Dilation of Geometric Networks

Keywords and Synonyms

Detour; Spanning ratio; Stretch factor              

Problem Definition

Notations

Let ${ G=(V,E) }$ be a plane geometric network, whose vertex set V is a finite set of point sites in ${ {\mathbb R}^2 }$, connected by an edge set E of non-crossing straight line segments with endpoints in V. For two points ${ p \not= q \in V }$ let ${ \xi_G(p,q) }$ denote a shortest path from p to q in G. Then

$$ \sigma(p,q) := \frac{|\xi_G(p,q)|}{|pq|} $$

(1)

is the detour one encounters when using network G, in order to get from p to q, instead of walking straight. Here, ${ |.| }$ denotes the Euclidean length.

The dilation of G is defined by

$$ \sigma(G) := \max_{p \not= q \in V}\sigma(p,q)\:. $$

(2)

This value is also known as the spanning ratio or the stretch factor of G. It should, however, not be confused with the geometric dilation of a network, where the points on the edges are also being considered, in addition to the vertices.

Given a finite set Sof points in the...