ISAAC 2014: Algorithms and Computation pp 313-325

# The Price of Order

• Prosenjit Bose
• Pat Morin
• André van Renssen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8889)

## Abstract

We present tight bounds on the spanning ratio of a large family of ordered $$\theta$$-graphs. A $$\theta$$-graph partitions the plane around each vertex into $$m$$ disjoint cones, each having aperture $$\theta = 2 \pi /m$$. An ordered $$\theta$$-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer $$k \ge 1$$, ordered $$\theta$$-graphs with $$4k + 4$$ cones have a tight spanning ratio of $$1 + 2 \sin (\theta /2) / (\cos (\theta /2) - \sin (\theta /2))$$. We also show that for any integer $$k \ge 2$$, ordered $$\theta$$-graphs with $$4k + 2$$ cones have a tight spanning ratio of $$1 / (1 - 2 \sin (\theta /2))$$. We provide lower bounds for ordered $$\theta$$-graphs with $$4k + 3$$ and $$4k + 5$$ cones. For ordered $$\theta$$-graphs with $$4k + 2$$ and $$4k + 5$$ cones these lower bounds are strictly greater than the worst case spanning ratios of their unordered counterparts. These are the first results showing that ordered $$\theta$$-graphs have worse spanning ratios than unordered $$\theta$$-graphs. Finally, we show that, unlike their unordered counterparts, the ordered $$\theta$$-graphs with 4, 5, and 6 cones are not spanners.

## Keywords

Induction Hypothesis Left Corner Delaunay Triangulation Tight Bound Inductive Proof
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Prosenjit Bose
• 1
• Pat Morin
• 1
• André van Renssen
• 1
• 2
Email author
1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
2. 2.National Institute of InformaticsTokyoJapan