Depth-13 Sorting Networks for 28 Channels

2025-11-06

AI Summary

This work addresses the problem of improving the depth upper bounds for 27- and 28-channel sorting networks. We propose a constructive method integrating reflection-symmetric architecture design, reuse of high-quality prefixes (16-channel and 12-channel), and greedy comparator extension, followed by formal verification and completion of remaining layers using a SAT solver. Our key contribution is the first reduction of the depth upper bound for 28-channel sorting networks from 14 to 13, concurrently improving the bound for 27-channel networks. The resulting depth-13 28-channel network is currently optimal, achieving significant gains in both construction efficiency and structural quality. This advance tightens the theoretical depth bounds for small-scale sorting networks and establishes a novel paradigm for combinatorial construction leveraging symmetry-aware design and formal methods.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Establishing new depth upper bounds for sorting networks
• Improving sorting network depth from 14 to 13 layers
• Constructing 28-channel networks using SAT solver optimization

Innovation

• Depth-13 sorting networks for 28 channels
• Combining prefixes from smaller networks symmetrically
• Using SAT solver to complete remaining layers