On the Fixed-Length-Burst Levenshtein Ball with Unit Radius

TL;DR

This paper investigates the structure and size of fixed-length Levenshtein balls with multiple consecutive deletions and insertions, providing explicit formulas, bounds, and concentration results for these combinatorial objects.

Contribution

It extends the analysis of Levenshtein balls to multiple consecutive edits, offering explicit formulas and bounds for their sizes, which was previously limited to single-edit cases.

Findings

Derived explicit formulas for the size of fixed-length burst Levenshtein balls.

Established extremal bounds for the minimum and maximum sizes.

Analyzed the expected size and its concentration properties.

Abstract

Consider a length-$n$ sequence $\bm{x}$ over a $q$-ary alphabet. The \emph{fixed-length Levenshtein ball} $\mathcal{L}_t(\bm{x})$ of radius $t$ encompasses all length-$n$ $q$-ary sequences that can be derived from $\bm{x}$ by performing $t$ deletions followed by $t$ insertions. Analyzing the size and structure of these balls presents significant challenges in combinatorial coding theory. Recent studies have successfully characterized fixed-length Levenshtein balls in the context of a single deletion and a single insertion. These works have derived explicit formulas for various key metrics, including the exact size of the balls, extremal bounds (minimum and maximum sizes), as well as expected sizes and their concentration properties. However, the general case involving an arbitrary number of $t$ deletions and $t$ insertions $(t>1)$ remains largely uninvestigated. This work systematically examines fixed-length Levenshtein balls with multiple deletions and insertions, focusing specifically on \emph{fixed-length burst Levenshtein balls}, where deletions occur consecutively, as do insertions. We provide comprehensive solutions for explicit cardinality formulas, extremal bounds (minimum and maximum sizes), expected size, and concentration properties surrounding the expected value.