On the size of the neighborhoods of a word

TL;DR

This paper derives exact formulas and bounds for the sizes of neighborhoods of words under Levenshtein distance, aiding the analysis of approximate pattern matching algorithms.

Contribution

It provides exact formulas for unary words and new upper bounds for neighborhoods of arbitrary words, confirming a conjecture on maximum neighborhood size.

Findings

Exact formulas for unary word neighborhoods.

New upper bounds for neighborhoods of arbitrary words.

Proof of a conjectured upper bound.

Abstract

The d-neighborhood of a word W in the Levenshtein distance is the set of all words at distance at most d from W. Generating the neighborhood of a word W, or related sets of words such as the condensed neighborhood or the super-condensed neighborhood has applications in the design of approximate pattern matching algorithms. It follows that bounds on the maximum size of the neighborhood of words of a given length can be used in the complexity analysis of such approximate pattern matching algorithms. In this note, we present exact formulas for the size of the condensed and super condensed neighborhoods of a unary word, a novel upper bound for the maximum size of the condensed neighborhood of an arbitrary word of a given length, and we prove a conjectured upper bound again for the maximum size of the condensed neighborhood of an arbitrary word of a given length.