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Sorting in One and Two Rounds Using t-Comparators

Authors Ran Gelles , Zvi Lotker , Frederik Mallmann-Trenn

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Author Details

Ran Gelles
  • Bar-Ilan University, Ramat Gan, Israel
Zvi Lotker
  • Bar-Ilan University, Ramat Gan, Israel
Frederik Mallmann-Trenn
  • King’s College London, UK

Funding

  • Gelles, Ran: Research supported in part by the United States-Israel Binational Science Foundation (BSF) through Grant No. 2020277.
  • Mallmann-Trenn, Frederik: Was funded by the EPSRC grant EP/W005573/1.

Acknowledgements

R. Gelles would like to thank Paderborn University and CISPA - Helmholtz Center for Information Security for hosting him while part of this research was done. The authors would also like to thank the anonymous reviewers for multiple helpful comments.

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Ran Gelles, Zvi Lotker, and Frederik Mallmann-Trenn. Sorting in One and Two Rounds Using t-Comparators. In 38th International Symposium on Distributed Computing (DISC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 319, pp. 27:1-27:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.DISC.2024.27

Abstract

We examine sorting algorithms for n elements whose basic operation is comparing t elements simultaneously (a t-comparator). We focus on algorithms that use only a single round or two rounds - comparisons performed in the second round depend on the outcomes of the first round comparators. Algorithms with a small number of rounds are well-suited to distributed settings in which communication rounds are costly.
We design deterministic and randomized algorithms. In the deterministic case, we show an interesting relation to design theory (namely, to 2-Steiner systems), which yields a single-round optimal algorithm for n = t^{2^k} with any k ≥ 1 and a variety of possible values of t. For some values of t, however, no algorithm can reach the optimal (information-theoretic) bound on the number of comparators. For this case (and any other n and t), we show an algorithm that uses at most three times as many comparators as the theoretical bound. 
We also design a randomized Las-Vegas two-round sorting algorithm for any n and t. Our algorithm uses an asymptotically optimal number of O(max(n^{3/2}/t²,n/t)) comparators, with high probability, i.e., with probability at least 1-1/n. The analysis of this algorithm involves the gradual unveiling of randomness, using a novel technique which we coin the binary tree of deferred randomness.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sorting and searching
  • Mathematics of computing → Probabilistic algorithms
  • Theory of computation → Distributed algorithms
Keywords
  • Sorting
  • Steiner-System
  • Round Complexity
  • Deferred Randomness

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