LIPIcs.APPROX-RANDOM.2024.67.pdf
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Explicit and Near-Optimal Construction of t-Rankwise Independent Permutations
Authors
Nicholas Harvey 
,
Arvin Sahami
-
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Volume:
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-09-16
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Author Details
- Department of Computer Science and Department of Mathematics, University of British Columbia, Vancouver, Canada
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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
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References
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