Quantiles, such as the median or percentiles, provide concise and useful information about the distribution of a collection of items, drawn from a linearly ordered universe. We study data structures, called quantile summaries, which keep track of all quantiles, up to an error of at most @math . That is, an @math -approximate quantile summary first processes a stream of items and then, given any quantile query @math , returns an item from the stream, which is a @math -quantile for some @math . We focus on comparison-based quantile summaries that can only compare two items and are otherwise completely oblivious of the universe. The best such deterministic quantile summary to date, by Greenwald and Khanna (ACM SIGMOD '01), stores at most @math items, where @math is the number of items in the stream. We prove that this space bound is optimal by providing a matching lower bound. Our result thus rules out the possibility of constructing a deterministic comparison-based quantile summary in space @math , for any function @math that does not depend on @math . A consequence of our results is also to show a lower bound for randomized algorithms.
Luo al @cite_4 compared quantile summaries experimentally and also provided a simple randomized algorithm with a good practical performance. This paper studies not only streaming algorithms for insertion-only streams (i.e., the cash register model), but also for turnstile streams, in which items may depart. Note that any algorithm for turnstile streams inherently relies on the bounded size of the universe. We refer the interested reader to the survey of Greenwald and Khanna @cite_11 for a description of both deterministic and randomized algorithms, together with algorithms for turnstile streams, the sliding window model, and distributed algorithms.
@cite_4 @cite_11
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We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass @math -approximation streaming algorithm can be easily obtained with space @math , where @math denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space @math is granted, for every @math . Specifically, for every @math , we show that in the one-pass turnstile streaming model, in order to compute a @math -approximation, space @math is required for constant error randomized algorithms, and, up to logarithmic factors, space @math is sufficient. Our lower bound result is proved in the simultaneous message model of communication and may be of independent interest. This paper resolves one of the longest standing basic problems in the streaming computational model. Namely, optimal construction of quantile sketches. An e approximate quantile sketch receives a stream of items x1,·,xn and allows one to approximate the rank of any query item up to additive error e n with probability at least 1-δ.The rank of a query x is the number of stream items such that xi ≤ x. The minimal sketch size required for this task is trivially at least 1 e.Felber and Ostrovsky obtain a O((1 e)log(1 e)) space sketch for a fixed δ.Without restrictions on the nature of the stream or the ratio between e and n, no better upper or lower bounds were known to date. This paper obtains an O((1 e)log log (1 δ)) space sketch and a matching lower bound. This resolves the open problem and proves a qualitative gap between randomized and deterministic quantile sketching for which an Ω((1 e)log(1 e)) lower bound is known. One of our contributions is a novel representation and modification of the widely used merge-and-reduce construction. This modification allows for an analysis which is both tight and extremely simple. The same technique was reported, in private communications, to be useful for improving other sketching objectives and geometric coreset constructions.