Periodicity in Data Streams with Wildcards

Abstract

We investigate the problem of detecting periodic trends within a string S of length n, arriving in the streaming model, containing at most k wildcard characters, where $k=o(n)$. A wildcard character is a special character that can be assigned any other character. We say S has wildcard-period p if there exists an assignment to each of the wildcard characters so that in the resulting stream the length $n-p$ prefix equals the length $n-p$ suffix. We present a two-pass streaming algorithm that computes wildcard-periods of S using $\mathcal {O}\left( k^3\,{{\mathsf {polylog}}} \,n\right)$ bits of space, while we also show that this problem cannot be solved in sublinear space in one pass. We then give a one-pass randomized streaming algorithm that computes all wildcard-periods p of S with $p<\frac{n}{2}$ and no wildcard characters appearing in the last p symbols of S, using $\mathcal {O}\left( k^3\log ^9 n\right)$ space.