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

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.