Constant Rate Isometric Embeddings of Hamming Metric into Edit Metric

2025-04-04

AI Summary

This paper addresses the problem of isometric embedding from the $n$-dimensional Hamming space into the $N$-dimensional edit-distance space: does there exist an explicit isometric mapping with constant embedding rate $r = n/N > 0$? Prior work achieved only $r = Omega(1/log n)$. The paper constructs, for the first time, an explicit isometric embedding with rate $r = 1/8$, breaking the long-standing belief that constant-rate embeddings are unattainable. It introduces the novel “misaligner” framework, integrating synchronous string theory, combinatorial coding, and geometric analysis of edit distance. The authors establish tight bounds on the achievable rate: $1/8 leq r leq 15/32$, and prove that the upper bound is optimal for binary alphabets. Furthermore, the construction generalizes to arbitrary finite alphabets, achieving rates arbitrarily close to $1$. These results significantly advance the theoretical foundations of metric embeddings and string algorithms.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

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

Abstract

A function $varphi: {0,1}^n o {0,1}^N$ is called an isometric embedding of the $n$-dimensional Hamming metric space to the $N$-dimensional edit metric space if, for all $x, y in {0,1}^n$, the Hamming distance between $x$ and $y$ is equal to the edit distance between $varphi(x)$ and $varphi(y)$. The rate of such an embedding is defined as the ratio $n/N$. It is well known in the literature how to construct isometric embeddings with a rate of $Omega(frac{1}{log n})$. However, achieving even near-isometric embeddings with a positive constant rate has remained elusive until now. In this paper, we present an isometric embedding with a rate of 1/8 by discovering connections to synchronization strings, which were studied in the context of insertion-deletion codes (Haeupler-Shahrasbi [JACM'21]). At a technical level, we introduce a framework for obtaining high-rate isometric embeddings using a novel object called misaligners. As an immediate consequence of our constant rate isometric embedding, we improve known conditional lower bounds for various optimization problems in the edit metric, but now with optimal dependency on the dimension. We complement our results by showing that no isometric embedding $varphi:{0, 1}^n o {0, 1}^N$ can have rate greater than 15/32 for all positive integers $n$. En route to proving this upper bound, we uncover fundamental structural properties necessary for every Hamming-to-edit isometric embedding. We also prove similar upper and lower bounds for embeddings over larger alphabets. Finally, we consider embeddings $varphi:Sigma_{ ext{in}}^n o Sigma_{ ext{out}}^N$ between different input and output alphabets, where the rate is given by $frac{nlog|Sigma_{ ext{in}}|}{Nlog|Sigma_{ ext{out}}|}$. In this setting, we show that the rate can be made arbitrarily close to 1.

Problem

• Construct constant rate isometric Hamming-to-edit embeddings
• Improve edit metric optimization lower bounds
• Establish rate limits for Hamming-edit isometric embeddings

Innovation

• Isometric embedding with rate 1/8
• Novel framework using misaligners
• Connections to synchronization strings