We first note that [EJS10] shows computing the period of a string in one-pass requiresΩ(n) space. Since the problem of periodicity for strings containing wild- cards is a generalization of exact periodicity, the same lower bound applies.
Periodicity in Data Streams with Wildcards 103
Theorem 7 (Implied from Theorem 3 from [EJS10] and Theorem 16 from [EGSZ17]). Given a string S with at most k wildcard characters, any one-pass streaming algorithm that computes the smallest wildcard-period requires Ω(n) space.
To show a lower bound that randomized streaming algorithm that computes all wildcard-periods ofS with probability at least 1−n1, even under the promise that the wildcard-periods are at most n/2, consider the following construction.
Define an infinite string 110112021303. . ., as in [GMSU16], and letνbe the prefix of length n4. DefineX to be the set of binary strings of length n4 with Hamming distance k2 from ν. For x∈ X, letYx be the set of binary strings of length n4 with either Δ(x, y) = k2 or Δ(x, y) = k2 + 1. Pick (x, y) uniformly at random from (X, Yx). Then Theorem 17 in [EGSZ17] shows a lower bound on the size of the sketches necessary to determine whether Δ(x, y) = k2 or Δ(x, y) = k2 + 1.
Theorem 8 [EGSZ17]. Any sketching function S that determines whether Δ(x, y) = k2 orΔ(x, y)> k2 fromS(x)andS(y), with probability at least1−n1 fork=o(√
n), usesΩ(klogn)space.
Suppose Alice hasy, along with the locations of the first k2 positionsiin which y[i]=x[i]. Alice replaces these locations with wildcard characters ⊥, runs the wildcard-period algorithm, and forwards the state of the algorithm to Bob, who hasx. Bob then continues running the algorithm onx◦x◦xto determine the wildcard-period of the string S(x, y) =y◦x◦x◦x. Observe that:
Lemma 5. If Δ(x, y) =k2, then the string S(x, y) =y◦x◦x◦xhas period n4. On the other hand, if Δ(x, y) =k2+ 1, then S(x, y) has period greater than n4. Combining Theorem8and Lemma5:
Theorem 9. For k = o(√
n) with k > 2, any one-pass randomized streaming algorithm that computes all wildcard-periods of an input stringS with probability at least1−1n requiresΩ(klogn)space, even under the promise that the wildcard- periods are at most n2.
Acknowledgements. We would like to thank the anonymous reviewers for their help- ful comments. The work was supported by the National Science Foundation under NSF Awards #1649515 and #1619081.
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