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On the size distribution of Levenshtein balls with radius one

Geyang Wang, Qi Wang

TL;DR

The paper investigates the distribution of fixed-length Levenshtein radius-one ball sizes in $\mathbb{Z}_m^n$, building on exact minimum, maximum, and average sizes established by Bar-Lev et al. It proves strong concentration of $|L_1(\boldsymbol{x})|$ around its mean using Azuma's inequality via a Doob martingale, deriving explicit tail bounds for both the binary and $m$-ary alphabets and providing the mean formula $\mathbb{E}[|L_1(\boldsymbol{x})|]$.$ The results quantify the variability of radius-one FLL balls and enhance understanding of synchronization-error-correcting measures in coding theory, with simulation results suggesting the bounds may be conservative and offering directions for sharper bounds and extensions to larger radii.$

Abstract

The fixed length Levenshtein (FLL) distance between two words $\mathbf{x,y} \in \mathbb{Z}_m^n$ is the smallest integer $t$ such that $\mathbf{x}$ can be transformed to $\mathbf{y}$ by $t$ insertions and $t$ deletions. The size of a ball in FLL metric is a fundamental but challenging problem. Very recently, Bar-Lev, Etzion, and Yaakobi explicitly determined the minimum, maximum and average sizes of the FLL balls with radius one. In this paper, based on these results, we further prove that the size of the FLL balls with radius one is highly concentrated around its mean by Azuma's inequality.

On the size distribution of Levenshtein balls with radius one

TL;DR

The paper investigates the distribution of fixed-length Levenshtein radius-one ball sizes in , building on exact minimum, maximum, and average sizes established by Bar-Lev et al. It proves strong concentration of around its mean using Azuma's inequality via a Doob martingale, deriving explicit tail bounds for both the binary and -ary alphabets and providing the mean formula .

Abstract

The fixed length Levenshtein (FLL) distance between two words is the smallest integer such that can be transformed to by insertions and deletions. The size of a ball in FLL metric is a fundamental but challenging problem. Very recently, Bar-Lev, Etzion, and Yaakobi explicitly determined the minimum, maximum and average sizes of the FLL balls with radius one. In this paper, based on these results, we further prove that the size of the FLL balls with radius one is highly concentrated around its mean by Azuma's inequality.
Paper Structure (8 sections, 10 theorems, 43 equations, 1 figure)

This paper contains 8 sections, 10 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The size distribution of the FLL balls with radius one
  4. The binary case
  5. The $m$-ary case
  6. Conclusion
  7. Proof of Eq. \ref{['equ: bounded difference']}
  8. Simulation results

Key Result

Lemma 2.2

Let $n>0$, $m>1$ both be integers. If $\boldsymbol{x}$ is picked uniformly at random, then we have

Figures (1)

  • Figure 1: $N = 5000$ for each experiment.

Theorems & Definitions (21)

  • Example 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3: Fixed-length Levenshtein (FLL) distance
  • Definition 2.4: FLL ball
  • Lemma 2.5
  • Lemma 2.6
  • Remark 2.7
  • Theorem 2.8
  • Theorem 2.9: Azuma's inequality
  • ...and 11 more