Blocksequences of k-local Words

TL;DR

This paper explores the concept of blocksequences in k-local words, introducing extended blocksequences to better understand their structure and providing combinatorial results related to these sequences.

Contribution

It introduces the notion of extended blocksequences for k-local words and presents new combinatorial results on their structure and properties.

Findings

Words with the same blocksequence are loosely connected

Extended blocksequences capture more detailed block information

New combinatorial results on extended blocksequences

Abstract

The locality of words is a relatively young structural complexity measure, introduced by Day et al. in 2017 in order to define classes of patterns with variables which can be matched in polynomial time. The main tool used to compute the locality of a word is called marking sequence: an ordering of the distinct letters occurring in the respective order. Once a marking sequence is defined, the letters of the word are marked in steps: in the ith marking step, all occurrences of the ith letter of the marking sequence are marked. As such, after each marking step, the word can be seen as a sequence of blocks of marked letters separated by blocks of non-marked letters. By keeping track of the evolution of the marked blocks of the word through the marking defined by a marking sequence, one defines the blocksequence of the respective marking sequence. We first show that the words sharing the same blocksequence are only loosely connected, so we consider the stronger notion of extended blocksequence, which stores additional information on the form of each single marked block. In this context, we present a series of combinatorial results for words sharing the extended blocksequence.