Periodicity in Data Streams with Wildcards

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.