Tight Tradeoffs for Real-Time Approximation of Longest Palindromes in Streams

TL;DR

This paper establishes tight lower bounds and presents efficient algorithms for approximating the longest palindrome in streaming data, balancing space complexity and approximation accuracy.

Contribution

It provides the first tight bounds on space requirements and introduces practical algorithms for real-time palindrome approximation in streams.

Findings

Lower bounds rule out sublinear space for Las Vegas algorithms.

Monte Carlo algorithms achieve near-optimal space complexity.

A deterministic algorithm finds short palindromes exactly.

Abstract

We consider computing a longest palindrome in the streaming model, where the symbols arrive one-by-one and we do not have random access to the input. While computing the answer exactly using sublinear space is not possible in such a setting, one can still hope for a good approximation guarantee. Our contribution is twofold. First, we provide lower bounds on the space requirements for randomized approximation algorithms processing inputs of length $n$. We rule out Las Vegas algorithms, as they cannot achieve sublinear space complexity. For Monte Carlo algorithms, we prove a lower bounds of $\Omega( M \log\min\{|\Sigma|,M\})$ bits of memory; here $M=n/E$ for approximating the answer with additive error $E$, and $M= \frac{\log n}{\log (1+\varepsilon)}$ for approximating the answer with multiplicative error $(1 + \varepsilon)$. Second, we design three real-time algorithms for this problem. Our Monte Carlo approximation algorithms for both additive and multiplicative versions of the problem use $O(M)$ words of memory. Thus the obtained lower bounds are asymptotically tight up to a logarithmic factor. The third algorithm is deterministic and finds a longest palindrome exactly if it is short. This algorithm can be run in parallel with a Monte Carlo algorithm to obtain better results in practice. Overall, both the time and space complexity of finding a longest palindrome in a stream are essentially settled.