The Inverse Lyndon Array: Definition, Properties, and Linear-Time Construction

TL;DR

This paper introduces the inverse Lyndon array, characterizes it using suffix arrays and LCE values, and provides a linear-time construction algorithm with practical efficiency demonstrated through experiments.

Contribution

It defines the inverse Lyndon array, establishes its theoretical properties, and develops a linear-time construction algorithm for general alphabets.

Findings

The inverse Lyndon array can be characterized via next greater suffix and border correction.

An O(n)-time algorithm for constructing the inverse Lyndon array is proposed.

Experimental results confirm practical linear-time performance similar to the standard Lyndon array.

Abstract

The Lyndon array stores, at each position of a word, the length of the longest maximal Lyndon subword starting at that position, and plays an important role in combinatorics on words, for example in the construction of fundamental data structures such as the suffix array. In this paper, we introduce the Inverse Lyndon Array, the analogous structure for inverse Lyndon words, namely words that are lexicographically greater than all their proper suffixes. Unlike standard Lyndon words, inverse Lyndon words may have non-trivial borders, which introduces a genuine theoretical difficulty. We show that the inverse Lyndon array can be characterized in terms of the next greater suffix array together with a border-correction term, and prove that this correction coincides with a longest common extension (LCE) value. Building on this characterization, we adapt the nearest-suffix framework underlying Ellert's linear-time construction of the Lyndon array to the inverse setting, obtaining an O(n)-time algorithm for general ordered alphabets. Finally, we discuss implications for suffix comparison and report experiments on random, structured, and real datasets showing that the inverse construction exhibits the same practical linear-time behavior as the standard one.