Online Detection of Repetitions with Backtracking

TL;DR

This paper introduces two efficient online algorithms for detecting repetitions with a specified exponent in strings, supporting backtracking and optimized for different alphabet sizes and memory constraints.

Contribution

The paper presents novel algorithms for online repetition detection that support backtracking and improve on memory efficiency compared to previous solutions.

Findings

First algorithm supports backtracking with $O(n ext{log} m)$ time and $O(m)$ space.

Second algorithm operates in $O(n ext{log}\sigma)$ time and $O(n)$ space, simpler and more memory-efficient.

Both algorithms effectively detect repetitions with exponent $ extgreater e$ in an online setting.

Abstract

In this paper we present two algorithms for the following problem: given a string and a rational $e > 1$, detect in the online fashion the earliest occurrence of a repetition of exponent $\ge e$ in the string. 1. The first algorithm supports the backtrack operation removing the last letter of the input string. This solution runs in $O(n\log m)$ time and $O(m)$ space, where $m$ is the maximal length of a string generated during the execution of a given sequence of $n$ read and backtrack operations. 2. The second algorithm works in $O(n\log\sigma)$ time and $O(n)$ space, where $n$ is the length of the input string and $\sigma$ is the number of distinct letters. This algorithm is relatively simple and requires much less memory than the previously known solution with the same working time and space. a string generated during the execution of a given sequence of $n$ read and backtrack operations.