Fast and Practical Algorithms for Computing All the Runs in a String

Abstract

A repetition in a string x is a substring ${ \bf{w}} = {\it \bf{u}}^e$ of x, maximum e ≥ 2, where u is not itself a repetition in w. A run in x is a substring ${\it \bf{w}} = {\it \bf{u}}^e{\it \bf{u^{*}}}$ of “maximal periodicity”, where ${\it \bf{u}}^e$ is a repetition and u ^* a maximum-length possibly empty proper prefix of u. A run may encode as many as $|{\it \bf{u}}|$ repetitions. The maximum number of repetitions in any string ${\it \bf{x}} = {\it \bf{x}}[1..n]$ is well known to be Θ(nlogn). In 2000 Kolpakov & Kucherov showed that the maximum number of runs in x is O(n); they also described a Θ(n)-time algorithm, based on Farach’s Θ(n)-time suffix tree construction algorithm (STCA), Θ(n)-time Lempel-Ziv factorization, and Main’s Θ(n)-time leftmost runs algorithm, to compute all the runs in x. Recently Abouelhoda et al. proposed a Θ(n)-time Lempel-Ziv factorization algorithm based on an “enhanced” suffix array — a suffix array together with other supporting data structures. In this paper we introduce a collection of fast space-efficient algorithms for computing all the runs in a string that appear in many circumstances to be superior to those previously proposed.

The work of the first and third authors was supported in part by grants from the Natural Sciences & Engineering Research Council of Canada.