ISAAC 2010: Algorithms and Computation pp 37-48

# Approximating the Average Stretch Factor of Geometric Graphs

• Siu-Wing Cheng
• Christian Knauer
• Stefan Langerman
• Michiel Smid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6506)

## Abstract

Let G be a geometric graph whose vertex set S is a set of n points in ℝd. The stretch factor of two distinct points p and q in S is the ratio of their shortest-path distance in G and their Euclidean distance. We consider the problem of approximating the sum of all $$n \choose 2$$ stretch factors determined by all pairs of points in S. We show that for paths, cycles, and trees, this sum can be approximated, within a factor of 1 + ε, in O(npolylog(n)) time. For plane graphs, we present a (2 + ε)-approximation algorithm with running time O(n5/3polylog(n)), and a (4 + ε)-approximation algorithm with running time O(n3/2polylog(n)).

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## Authors and Affiliations

• Siu-Wing Cheng
• 1
• Christian Knauer
• 2
• Stefan Langerman
• 3
• Michiel Smid
• 4
1. 1.Department of Computer Science and EngineeringHKUSTHong Kong
2. 2.Institute of Computer ScienceUniversität BayreuthGermany
3. 3.Département d’InformatiqueUniversité Libre de BruxellesBelgium
4. 4.School of Computer ScienceCarleton UniversityOttawaCanada