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

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 bound of varOmega ( M log min {|varSigma |,M}) bits of memory; here M=n/E for approximating the answer with additive error E, and M= log n/log (1+varepsilon ) for approximating the answer with multiplicative error (1 + varepsilon ). Second, we design four real-time algorithms for this problem. Three of them are Monte Carlo approximation algorithms for additive error, “small” and “big” multiplicative errors, respectively. Each algorithm uses mathcal {O}(M) words of memory. Thus the obtained lower bounds are asymptotically tight up to a logarithmic factor. The fourth 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.