A Minimal Periods Algorithm with Applications

TL;DR

This paper extends Kosaraju's linear-time algorithm for minimal squares to compute minimal k-th powers with larger periods in words, providing detailed proofs and applications to pseudo-pattern detection.

Contribution

It generalizes a known algorithm to handle arbitrary exponents and periods, with detailed correctness proofs and practical applications.

Findings

Algorithm computes minimal k-th powers in O(k|w|) time

Provides detailed proof of correctness

Enables detection of pseudo-patterns in words

Abstract

Kosaraju in ``Computation of squares in a string'' briefly described a linear-time algorithm for computing the minimal squares starting at each position in a word. Using the same construction of suffix trees, we generalize his result and describe in detail how to compute in O(k|w|)-time the minimal k-th power, with period of length larger than s, starting at each position in a word w for arbitrary exponent $k\geq2$ and integer $s\geq0$. We provide the complete proof of correctness of the algorithm, which is somehow not completely clear in Kosaraju's original paper. The algorithm can be used as a sub-routine to detect certain types of pseudo-patterns in words, which is our original intention to study the generalization.