2D Lyndon Words and Applications

TL;DR

This paper extends the concept of Lyndon words to two dimensions, introducing 2D Lyndon words to analyze matrix periodicity and improve 2D pattern matching efficiency.

Contribution

It defines 2D Lyndon words and develops linear-time algorithms for their computation, enabling efficient 2D pattern matching and periodicity analysis.

Findings

Linear time algorithm for 2D Lyndon word computation

Efficient 2D dictionary matching algorithm

Handling exponential period least common multiples

Abstract

A Lyndon word is a primitive string which is lexicographically smallest among cyclic permutations of its characters. Lyndon words are used for constructing bases in free Lie algebras, constructing de Bruijn sequences, finding the lexicographically smallest or largest substring in a string, and succinct suffix-prefix matching of highly periodic strings. In this paper, we extend the concept of the Lyndon word to two dimensions. We introduce the 2D Lyndon word and use it to capture 2D horizontal periodicity of a matrix in which each row is highly periodic, and to efficiently solve 2D horizontal suffix-prefix matching among a set of patterns. This yields a succinct and efficient algorithm for 2D dictionary matching. We present several algorithms that compute the 2D Lyndon word that represents a matrix. The final algorithm achieves linear time complexity even when the least common multiple of the periods of the rows is exponential in the matrix width.