The price of order

Abstract

We present tight bounds on the spanning ratio of a large family of ordered θ-graphs. Aθ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2π/m. An ordered θ-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 ≥ 1, ordered θ-graphs with 4k + 4 cones have a tight spanning ratio of 1 + 2 sin(θ/2)/(cos(θ/2) − sin(θ/2)). We also show that for any integer k ≥ 2, ordered θ-graphs with 4k + 2 cones have a tight spanning ratio of 1/(1 − 2 sin(θ/2)). We provide lower bounds for ordered θ-graphs with 4k+3 and 4k+5 cones. For ordered θ-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 θ-graphs have worse spanning ratios than unordered θ-graphs. Finally, we show that, unlike their unordered counterparts, the ordered θ- graphs with 4, 5, and 6 cones are not spanners.