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

2026-03-18

AI Summary

This work introduces the inverse Lyndon array, a novel structure that characterizes inverse Lyndon words—strings lexicographically greater than all their proper suffixes. Addressing the theoretical challenge posed by nontrivial borders in such words, the study establishes an intrinsic connection between the inverse Lyndon array and longest common extension (LCE) queries by introducing the “next greater suffix array” along with a border correction term. Building upon Ellert’s linear-time suffix array framework, the authors devise the first O(n)-time construction algorithm for the inverse Lyndon array. Experimental evaluation demonstrates that the algorithm achieves practical linear performance across random, structured, and real-world datasets, offering a new tool for combinatorics on words and efficient suffix-based data structure construction.

Technology Category

Natural Language Processing: Lexical Semantics and MorphologySearch and Optimization: Distributed SearchConstraint Satisfaction and Optimization: Applications

Application Category

Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deploymentsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Inverse Lyndon Array
• Lyndon words
• combinatorics on words
• suffix array
• border

Innovation

• Inverse Lyndon Array
• Lyndon words
• linear-time algorithm
• longest common extension
• suffix array