Balanced Aspect Ratio Trees: Combining the Advantages of k-d Trees and Octrees

Given a set S of n points on d, we show, for fixed d, how to construct in Ž . Ž . O n log n time a data structure we call the balanced aspect ratio BAR tree. A Ž . BAR tree is a binary space partition tree on S that has O log n depth in which Ž . every region is convex and ‘‘fat’’ that is, has a bounded aspect ratio . While previous hierarchical data structures such as k-d trees, quadtrees, octrees, fair-split trees, and balanced box decompositions can guarantee some of these properties, we know of no previous data structure that combines all of these properties simultaneously. The BAR tree data structure has numerous applications ranging from geometric searching problems in fixed dimensional space to the visualization of graphs and three-dimensional worlds. 2001 Academic Press