Towards Logarithmic Search Time Complexity for R-Trees

Abstract

Index structures are frequently used to reduce search times in large databases. With index structures like the B-Tree search time grows only logarithmic with the size of a database for several types of searches. This means that the search time is almost constant for very large database systems even if their size grows significantly. Conventional index structures however do not well support searches specifying lower and/or upper bounds for more than one attribute (multidimensional range searches). Therefore R-Trees are increasingly used in this application context. A typical application domain of R-Trees are spatial database systems with two dimensional search conditions specifying upper and lower bounds for longitude and latitude values. Unfortunately R-Tree efficiency does not meet the expectations in many cases. Theoretical analysis of this problem showed, that search time grows much faster than logarithmic for two and more dimensional range searches in contrary to the one dimensional case. In this paper we prove that a logarithmic search complexity can be achieved for two dimensions, if the form of nodes is optimized relative to the form of search conditions. Based on this result, the paper investigates the form of nodes generated by different existing tree packing methods. Since existing methods fail to ensure the required form, a new tree packing method is proposed which improves the chance to meet the identified requirements.

Keywords

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