. CoRR, abs/1110.1821, 2011. URL:
  18. Gerd Mockenhaupt and Terence Tao. Restriction and Kakeya phenomena for finite fields. Duke Math. J., 121(1):35-74, 2004. URL:
  19. H. J. Ryser. Combinatorial Mathematics. Number 14 in The Carus Mathematical Monographs. Mathematical Association of America, 1963. Google Scholar
  20. Shubhangi Saraf and Madhu Sudan. An improved lower bound on the size of Kakeya sets over finite fields. Anal. PDE, 1(3):375-379, 2008. URL:
  21. Gérald Tenenbaum. Introduction to Analytic and Probabilistic Number Theory, volume 163 of Graduate Studies in Mathematics. American Mathematical Society, 3rd edition, 2015. Google Scholar
  22. Leslie G. Valiant. Completeness classes in algebra. In Michael J. Fischer, Richard A. DeMillo, Nancy A. Lynch, Walter A. Burkhard, and Alfred V. Aho, editors, Proceedings of the 11h Annual ACM Symposium on Theory of Computing, April 30 - May 2, 1979, Atlanta, Georgia, USA, pages 249-261. ACM, 1979. URL:
  23. Leslie G. Valiant. The complexity of computing the permanent. Theor. Comput. Sci., 8:189-201, 1979. URL:
  24. Joachim von zur Gathen and Jürgen Gerhard. Modern Computer Algebra. Cambridge University Press, Cambridge, third edition, 2013. URL:
  25. Richard Ryan Williams. Strong ETH breaks with merlin and arthur: Short non-interactive proofs of batch evaluation. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 2:1-2:17. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL:
  26. Thomas Wolff. Recent work connected with the Kakeya problem. In Prospects in Mathematics (Princeton, NJ, 1996), pages 129-162. Amer. Math. Soc., Providence, RI, 1999. Google Scholar
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