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.
Other results arise when relaxing the requirement for correctness under adversarial order to assuming that the input arrives in a random order. For random-order streams, Guha and McGregor @cite_13 studied algorithms for exact and approximate selection of quantiles. Among other things, they gave an algorithm for finding the exact @math -quantile in space @math using @math passes over a random-order stream, while with @math memory we need to do @math passes on the worst-case stream. The Shifting Sands algorithm @cite_9 reduce the magnitude of the error from @math to @math . Since our lower bound relies on carefully constructing an adversarial input sequence, it does not apply to this random order model.
@cite_9 @cite_13
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When trying to process a data stream in small space, how important is the order in which the data arrive? Are there problems that are unsolvable when the ordering is worst case, but that can be solved (with high probability) when the order is chosen uniformly at random? If we con- sider the stream as if ordered by an adversary, what happens if we restrict the power of the adversary? We study these questions in the context of quantile estimation, one of the most well studied problems in the data-stream model. Our results include an O(polylog n)-space, O(log logn)-pass algorithm for exact selection in a randomly ordered stream of n elements. This resolves an open question of Munro and Paterson (Theoret. Comput. Sci., 23 (1980), pp. 315-323). We then demonstrate an exponential separation between the random-order and adversarial-order models: using O(polylog n) space, exact selection requires Ω(log n log logn) passes in the adversarial-order model. This lower bound, in contrast to previous results, applies to fully general randomized algorithms and is estab- lished via a new bound on the communication complexity of a natural pointer-chasing style problem. We also prove the first fully general lower bounds in the random-order model: finding an element with rank n 2 ± n δ in the single-pass random-order model with probability at least 9 10 requires Ω( � n 1−3δ logn) space. In most algorithmic applications which compare two distributions, information theoretic distances are more natural than standard l p norms. In this paper we design streaming and sublinear time property testing algorithms for entropy and various information theoretic distances. posed the problem of property testing with respect to the Jensen-Shannon distance. We present optimal algorithms for estimating bounded, symmetric f-divergences (including the Jensen-Shannon divergence and the Hellinger distance) between distributions in various property testing frameworks. Along the way, we close a (log n) H gap between the upper and lower bounds for estimating entropy H, yielding an optimal algorithm over all values of the entropy. In a data stream setting (sublinear space), we give the first algorithm for estimating the entropy of a distribution. Our algorithm runs in polylogarithmic space and yields an asymptotic constant factor approximation scheme. An integral part of the algorithm is an interesting use of an F 0 (the number of distinct elements in a set) estimation algorithm; we also provide other results along the space time approximation tradeoff curve.Our results have interesting structural implications that connect sublinear time and space constrained algorithms. The mediating model is the random order streaming model, which assumes the input is a random permutation of a multiset and was first considered by Munro and Paterson in 1980. We show that any property testing algorithm in the combined oracle model for calculating a permutation invariant functions can be simulated in the random order model in a single pass. This addresses a question raised by regarding the relationship between property testing and stream algorithms. Further, we give a polylog-space PTAS for estimating the entropy of a one pass random order stream. This bound cannot be achieved in the combined oracle (generalized property testing) model. 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. We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger.