On Searching Zimin Patterns

TL;DR

This paper introduces an efficient linear-time algorithm for computing the Zimin type of words and explores its applications in pattern searching, including in Fibonacci words, with implications for pattern avoidability and bounds on pattern occurrence.

Contribution

It presents a linear-time on-line algorithm for computing Zimin types, relates Zimin types in Fibonacci words to Fibonacci numeration, and establishes bounds for pattern occurrence in k-ary words.

Findings

Linear-time algorithm for Zimin type computation

Logarithmic-time detection of Zimin patterns in Fibonacci prefixes

Bounds on pattern occurrence in k-ary words

Abstract

In the area of pattern avoidability the central role is played by special words called Zimin patterns. The symbols of these patterns are treated as variables and the rank of the pattern is its number of variables. Zimin type of a word $x$ is introduced here as the maximum rank of a Zimin pattern matching $x$. We show how to compute Zimin type of a word on-line in linear time. Consequently we get a quadratic time, linear-space algorithm for searching Zimin patterns in words. Then we how the Zimin type of the length $n$ prefix of the infinite Fibonacci word is related to the representation of $n$ in the Fibonacci numeration system. Using this relation, we prove that Zimin types of such prefixes and Zimin patterns inside them can be found in logarithmic time. Finally, we give some bounds on the function $f(n,k)$ such that every $k$-ary word of length at least $f(n,k)$ has a factor that matches the rank $n$ Zimin pattern.