Discrete & Computational Geometry

, Volume 53, Issue 2, pp 296–326 | Cite as

Average Stretch Factor: How Low Does It Go?

  • Vida Dujmović
  • Pat MorinEmail author
  • Michiel Smid


In a geometric graph \(G\), the stretch factor between two vertices \(u\) and \(w\) is the ratio between the Euclidean length of the shortest path from \(u\) to \(w\) in \(G\) and the Euclidean distance between \(u\) and \(w\). The average stretch factor of \(G\) is the average stretch factor taken over all pairs of vertices in \(G\). We show that, for any constant dimension \(d\) and any set \(V\) of \(n\) points in \(\mathbb {R}^d\), there exists a geometric graph with vertex set \(V\) that has \(O(n)\) edges and that has average stretch factor \(1+ o_n(1)\). More precisely, the average stretch factor of this graph is \(1+O\big ((\log n/n)^{1/(2d+1)}\big )\). We complement this upper bound with a lower bound: There exist \(n\)-point sets in \(\mathbb {R}^2\) for which any graph with \(O(n)\) edges has average stretch factor \(1+\Omega (1/\sqrt{n})\). Bounds of this type are not possible for the more commonly studied worst-case stretch factor. In particular, there exist point sets \(V\) such that any graph with worst-case stretch factor \(1+o_n(1)\) has a superlinear number of edges.


Spanners Average stretch Average dilation 



The authors of this paper are partly funded by NSERC and CFI. The authors are indebted to Shay Solomon for providing helpful feedback on an earlier version of this paper. This research was funded by NSERC and the Ontario Ministry of Research and Innovation.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Computer Science and Electrical EngineeringUniversity of OttawaOttawaCanada
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada

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