On the dilation spectrum of paths, cycles, and trees

Authors:
Rolf Klein;Christian Knauer;Giri Narasimhan;Michiel Smid
Affiliations:
Institute of Computer Science I, Universität Bonn, Bonn, Germany;Institute of Computer Science, Freie Universität Berlin, Berlin, Germany;School of Computing and Information Sciences, Florida International University, Miami, FL, USA;School of Computer Science, Carleton University, Ottawa, Ontario, Canada
Venue:
Computational Geometry: Theory and Applications
Year:
2009

Citing 13
Cited 0

Fast algorithms for shortest paths in planar graphs, with applications

SIAM Journal on Computing
A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields

Journal of the ACM (JACM)
Dealing with higher dimensions: the well-separated pair decomposition and its applications

Dealing with higher dimensions: the well-separated pair decomposition and its applications
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
Geometric Spanner Networks

Geometric Spanner Networks
Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

Discrete & Computational Geometry
Computing a minimum-dilation spanning tree is NP-hard

Computational Geometry: Theory and Applications
Improving the Stretch Factor of a Geometric Network by Edge Augmentation

SIAM Journal on Computing
Computing the Maximum Detour of a Plane Graph in Subquadratic Time

ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Computing Best and Worst Shortcuts of Graphs Embedded in Metric Spaces

ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Algorithms for graphs of bounded treewidth via orthogonal range searching

Computational Geometry: Theory and Applications
Introduction to Algorithms, Third Edition

Introduction to Algorithms, Third Edition

Quantified Score

Hi-index	0.00

Visualization

Abstract

Let G be a graph with n vertices which is embedded in Euclidean space R^d. For any two vertices of G, their dilation is defined to be the ratio of the length of a shortest connecting path in G to the Euclidean distance between them. In this paper, we study the spectrum of the dilation, over all pairs of vertices of G. For paths, cycles, and trees in R^2, we present O(n^3^/^2^+^@e)-time randomized algorithms that compute, for a given value @k1, the exact number of vertex pairs of dilation at most @k. Then we present deterministic algorithms that approximate the number of vertex pairs of dilation at most @k to within a factor of 1+@e. They run in O(nlog^2n) time for paths and cycles, and in O(nlog^3n) time for trees, in any constant dimension d.