Online Square Detection

TL;DR

This paper introduces an efficient online algorithm for detecting the first occurrence of a square in a string as characters are received, with improved memory efficiency for ordered alphabets.

Contribution

It presents a simple, memory-efficient online square detection algorithm with optimal time complexity for ordered alphabets, and an alternative solution for unordered alphabets.

Findings

O(n log σ) time and linear space for ordered alphabet

Simpler and more memory-efficient than previous algorithms

O(n log n) time solution for unordered alphabet

Abstract

The online square detection problem is to detect the first occurrence of a square in a string whose characters are provided as input one at a time. Recall that a square is a string that is a concatenation of two identical strings. In this paper we present an algorithm solving this problem in $O(n\log\sigma)$ time and linear space on ordered alphabet, where $\sigma$ is the number of different letters in the input string. Our solution is relatively simple and does not require much memory unlike the previously known online algorithm with the same working time. Also we present an algorithm working in $O(n\log n)$ time and linear space on unordered alphabet, though this solution does not outperform the previously known result with the same time bound.