We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of integers in {1,..., U} under an arbitrary sequence of n insertions and deletions, with O(log log U) expected query time and expected amortized update time, and O(n) space. The query bound is optimal in U for linear-space structures and improves previous near-O((log log U) 2) methods. The same method solves a fundamental problem from computational geometry: point location in orthogonal planar subdivisions (where edges are vertical or horizontal). We obtain the first static data structure achieving O(log log U) worst-case query time and linear space. This result is again optimal in U for linear-space structures and improves the previous O((log log U) 2) method by de Berg, Snoeyink, and van Kreveld (1992). The same result also holds for higherdimensional subdivisions that are orthogonal binary space partitions, and for certain nonorthogonal planar subdivisions such as triangulations without small angles. Many geometric applications follow, including improved query times for orthogonal range reporting for dimensions ≥ 3 on the RAM. Our key technique is an interesting new van-Emde-Boas–style recursion that alternates between two strategies, both quite simple.

A binary space partition tree is a data structure for the representation of a set of objectsin space. It found an increasing number of applications over the last decades. In recent years, intensifying research focused on its combinatorial properties, which affect directly the efficiency of applications. Important advances were made on binary space partitions for disjoint line segments in the plane and for axis-aligned objects in higher dimensions. New research directions were also initiated on some realistic polygonal scenes and on kinetic binary space partitions. This paper attempts to give an overview of these results and reiterates some of the most pressing open problems.