Tight tradeoffs for approximating palindromes in streams

TL;DR

This paper investigates the space complexity of streaming algorithms for approximating the longest palindrome in a text, establishing tight bounds and providing an efficient Monte Carlo algorithm that nearly matches these bounds.

Contribution

It proves lower bounds for additive and multiplicative approximation of longest palindromes in streams and introduces a near-optimal Monte Carlo algorithm, settling the space complexity question.

Findings

No sublinear space Las Vegas algorithms exist for this problem.

Lower bounds of rac{n}{E} and rac{log n}{log(1+b5)} bits for additive and multiplicative approximations.

A Monte Carlo algorithm matching these bounds up to a logarithmic factor.

Abstract

We consider computing the longest palindrome in a text of length $n$ in the streaming model, where the characters arrive one-by-one, and we do not have random access to the input. While computing the answer exactly using sublinear memory is not possible in such a setting, one can still hope for a good approximation guarantee. We focus on the two most natural variants, where we aim for either additive or multiplicative approximation of the length of the longest palindrome. We first show that there is no point in considering Las Vegas algorithms in such a setting, as they cannot achieve sublinear space complexity. For Monte Carlo algorithms, we provide a lowerbound of $\Omega(\frac{n}{E})$ bits for approximating the answer with additive error $E$, and $\Omega(\frac{\log n}{\log(1+\varepsilon)})$ bits for approximating the answer with multiplicative error $(1+\varepsilon)$ for the binary alphabet. Then, we construct a generic Monte Carlo algorithm, which by choosing the parameters appropriately achieves space complexity matching up to a logarithmic factor for both variants. This substantially improves the previous results by Berenbrink et al. (STACS 2014) and essentially settles the space complexity.