Computing The Maximum Exponent in a Stream

Abstract

We consider the streaming version of the following problem: given an input string s of length n, find the maximum exponent of a substring of s. We prove that any algorithm deciding, w.h.p., whether a string contains a square, uses memory of size $\varOmega (n)$, and thus does not satisfy the limitations of the streaming model. Thus the considered problem has no exact solution in the streaming model. Our main result is a Monte Carlo algorithm which computes the maximum exponent up to an additive error $\varepsilon <1/2$: it outputs a number $\alpha$ such that s has a substring of exponent $\alpha$ but no substrings of exponent $\alpha +\varepsilon$ or higher. The algorithm uses $\mathcal {O}(\frac{\log ^2 n}{\varepsilon })$ words of memory and performs $\mathcal {O}(\log n)$ operations, including dictionary operations, per input symbol.