Sliding window order statistics in sublinear space
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We extend the multi-pass streaming model to sliding window problems, and address the problem of computing order statistics on fixed-size sliding windows, in the multi-pass streaming model as well as the closely related communication complexity model. In the 2-pass streaming model, we show that on input of length N with values in range [0,R] and a window of length K, sliding window minimums can be computed in O(√(N)). We show that this is nearly optimal (for any constant number of passes) when R ≥ K, but can be improved when R = o(K) to O(√(NR/K)). Furthermore, we show that there is an (l+1)-pass streaming algorithm which computes l^th-smallest elements in O(l^3/2√(N)) space. In the communication complexity model, we describe a simple O(pN^1/p) algorithm to compute minimums in p rounds of communication for odd p, and a more involved algorithm which computes the l^th-smallest elements in O(pl^2 N^1/(p-2l-1)) space. Finally, we prove that the majority statistic on boolean streams cannot be computed in sublinear space, implying that l^th-smallest elements cannot be computed in space both sublinear in N and independent of l.
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