The frequency moments of a sequence containing mi elements of type i, for 1 ≤ i ≤ n, are the numbers Fk = �n i=1 mki. We consider the space complexity of randomized algorithms that approximate the numbers Fk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbers F0, F1 and F2 can be approximated in logarithmic space, whereas the approximation of Fk for k ≥ 6 requires nΩ(1) space. Applications to data bases are mentioned as well.

In large data recording and warehousing environments, it is often advantageous to provide fast, approximate answers to queries. The goal is to provide an estimated response in orders of magnitude less time than the time to compute an exact answer, by avoiding or minimizing the number of accesses to the base data. This paper presents the Approximate QUery Answering (AQUA) System, for fast, highlyaccurate approximate answers to queries. Aqua provides approximate answers using small, precomputed synopses (samples, counts, etc.) of the underlying base data. An important feature of Aqua is that it provides accuracy guarantees without any a priori assumptions on either the data distribution, the order in which the base data is loaded, or the layout of the data on the disks. Currently, the system provides fast approximate answers for queries with selects, aggregates, group bys and/or joins (especially, the multi-way foreign key joins that are popular in OLAP). We present several ne...

Viswanath Poosala z In large data recording and warehousing environments, it is often advantageous to provide fast, approximate answers to queries, whenever possible. The goal is to provide an estimated response in orders of magnitude less time than the time to compute an exact answer, by avoiding or minimizing the number of accesses to the base data. This white paper describes the Approximate QUery Answering (AQUA) Project underway in the Information Sciences Research Center at Bell Labs. We present a framework for an approximate query engine that observes new data as it arrives and maintains small synopsis data structures on that data. These data structures are used to provide fast, approximate answers to a broad class of queries. We describe metrics for evaluating approximate query answers. We also present new synopsis data structures, and new techniques for approximate query answers. We report on the goals and status of the Aqua project, and plans for future work.

A multidatabase system (MDBS) integrates information from autonomous local databases managed by different database management systems (MDBS) in a distributed environment. A number of challenges are raised for query optimization in such an MDBS. One of the major challenges is that some local optimization information may not be available at the global level. We recently proposed a query sampling method to drive cost estimation formulas for local databases in an MDBS [22] . To use the derived formulas to estimate the costs of queries, we need to know the selectivities of the qualifications of the queries. Unfortunately, existing methods for estimating selectivities cannot be used efficiently in an MDBS environment. This paper discusses difficulties of estimating selectivities in an MDBS. Based on the discussion, this paper presents an integrated method to estimate selectivities in an MDBS. The method integrates and extends several existing methods so that they can be used in an MDBS eff...

Abstract. We address the problem of answering Web ontology queries efficiently. An ontology is formalized as a Deductive Ontology Base (DOB), a deductive database that comprises the ontology’s inference axioms and facts, and we present a cost-based query optimization technique for DOB. A hybrid cost model is proposed to estimate the cost and cardinality of basic and inferred facts. Cardinality and cost of inferred facts are estimated using an adaptive sampling technique, while techniques of traditional relational cost models are used for estimating the cost of basic facts and conjunctive ontology queries. Finally, we implement a dynamic-programming optimization algorithm to identify query evaluation plans that minimize the number of intermediate inferred facts. We modeled a subset of the Web ontology language OWL Lite as a DOB, and performed an experimental study to analyze the predictive capacity of our cost model and the benefits of the query optimization technique. Our study has been conducted over synthetic and real-world OWL ontologies, and shows that the techniques are accurate and improve query performance. 1

space complexity of approximating the frequency moments