Reduction ratio of the IS-algorithm: worst and random cases

TL;DR

This paper analyzes the reduction ratio of the IS-algorithm for suffix array computation, providing worst-case and average-case insights, including the expected number of recursive calls for random inputs.

Contribution

It offers a detailed analysis of the reduction ratio in the IS-algorithm, including worst-case scenarios and probabilistic behavior for random words, which was previously unexplored.

Findings

Exact results for worst-case reduction ratios.

Strong convergence phenomena in random cases.

Expected recursive calls for random words are about log(log(n)).

Abstract

We study the IS-algorithm, a well-known linear-time algorithm for computing the suffix array of a word. This algorithm relies on transforming the input word $w$ into another word, called the reduced word of $w$, that will be at least twice shorter; then, the algorithm recursively computes the suffix array of the reduced word. In this article, we study the reduction ratio of the IS-algorithm, i.e., the ratio between the lengths of the input word and the word obtained after reducing $k$ times the input word. We investigate both worst cases, in which we find precise results, and random cases, where we prove some strong convergence phenomena. Finally, we prove that, if the input word is a randomly chosen word of length $n$, we should not expect much more than $\log(\log(n))$ recursive function calls.