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On the Stretch Factor of Polygonal Chains

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Acknowledgements

This work was initiated at the Fields Workshop on Discrete and Computational Geometry, held July 31 - August 4, 2017, at Carleton University. The authors thank the organizers and all participants of the workshop for inspiring discussions and for providing a great research atmosphere. This problem was initially posed by Rolf Klein in 2005. We would like to thank Rolf Klein and Christian Knauer for interesting discussions on the stretch factor and related topics.

Cite As

Ke Chen, Adrian Dumitrescu, Wolfgang Mulzer, and Csaba D. Tóth. On the Stretch Factor of Polygonal Chains. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.MFCS.2019.56

Abstract

Let P=(p_1, p_2, ..., p_n) be a polygonal chain. The stretch factor of P is the ratio between the total length of P and the distance of its endpoints, sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|. For a parameter c >= 1, we call P a c-chain if |p_ip_j|+|p_jp_k| <= c|p_ip_k|, for every triple (i,j,k), 1 <= i<j<k <= n. The stretch factor is a global property: it measures how close P is to a straight line, and it involves all the vertices of P; being a c-chain, on the other hand, is a fingerprint-property: it only depends on subsets of O(1) vertices of the chain. We investigate how the c-chain property influences the stretch factor in the plane: (i) we show that for every epsilon > 0, there is a noncrossing c-chain that has stretch factor Omega(n^{1/2-epsilon}), for sufficiently large constant c=c(epsilon); (ii) on the other hand, the stretch factor of a c-chain P is O(n^{1/2}), for every constant c >= 1, regardless of whether P is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain P in R^2 with n vertices, the minimum c >= 1 for which P is a c-chain in O(n^{2.5} polylog n) expected time and O(n log n) space.

Subject Classification

ACM Subject Classification
• Mathematics of computing → Paths and connectivity problems
• Theory of computation → Computational geometry
Keywords
• polygonal chain
• vertex dilation
• Koch curve
• recursive construction

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