Estimating quantiles, like the median or percentiles, is a fundamental task in data mining and data science. A (streaming) quantile summary is a data structure that can process a set S of n elements in a streaming fashion and at the end, for any phi in (0,1], return a phi-quantile of S up to an eps error, i.e., return a phi'-quantile with phi'=phi +- eps. We are particularly interested in comparison-based summaries that only compare elements of the universe under a total ordering and are otherwise completely oblivious of the universe. The best known deterministic quantile summary is the 20-year old Greenwald-Khanna (GK) summary that uses O((1/eps) log(eps n)) space [SIGMOD'01]. This bound was recently proved to be optimal for all deterministic comparison-based summaries by Cormode and Vesle\'y [PODS'20]. In this paper, we study weighted quantiles, a generalization of the quantiles problem, where each element arrives with a positive integer weight which denotes the number of copies of that element being inserted. The only known method of handling weighted inputs via GK summaries is the naive approach of breaking each weighted element into multiple unweighted items and feeding them one by one to the summary, which results in a prohibitively large update time (proportional to the maximum weight of input elements). We give the first non-trivial extension of GK summaries for weighted inputs and show that it takes O((1/eps) log(eps n)) space and O(log(1/eps)+ log log(eps n)) update time per element to process a stream of length n (under some quite mild assumptions on the range of weights and eps). En route to this, we also simplify the original GK summaries for unweighted quantiles.
翻译:(1/ imassing quantiles) 像中位数或百分位数一样, 估计量是数据挖掘和数据科学中的一项基本任务。 一个( 流) 量摘要是一个数据结构, 它能以流态方式处理一组Sn元素, 在结尾处, 任何phi (0, 1) 将S 的双量摘要返回到 eps 错误, 也就是说, 返回一个phi' 量和 phi=phi +- eps。 我们特别感兴趣的是基于比较的摘要的汇总, 该摘要只比较宇宙在总顺序下的各个元素, 而不是完全忽略宇宙的。 最已知的确定性量化摘要是20年前的Greenwald- Khanna (GK) 概要, 将Si- (1/ eps) logn( enn) 空间返回到 epepsnicial 错误, 即返回一个基于 cormode和 Vesle=y [PO] 的所有确定性比较性摘要。 在本文中, 我们研究 eqal Qeqrial Qreal realalalalalal made ma 中, made made max max i 。</s>