Periodicity in Streams

Abstract

In this work we study sublinear space algorithms for detecting periodicity over data streams. A sequence of length n is said to be periodic if it consists of repetitions of a block of length p for some $p \leq \frac{n}{2}$. In the first part of this paper, we give a 1-pass randomized streaming algorithm that uses O(log² n) space and reports the shortest period if the given stream is periodic. At the heart of this result is a 1-pass O(lognlogm) space streaming pattern matching algorithm. This algorithm uses similar ideas to Porat and Porat’s algorithm in FOCS 2009 but it does not need an offline pre-processing stage and is simpler.

In the second part, we study distance to p-periodicity under the Hamming metric, where we estimate the minimum number of character substitutions needed to make a given sequence p-periodic. In streaming terminology, this problem can be described as computing the cascaded aggregate $L_1\circ F_1^{res(1)}$ over a matrix $A_{p \times \lfloor\frac{n}{p}\rfloor}$ given in column ordering. For this problem, we present a randomized streaming algorithm with approximation factor 2 + ε that takes $\tilde{O}(\frac{1}{\epsilon^2})$ space. We also show a 1 + ε randomized streaming algorithm which uses $\tilde{O}(\frac{1}{\epsilon^{5.5}}p^{1/2})$ space.