A Generalized Hamming Distance of Sequence Patterns

TL;DR

This paper introduces a generalized Hamming distance for sequence patterns, considering equivalence classes under symbol relabeling, and provides methods to compute maximal and pairwise distances.

Contribution

It defines a new distance measure for sequence patterns based on equivalence classes and offers formulas for maximal and pairwise distances.

Findings

Derived the maximal distance for sets of sequence patterns.

Provided exact calculation methods for the distance between two sequence patterns.

Extended Hamming distance to equivalence classes of sequences.

Abstract

We define sequence patterns of length $n$ and level $\ell$ to be equivalence classes of sequences that have $n$ elements from the set of $\ell$ integer symbols $\{1,2,\ldots,\ell\}$ with no restriction on repetition, where the equivalence relation is induced by symbol relabeling without swapping positions of symbols. We define a distance for a set of $k$ sequence patterns of length $n$ and level $\ell$ by generalizing the Hamming distance between sequences. We compute the maximal distance for $k$ sequence patterns of length $n$ and level $\ell$ and demonstrate how to calculate the exact distance between a pair of length-$n$ level-$\ell$ sequence patterns.