Approximation and streaming algorithms for histogram construction problems

ABSTRACT

Histograms and related synopsis structures are popular techniques for approximating data distributions. These have been successful in query optimization and a variety of applications, including approximate querying, similarity searching, and data mining, to name a few. Histograms were a few of the earliest synopsis structures proposed and continue to be used widely. The histogram construction problem is to construct the best histogram restricted to a space bound that reflects the data distribution most accurately under a given error measure.The histograms are used as quick and easy estimates. Thus, a slight loss of accuracy, compared to the optimal histogram under the given error measure, can be offset by fast histogram construction algorithms. A natural question arises in this context: Can we find a fast near optimal approximation algorithm for the histogram construction problem? In this article, we give the first linear time (1+ε)-factor approximation algorithms (for any ε > 0) for a large number of histogram construction problems including the use of piecewise small degree polynomials to approximate data, workloads, etc. Several of our algorithms extend to data streams.Using synthetic and real-life data sets, we demonstrate that in many scenarios the approximate histograms are almost identical to optimal histograms in quality and are significantly faster to construct.

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INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.2 Approximation

  H. Information Systems
  H.2 DATABASE MANAGEMENT

General Terms:
Algorithms, Performance, Theory

Keywords:
Data Streams, approximation algorithm, histograms