32 Proximity Algorithms

The notion of distance is fundamental to many aspects of computational geometry. A classic approach to characterize the distance properties of planar (and highdimensional) point sets that has been studied since the early 1980s are proximity graphs (Section 32.1). Proximity graphs are geometric graphs in which two vertices p, q are connected by an edge (p, q) if and only if a certain exclusion region for p, q contains no points from the vertex set. Depending on the specific exclusion region, many variants of proximity graphs can be defined, such as relative neighborhood graphs, Delaunay triangulations, β-skeletons, empty-strip graphs, etc. Since proximity graphs encode interesting information on the intrinsic structure of the point set, they have found many applications. From an algorithmic point of view, it is extremely useful to have a compact representation of the distance structure of a point set. The well-separated pair decomposition (WSPD) offers one way to achieve this (Section 32.2). WSPDs have numerous algorithmic applications, and the notion generalizes to certain non-Euclidean metrics. Furthermore, several variants of the WSPD have been developed to address its shortcomings, e.g., semi-separated pair decompositions and (α, β)-pair decompositions. Geometric spanners provide another means to approximate the complete Euclidean metric (Section 32.3). Here, the distance function is approximated by the shortest path distance in a sparse geometric graph. There are four basic constructions for geometric spanners: the greedy spanner, the Yao graph, the Θ-graph and the WSPD-spanner. To optimize various parameters, many variants have been defined, and the notion can be generalized beyond the Euclidean setting. Finally, we discuss work on making proximity structures dynamic, allowing for insertions and deletions of points (Section 32.4). The fundamental problem here is the dynamic nearest neighbor problem, which serves as a starting point for other structures. Additionally, there are several results on making geometric spanners dynamic.