Self-spatial join selectivity estimation using fractal concepts

The problem of selectivity estimation for queries of nontraditional databases is still an open issue. In this article, we examine the problem of selectivity estimation for some types of spatial queries in databases containing real data. We have shown earlier [Faloutsos and Kamel 1994] that real point sets typically have a nonuniform distribution, violating consistently the uniformity and independence assumptions. Moreover, we demonstrated that the theory of fractals can help to describe real point sets. In this article we show how the concept of fractal dimension, i.e., (noninteger) dimension, can lead to the solution for the selectivity estimation problem in spatial databases. Among the infinite family of fractal dimensions, we consider here the Hausdorff fractal dimension D(0) and the "Correlation" fractal dimension D(2). Specifically, we show that (a) the average number of neighbors for a given point set follows a power law, with Da as exponent, and (b) the average number of nonempty range queries follows a power law with E - D(0) as exponent (E is the dimension of the embedding space). We present the formulas to estimate the selectivity for "biased" range queries, for self-spatial joins, and for the average number of nonempty range queries. The result of some experiments on real and synthetic point sets are shown. Our formulas achieve very low relative errors, typically about 10%, versus 40%-100% of the formulas that are based on the uniformity and independence assumptions.