Binary Space Partitions: Recent Developments

Summary

A binary space partition tree is a data structure for the representation of a set of objects in 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.

1. Introduction

The binary space partition tree is a geometric data structure obtained by a recursive partitioning scheme, called binary space partition (for short, BSP) over a set of input objects: The space is partitioned along a hyper plane into two half-spaces, then either half-space is partitioned recursively until every subproblem contains only a trivial fraction of the input objects. The concept of BSP has emerged from the computer graphics community in the seventies. It was originally designed to assist efficient hidden-surface removal algorithms for moving viewpoints, but it has later found widespread applications in many areas of computational and combinatorial geometry.

In many of the applications, the bottle neck of the space complexity is the size of the BSP tree they rely on. Combinatorial research focused on determining the worst case complexity of BSPs for certain classes of inputs.