Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Abstract

Abstract. Given a Euclidean graph G in R d with n vertices and m edges, we consider the problem of adding an edge to G such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a graph with positive edge weights runs in O(nm + n 2 log n) time, resulting in a trivial O(n 3 m + n 4 log n)-time algorithm for computing the optimal edge. First, we show that a simple modification yields the optimal solution in O(n 4 ) time using O(n 2 ) space. To reduce the running time we consider several approximation algorithms.Key words. computational geometry, approximation algorithms, geometric networks AMS subject classifications. 65D18, 68U05, 68Q25 DOI. 10.1137/050635675 1. Introduction. Consider a set V of n points in R d . A network on V can be modeled as an undirected graph G with vertex set V of size n and an edge set E of size m where every edge (u, v) has a positive weight w (u, v). A Euclidean network is a geometric network where the weight of the edge (u, v) is equal to the Euclidean distance |uv| between its two endpoints u and v.For two vertices u, v in a weighted graph G we use δ G (u, v) to denote a shortest path between u and v in G, and the length of the path is denoted by d G (u, v). Consider a weighted graph G = (V, E) and a graph G = (V, E ) on the same vertex set but with edge set E ⊆ E. We say that G is a t-spanner of G if for each pair of vertices (u, v). The minimum t such that G is a t-spanner for V is called the stretch factor, or dilation, of G.We say that a Euclidean network G = (V, E) is a t-spanner if G = (V, E) is a t-spanner of the complete network on V . In other words, for any two points p, q ∈ V the graph distance in G is at most t times the Euclidean distance between the two points.Complete graphs represent ideal communication networks, but they are expensive to build; sparse spanners represent low-cost alternatives. The weight of the spanner network is a measure of its sparseness; other sparseness measures include the number of edges, the maximum degree, and the number of Steiner points. Spanners for complete Euclidean graphs as well as for arbitrary weighted graphs find applications in robotics, network topology design, distributed systems, design of parallel machines, and many other areas. Recently spanners found interesting practical applications

Computing the dilation of edge-augmented graphs in metric spaces

“…Farshi et al [3] considered the following problem: given a graph G = (V , E) with n vertices embedded in a metric space, find a vertex pair (u, v) ∈ V × V (called a shortcut) such that the dilation of G ∪ {(u, v)} is minimized. They gave a trivial O (n 4 ) time and O (n 2 ) space algorithm for this problem together with various approximation algorithms.…”

Improving Shortest Paths in the Delaunay Triangulation

“…Similarly, our problem is also related to the problem of adding edges to a network in order to improve its dilation (i.e. the maximum over all pairs of different vertices of the ratio between their network-distance and their geometric distance), 11,12 although here we are mostly concerned with improving the network distance between only one pair of points.…”

Geographic Information Science

“…Farshi et al [7] show that it is possible to compute, for a given geometric graph, the edge that results in the largest dilation reduction in O(n 4 ) time. This was later improved by Wulff-Nilsen [20] to O(n 3 log n) time.…”