Fast algorithms for shortest paths in planar graphs, with applications
SIAM Journal on Computing
Introduction to algorithms
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Approximating the Stretch Factor of Euclidean Graphs
SIAM Journal on Computing
Computing the Maximum Detour and Spanning Ratio of Planar Paths, Trees, and Cycles
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
Finding the best shortcut in a geometric network
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Geometric Spanner Networks
Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D
Discrete & Computational Geometry
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Let G be a graph embedded in Euclidean space. For any two vertices of G their dilation denotes the length of a shortest connecting path in G, divided by their Euclidean distance. In this paper we study the spectrum of the dilation, over all pairs of vertices of G. For paths, trees, and cycles in 2D we present O(n3/2+ε) randomized algorithms that compute, for a given value κ ≥ 1, the exact number of vertex pairs of dilation κ. Then we present deterministic algorithms that approximate the number of vertex pairs of dilation κ up to an 1+η factor. They run in time O(n log2n) for chains and cycles, and in time O(n log3n) for trees, in any constant dimension.