Clique-Based Deletion-Correcting Codes via Penalty-Guided Clique Search

2026-02-09

AI Summary

This work addresses the problem of correcting up to $d$ deletion errors in the binary deletion channel by formulating the construction of $d$-deletion-correcting codes as a maximum clique problem. The authors introduce, for the first time, a Penalty-Guided Clique Search (PGCS) algorithm to efficiently construct large codebooks at scale. Furthermore, they design a low-complexity exact decoder based on the Longest Common Subsequence (LCS) distance, incorporating symbol-count filtering and early termination strategies to significantly reduce decoding overhead. Experimental results for block lengths 8–14 and deletion counts 1–3 demonstrate that the constructed codebooks outperform existing graph-based heuristic methods, with some achieving known optimality, thereby validating the efficacy and superiority of the proposed approach.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search

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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the construction of $d$-deletion-correcting binary codes by formulating the problem as a Maximum Clique Problem (MCP). In this formulation, vertices represent candidate codewords and edges connect pairs whose longest common subsequence (LCS) distance guarantees correction of up to $d$ deletions. A valid codebook corresponds to a clique in the resulting graph, and finding the largest codebook is equivalent to identifying a maximum clique. While MCP-based formulations for deletion-correcting codes have previously been explored, we demonstrate that applying Penalty-Guided Clique Search (PGCS), a lightweight stochastic clique-search heuristic inspired by Dynamic Local Search (DLS), consistently yields larger codebooks than existing graph-based heuristics, including minimum-degree and coloring methods, for block lengths $n = 8,9,\dots,14$ and deletion parameters $d = 1,2,3$. In several finite-length regimes, the resulting codebooks match known optimal sizes and outperform classical constructions such as Helberg codes. For decoding under segmented reception, where codeword boundaries are known, we propose an optimized LCS-based decoder that exploits symbol-count filtering and early termination to substantially reduce the number of LCS evaluations while preserving exact decoding guarantees. These optimizations lead to significantly lower average-case decoding complexity than the baseline $O(|C| n^2)$ approach.

Problem

• deletion-correcting codes
• maximum clique problem
• codebook construction
• LCS distance
• binary codes

Innovation

• Penalty-Guided Clique Search
• deletion-correcting codes
• Maximum Clique Problem
• LCS-based decoding
• codebook construction