Explicit and Near-Optimal Construction of t-Rankwise Independent Permutations

Harvey, Nicholas; Sahami, Arvin

doi:10.4230/LIPIcs.APPROX/RANDOM.2024.67

File

LIPIcs.APPROX-RANDOM.2024.67.pdf

Filesize: 0.69 MB
12 pages

Document Identifiers

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.67
URN: urn:nbn:de:0030-drops-210600

Author Details

Nicholas Harvey

Department of Computer Science and Department of Mathematics, University of British Columbia, Vancouver, Canada

Arvin Sahami

Department of Computer Science and Department of Mathematics, University of British Columbia, Vancouver, Canada

Cite AsGet BibTex

Nicholas Harvey and Arvin Sahami. Explicit and Near-Optimal Construction of t-Rankwise Independent Permutations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 67:1-67:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.67

Abstract

Letting t ≤ n, a family of permutations of [n] = {1,2,…, n} is called t-rankwise independent if for any t distinct entries in [n], when a permutation π is sampled uniformly at random from the family, the order of the t entries in π is uniform among the t! possibilities. Itoh et al. show a lower bound of (n/2)^⌊t/4⌋ for the number of members in such a family, and provide a construction of a t-rankwise independent permutation family of size n^O(t^2/ln(t)). We provide an explicit, deterministic construction of a t-rankwise independent family of size n^O(t) for arbitrary parameters t ≤ n. Our main ingredient is a way to make the elements of a t-independent family "more injective", which might be of independent interest.

Subject Classification

ACM Subject Classification

Theory of computation → Pseudorandomness and derandomization

Keywords

Rankwise independent permutations

Metrics

Access Statistics
Total Accesses (updated on a weekly basis)

0

PDF Downloads

0

Metadata Views

References

Andrei Z. Broder, Moses Charikar, Alan M. Frieze, and Michael Mitzenmacher. Min-wise independent permutations. Journal of Computer and System Sciences, 60(3):630-659, 2000. URL: https://doi.org/10.1006/jcss.1999.1690.
J. Lawrence Carter and Mark N. Wegman. Universal classes of hash functions. Journal of Computer and System Sciences, 18:143-154, 1979. URL: https://doi.org/10.1016/0022-0000(79)90044-8.
Venkat Guruswami, Atri Rudra, and Madhu Sudan. Essential Coding Theory, 2018. Manuscript.
Nicholas Harvey and Arvin Sahami. Explicit orthogonal arrays and universal hashing with arbitrary parameters. In Proceedings of the ACM Symposium on Theory of Computation (STOC), 2024.
Toshiya Itoh, Yoshinori Takei, and Jun Tarui. On permutations with limited independence. In Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '00, pages 137-146, USA, 2000. Society for Industrial and Applied Mathematics.
Enrico Iurlano. Growth of the perfect sequence covering array number. Des. Codes Cryptography, 91(4):1487-1494, December 2022. URL: https://doi.org/10.1007/s10623-022-01168-3.
Greg Kuperberg, Shachar Lovett, and Ron Peled. Probabilistic existence of rigid combinatorial structures. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 1091-1106, 2012. URL: https://doi.org/10.1145/2213977.2214075.
Rajeev Motwani and Prabhakar Raghavan. Randomized Algorithms. Cambridge University Press, 1995.
Salil P. Vadhan. Pseudorandomness. Foundations and Trends® in Theoretical Computer Science, 7(1–3):1-336, 2012. URL: https://doi.org/10.1561/0400000010.
Raphael Yuster. Perfect sequence covering arrays. Des. Codes Cryptography, 88(3):585-593, March 2020. URL: https://doi.org/10.1007/s10623-019-00698-7.