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 that S has wildcard-period p if there exists an assignment to each of the wildcard characters so that in the resulting stream the prefix of length n − p equals the suffix of length n − p. We present a two-pass streaming algorithm that computes wildcard-periods of S using \(\mathcal {O}(k^{3} \text {polylog} n)\) bits of space, while we also show that this problem cannot be solved in sublinear space in one pass. We also 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 \(\mathcal {O}(k^{3}\log ^{9} n)\) space.

中文翻译：

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带通配符的数据流中的周期性

我们研究了在长度为n的字符串S中检测周期性趋势的问题，该趋势到达流模型中，该模型最多包含k个通配符，其中k = o（n）。通配符是可以分配给其他任何字符的特殊字符。我们说小号有通配符周期p，如果存在一个分配给每个通配符，使得所得流长的前缀ñ - p等于长度的后缀ñ - p。我们提出了一种两遍式流算法，该算法可计算通配符周期的S使用\（\ mathcal {O}（k ^ {3} \ text {polylog} n）\）的空间位，而我们也证明了这一问题不能在子线性空间中一次性解决。我们还给出了一种单程随机流算法，该算法使用\（p <\ frac {n} {2} \）计算S的所有通配符周期p，并且使用\（在S的最后p个符号中不出现通配符\ mathcal {O}（k ^ {3} \ log ^ {9} n）\）空间。