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Depth-13 Sorting Networks for 28 Channels

Chengu Wang

TL;DR

This work tightens the depth upper bound for sorting networks on small even channel counts by achieving a depth-$13$ network for $n=28$ (and a corresponding result for $n=27$). The authors leverage reflection symmetry to prune the search space, construct high-quality $16$-channel and $12$-channel prefixes, and greedily extend to 6 layers before solving the remaining layers with a SAT solver. Key contributions include a complete depth-$13$ construction for $28$ channels, a demonstration that such networks can be found within a few tens of minutes on a standard desktop, and a practical framework combining generate-and-prune with SAT solving. The results have direct implications for efficient hardware implementations of oblivious sorting and related cryptographic protocols, where small-depth networks are advantageous.

Abstract

We establish new depth upper bounds for sorting networks on 27 and 28 channels, improving the previous best bound of 14 to 13. Our 28-channel network is constructed with reflectional symmetry by combining high-quality prefixes of 16- and 12-channel networks, extending them greedily one comparator at a time, and using a SAT solver to complete the remaining layers.

Depth-13 Sorting Networks for 28 Channels

TL;DR

This work tightens the depth upper bound for sorting networks on small even channel counts by achieving a depth- network for (and a corresponding result for ). The authors leverage reflection symmetry to prune the search space, construct high-quality -channel and -channel prefixes, and greedily extend to 6 layers before solving the remaining layers with a SAT solver. Key contributions include a complete depth- construction for channels, a demonstration that such networks can be found within a few tens of minutes on a standard desktop, and a practical framework combining generate-and-prune with SAT solving. The results have direct implications for efficient hardware implementations of oblivious sorting and related cryptographic protocols, where small-depth networks are advantageous.

Abstract

We establish new depth upper bounds for sorting networks on 27 and 28 channels, improving the previous best bound of 14 to 13. Our 28-channel network is constructed with reflectional symmetry by combining high-quality prefixes of 16- and 12-channel networks, extending them greedily one comparator at a time, and using a SAT solver to complete the remaining layers.

Paper Structure

This paper contains 8 sections, 8 theorems, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Redundant Prefixes
  3. Construction of Depth-13 Sorting Network for 28 Channels
  4. High-Quality 16-Channel and 12-Channel 5-Layer Prefixes
  5. Extending to 6 Layers
  6. SAT Solving the Remaining Layers
  7. Implementation and Running Time
  8. Conclusion

Key Result

Lemma 1

If a comparator network on $n$ channels sorts all $2^n$ Boolean sequences into non-decreasing order, then it correctly sorts all sequences of arbitrary (comparable) values. knuth1997artv3

Figures (3)

  • Figure 1: A sorting network for 4 channels of depth 3.
  • Figure 2: A sorting network for 28 channels with 13 layers
  • Figure 3: The first 5 layers of Van Voorhis's sorting network for 16 channels

Theorems & Definitions (9)

  • Lemma 1: Zero-One Principle
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Corollary 1
  • Corollary 2
  • Theorem 1
  • Corollary 3