Applications of Geometric Spanner Networks

Abstract

PROBLEM DEFINITIONGiven a geometric graph in d-dimensional space, it is useful to preprocess it so that distance queries, exact or approximate, can be answered efficiently. Algorithms that can report distance queries in constant time are also referred to as "distance oracles". With unlimited preprocessing time and space, it is clear that exact distance oracles can be easily designed. This entry sheds light on the design of approximate distance oracles with limited preprocessing time and space for the family of geometric graphs with constant dilation. Notation and DefinitionsIf p and q are points in R d , then we use the notation |pq| to denote the Euclidean distance between p and q; we use the notation δ G (p, q) to denote the Euclidean length of a shortest path between p and q in a geometric network G. Given a constant t > 1, a graph G with vertex set S is a t-spanner for S if δ G (p, q) ≤ t|pq| for any two points p and q of S. A t-spanner network is said to have dilation (or stretch factor) t. We also define a (1 + ε)-approximate shortest path between p and q to be any path in G between p and q having length ∆, whereFor a comprehensive overview of geometric spanners, see the book by Narasimhan and Smid [23].All networks considered in this entry are simple and undirected. The model of computation used is the traditional algebraic computation tree model with the added power of indirect addressing. In particular, the algorithms presented here do not use the non-algebraic floor function as a unit-time operation. The problem is formalized below.Problem 1 (Distance Oracle). Given an arbitrary real constant > 0, and a geometric graph G in d-dimensional Euclidean space with constant dilation t, design a data structure that answers (1 + )-approximate shortest path length queries in constant time.The data structure can also be applied to solve several other problems. These include (a) the problem of reporting approximate distance queries between vertices in a planar polygonal domain with "rounded" obstacles, (b) query versions of closest pair problems, and (c) the efficient computation of the approximate dilations of geometric graphs. Survey of Related Research