Computing the stretch factor and maximum detour of paths, trees, and cycles in the normed space

Christian Wulff-Nilsen; Ansgar Grüne; Rolf Klein; Elmar Langetepe; D. T. Lee; Tien Ching Lin; Sheung Hung Poon; Teng Kai Yu

doi:10.1142/S0218195912600035

Computing the stretch factor and maximum detour of paths, trees, and cycles in the normed space

Christian Wulff-Nilsen, Ansgar Grüne, Rolf Klein, Elmar Langetepe, D. T. Lee, Tien Ching Lin^*, Sheung Hung Poon, Teng Kai Yu

^*Corresponding author for this work

Department of Computer Science

Abstract

The stretch factor and maximum detour of a graph G embedded in a metric space measure how well G approximates the minimum complete graph containing G and the metric space, respectively. In this paper we show that computing the stretch factor of a rectilinear path in L ₁ plane has a lower bound of Ω(n log n) in the algebraic computation tree model and describe a worst-case O(σn log ² n) time algorithm for computing the stretch factor or maximum detour of a path embedded in the plane with a weighted fixed orientation metric defined by σ < 2 vectors and a worst-case O(n log ^d n) time algorithm to d < 3 dimensions in L ₁-metric. We generalize the algorithms to compute the stretch factor or maximum detour of trees and cycles in O(σn log ^d+1 n) time. We also obtain an optimal O(n) time algorithm for computing the maximum detour of a monotone rectilinear path in L ₁ plane.

Original language	English
Journal	International Journal of Computational Geometry and Applications
Volume	22
Issue number	1
Pages (from-to)	45-60
Number of pages	16
ISSN	0218-1959
DOIs	https://doi.org/10.1142/S0218195912600035
Publication status	Published - 2012

Keywords

cycle
dilation
L metric
maximum detour
Rectilinear path
spanning ratio
stretch factor
tree
weighted fixed orientation metric

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10.1142/S0218195912600035

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title = "Computing the stretch factor and maximum detour of paths, trees, and cycles in the normed space",

abstract = "The stretch factor and maximum detour of a graph G embedded in a metric space measure how well G approximates the minimum complete graph containing G and the metric space, respectively. In this paper we show that computing the stretch factor of a rectilinear path in L 1 plane has a lower bound of Ω(n log n) in the algebraic computation tree model and describe a worst-case O(σn log 2 n) time algorithm for computing the stretch factor or maximum detour of a path embedded in the plane with a weighted fixed orientation metric defined by σ < 2 vectors and a worst-case O(n log d n) time algorithm to d < 3 dimensions in L 1-metric. We generalize the algorithms to compute the stretch factor or maximum detour of trees and cycles in O(σn log d+1 n) time. We also obtain an optimal O(n) time algorithm for computing the maximum detour of a monotone rectilinear path in L 1 plane.",

keywords = "cycle, dilation, L metric, maximum detour, Rectilinear path, spanning ratio, stretch factor, tree, weighted fixed orientation metric",

author = "Christian Wulff-Nilsen and Ansgar Gr{\"u}ne and Rolf Klein and Elmar Langetepe and Lee, {D. T.} and Lin, {Tien Ching} and Poon, {Sheung Hung} and Yu, {Teng Kai}",

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AU - Wulff-Nilsen, Christian

AU - Grüne, Ansgar

AU - Klein, Rolf

AU - Langetepe, Elmar

AU - Lee, D. T.

AU - Lin, Tien Ching

AU - Poon, Sheung Hung

AU - Yu, Teng Kai

PY - 2012

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AB - The stretch factor and maximum detour of a graph G embedded in a metric space measure how well G approximates the minimum complete graph containing G and the metric space, respectively. In this paper we show that computing the stretch factor of a rectilinear path in L 1 plane has a lower bound of Ω(n log n) in the algebraic computation tree model and describe a worst-case O(σn log 2 n) time algorithm for computing the stretch factor or maximum detour of a path embedded in the plane with a weighted fixed orientation metric defined by σ < 2 vectors and a worst-case O(n log d n) time algorithm to d < 3 dimensions in L 1-metric. We generalize the algorithms to compute the stretch factor or maximum detour of trees and cycles in O(σn log d+1 n) time. We also obtain an optimal O(n) time algorithm for computing the maximum detour of a monotone rectilinear path in L 1 plane.

KW - cycle

KW - dilation

KW - L metric

KW - maximum detour

KW - Rectilinear path

KW - spanning ratio

KW - stretch factor

KW - tree

KW - weighted fixed orientation metric

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JO - International Journal of Computational Geometry and Applications

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Computing the stretch factor and maximum detour of paths, trees, and cycles in the normed space

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