Linear Time Runs over General Ordered Alphabets

Abstract

A run in a string is a maximal periodic substring. For example, the string bananatree contains the runs anana = (an) 3/2 and ee = e 2 . There are less than n runs in any length-n string, and computing all runs for a string over a linearly-sortable alphabet takes O(n) time (Bannai et al., SODA 2015). Kosolobov conjectured that there also exists a linear time runs algorithm for general ordered alphabets (Inf. Process. Lett. 2016). The conjecture was almost proven by Crochemore et al., who presented an O(nα(n)) time algorithm (where α(n) is the extremely slowly growing inverse Ackermann function). We show how to achieve O(n) time by exploiting combinatorial properties of the Lyndon array, thus proving Kosolobov's conjecture. ACM Subject Classification Theory of computation → Design and analysis of algorithmsKeywords and phrases String algorithms, Lyndon array, runs, longest common extension, general ordered alphabets, combinatorics on words Supplementary Material https://github.com/jonas-ellert/linear-time-runs/