Balanced aspect ratio trees revisited

Spatial databases support a variety of geometric queries on point data such as range searches, nearest neighbor searches, etc. Balanced Aspect Ratio (BAR) trees are hierarchical space decomposition structures that are general-purpose and space-efficient, and, in addition, enjoy a worst case performance poly-logarithmic in the number of points for approximate queries. They maintain limits on their depth, as well as on the aspect ratio (intuitively, how skinny the regions can be). BAR trees were initially developed for 2 dimensional spaces and a fixed set of partitioning planes, and then extended to d dimensional spaces and more general partitioning planes. Here we revisit 2 dimensional spaces and show that, for any given set of 3 partitioning planes, it is not only possible to construct such trees, it is also possible to derive a simple closed-form upper bound on the aspect ratio. This bound, and the resulting algorithm, are much simpler than what is known for general BAR trees. We call the resulting BAR trees Parameterized BAR trees and empirically evaluate them for different partitioning planes. Our experiments show that our theoretical bound converges to the empirically obtained values in the lower ranges, and also make a case for using evenly oriented partitioning planes.