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Towards Simpler Sorting Networks and Monotone Circuits for Majority

Authors Natalia Dobrokhotova-Maikova , Alexander Kozachinskiy , Vladimir Podolskii

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Author Details

Natalia Dobrokhotova-Maikova
  • Yandex, Tel Aviv, Israel
Alexander Kozachinskiy
  • IMFD & CENIA, Santiago, Chile
Vladimir Podolskii
  • Tufts University, Medford, MA, USA

Funding

  • Kozachinskiy, Alexander: funded by the National Center for Artificial Intelligence CENIA FB210017, Basal ANID, and by the Millennium Science Initiative Program - Code ICN17002.

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Natalia Dobrokhotova-Maikova, Alexander Kozachinskiy, and Vladimir Podolskii. Towards Simpler Sorting Networks and Monotone Circuits for Majority. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 50:1-50:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.50

Abstract

In this paper, we study the problem of computing the majority function by low-depth monotone circuits and a related problem of constructing low-depth sorting networks. We consider both the classical setting with elementary operations of arity 2 and the generalized setting with operations of arity k, where k is a parameter. For both problems and both settings, there are various constructions known, the minimal known depth being logarithmic. However, there is currently no known efficient deterministic construction that simultaneously achieves sub-log-squared depth, simplicity, and has a potential to be used in practice. In this paper we make progress towards resolution of this problem.
For computing majority by standard monotone circuits (gates of arity 2) we provide an explicit monotone circuit of depth O(log₂^{5/3} n). The construction is a combination of several known and not too complicated ideas. Essentially, for this result we gradually derandomize the construction of Valiant (1984). 
As one of the intermediate steps in our result we need an efficient construction of a sorting network with gates of arity k for arbitrary fixed k. For this we provide a new sorting network architecture inspired by representation of inputs as a high-dimensional cube. As a result we obtain a simple construction that improves previous upper bound of 4 log_k² n to 2 log_k² n. We prove the similar bound for the depth of the circuit computing majority of n bits consisting of gates computing majority of k bits. Note, that for both problems there is an explicit construction of depth O(log_k n) known, but the construction is complicated and the constant hidden in O-notation is huge.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
Keywords
  • Sorting networks
  • constant depth
  • lower bounds
  • threshold circuits

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