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.
If we tolerate randomization and relax the requirement for worst-case error guarantees, it is possible to design quantile summaries with space close to @math . After a sequence of improvements @cite_0 @cite_3 @cite_4 @cite_10 , Karnin, Lang, and Liberty @cite_14 designed a randomized comparison-based quantile summary with space bounded by @math , where @math is the probability of not returning an @math -approximate @math -quantile for some @math . The algorithm combines careful sampling, and uses the Greenwald-Khanna summary as a subroutine. They also provide a reduction to transform the deterministic @math lower bound into a randomized lower bound of @math , implying optimality of their approach in its model. We discuss further how the deterministic and randomized lower bounds relate in .
@cite_14 @cite_4 @cite_3 @cite_0 @cite_10
2963274201 2279830512 2950958145 2114944618
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. Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish connections between the deterministic and randomized complexities in the LOCAL model. Each result has a very compelling take-away message: 1. Fast @math -coloring of trees requires random bits: Building on the recent lower bounds of , we prove that the randomized complexity of @math -coloring a tree with maximum degree @math is @math , whereas its deterministic complexity is @math for any @math . This also establishes a large separation between the deterministic complexity of @math -coloring and @math -coloring trees. 2. Randomized lower bounds imply deterministic lower bounds: We prove that any deterministic algorithm for a natural class of problems that runs in @math rounds can be transformed to run in @math rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires @math time deterministically. 3. Deterministic lower bounds imply randomized lower bounds: We prove that the randomized complexity of any natural problem on instances of size @math is at least its deterministic complexity on instances of size @math . This shows that a deterministic @math lower bound for any problem implies a randomized @math lower bound. It also illustrates that the graph shattering technique is absolutely essential to the LOCAL model. We consider the problem of selecting the best subset of exactly @math columns from an @math matrix @math . We present and analyze a novel two-stage algorithm that runs in @math time and returns as output an @math matrix @math consisting of exactly @math columns of @math . In the first (randomized) stage, the algorithm randomly selects @math columns according to a judiciously-chosen probability distribution that depends on information in the top- @math right singular subspace of @math . In the second (deterministic) stage, the algorithm applies a deterministic column-selection procedure to select and return exactly @math columns from the set of columns selected in the first stage. Let @math be the @math matrix containing those @math columns, let @math denote the projection matrix onto the span of those columns, and let @math denote the best rank- @math approximation to the matrix @math . Then, we prove that, with probability at least 0.8, @math This Frobenius norm bound is only a factor of @math worse than the best previously existing existential result and is roughly @math better than the best previous algorithmic result for the Frobenius norm version of this Column Subset Selection Problem (CSSP). We also prove that, with probability at least 0.8, @math This spectral norm bound is not directly comparable to the best previously existing bounds for the spectral norm version of this CSSP. Our bound depends on @math , whereas previous results depend on @math ; if these two quantities are comparable, then our bound is asymptotically worse by a @math factor. In a recent paper [MRL98], we had described a general framework for single pass approximate quantile finding algorithms. This framework included several known algorithms as special cases. We had identified a new algorithm, within the framework, which had a significantly smaller requirement for main memory than other known algorithms. In this paper, we address two issues left open in our earlier paper. First, all known and space efficient algorithms for approximate quantile finding require advance knowledge of the length of the input sequence. Many important database applications employing quantiles cannot provide this information. In this paper, we present a novel non-uniform random sampling scheme and an extension of our framework. Together, they form the basis of a new algorithm which computes approximate quantiles without knowing the input sequence length. Second, if the desired quantile is an extreme value (e.g., within the top 1 of the elements), the space requirements of currently known algorithms are overly pessimistic. We provide a simple algorithm which estimates extreme values using less space than required by the earlier more general technique for computing all quantiles. Our principal observation here is that random sampling is quantifiably better when estimating extreme values than is the case with the median.