Light Orthogonal Networks with Constant Geometric Dilation

Applications of Geometric Spanner Networks

Given 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 th...

Algorithms for graphs of bounded treewidth via orthogonal range searching

We show that, for any fixed constant k ≥ 3, the sum of the distances between all pairs of vertices of an abstract graphs with n vertices and treewidth at most k can be computed in O(n logk−1 n) time. We also show that, for any fixed constant k ≥ 2, the dilation of a geometric graph (i.e., a graph drawn in the plane with straight-line segments) with n vertices and treewidth at most k can be comp...

On Geometric Dilation and Halving Chords

Let G be an embedded planar graph whose edges may be curves. The detour between two points, p and q (on edges or vertices) of G, is the ratio between the shortest path in G between p and q and their Euclidean distance. The supremum over all pairs of points of all these ratios is called the geometric dilation of G. Our research is motivated by the problem of designing graphs of low dilation. We ...

I/O-Efficiently Pruning Dense Spanners

Given a geometric graph G = (S, E) in R with constant dilation t, and a positive constant ε, we show how to construct a (1+ ε)spanner of G with O(|S|) edges using O(sort(|E|)) I/O operations.

Construction of Compactly Supported Nonseparable Orthogonal Wavelets of L^2(R^n)