The Price of Order
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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.
KeywordsInduction Hypothesis Left Corner Delaunay Triangulation Tight Bound Inductive Proof
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