Geometric Dilation of Geometric Networks

Keywords and Synonyms

Detour; Spanning ratio; Stretch factor                

Problem Definition

Urban street systems can be modeled by plane geometric networks ${ G=(V,E) }$ whose edges ${ e \in E }$ are piecewise smooth curves that connect the vertices ${ v \in V \subset {\mathbb R}^2 }$. Edges do not intersect, except at common endpoints in V. Since streets are lined with houses, the quality of such a network can be measured by the length of the connections it provides between two arbitrary points p and q on G.

Let ${ \xi_G(p,q) }$ denote a shortest path from p to q in G. Then

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

(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 geometric dilation of network G is defined by

$$ { \delta(G) := \sup_{p \not= q \in G}\delta(p,q). } $$

(2)

This definition differs from the notion of stretch factor (or: spanning ratio) used in the context...