Approximating ranks, quantiles, and distributions over streaming data is a central task in data analysis and monitoring. Given a stream of n items from a data universe U ( equipped with a total order ) , the task is to compute a sketch (data structure) of size poly(\log(n), 1/\varepsilon). Given the sketch and a query item y \in U, one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than y. Most works to date focused on additive \varepsilon n error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This paper investigates multiplicative (1\pm\varepsilon)-error approximations to the rank. The motivation stems from practical demand to understand the tails of distributions, and hence for sketches to be more accurate near extreme values. The most space-efficient algorithms that can be derived from prior work store either O(\log(\varepsilon^2 n)/\varepsilon^2) or O(\log^3(\varepsilon n)/\varepsilon) universe items. This paper presents a  sketch of size O(\log^{1.5}(\varepsilon n)/\varepsilon) (ignoring \text{poly(\log\log n, \log(1/\varepsilon)) factors) } that achieves a 1\pm\varepsilon multiplicative error guarantee, without prior knowledge of the stream length or dependence on the size of the data universe. This is within a O(\log(\varepsilon n)) factor of optimal  .
Approximating ranks, quantiles, and distributions over streaming data is a central task in data analysis and monitoring. Given a stream of n items from a data universe U equipped with a total order , the task is to compute a sketch (data structure) of size poly(\log(n), 1/\varepsilon). Given the sketch and a query item y \in U, one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than y. Most works to date focused on additive \varepsilon n error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This paper investigates multiplicative (1\pm\varepsilon)-error approximations to the rank. The motivation stems from practical demands to understand the tails of distributions, and hence for sketches to be more accurate near extreme values. The most space-efficient algorithms that can be derived from prior work store either O(\log(\varepsilon^2 n)/\varepsilon^2) or O(\log^3(\varepsilon n)/\varepsilon) universe items. This paper presents a randomized sketch of size O(\log^{1.5}(\varepsilon n)/\varepsilon) (\log\log n, \log(1/\varepsilon)) factors) } that achieves a 1\pm\varepsilon multiplicative error guarantee, without prior knowledge of the stream length or dependence on the size of the data universe. This is within a O(\log(\varepsilon n)) factor of optimal . Moreover, our sketch is fully mergeable, which makes it suitable for a parallel or distributed environment .
D
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D
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R
demand
demands
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randomized
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(ignoring \text{poly
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. Moreover, our sketch is fully mergeable, which makes it suitable for a parallel or distributed environment