Towards Tight Bounds on Theta-Graphs

We present improved upper and lower bounds on the spanning ratio of θgraphs with at least six cones. Given a set of points in the plane, a θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2π/m, and adds an edge to the ‘closest’ vertex in each cone. We show that for any integer k ≥ 1, θ-graphs with 4k+ 2 cones have a spanning ratio of 1 + 2 sin(θ/2) and we provide a matching lower bound, showing that this spanning ratio tight. Next, we show that for any integer k ≥ 1, θ-graphs with 4k + 4 cones have spanning ratio at most 1 + 2 sin(θ/2)/(cos(θ/2) − sin(θ/2)). We also show that θ-graphs with 4k+3 and 4k+5 cones have spanning ratio at most cos(θ/4)/(cos(θ/2) − sin(3θ/4)). This is a significant improvement on all families of θ-graphs for which exact bounds are not known. For example, the spanning ratio of the θ-graph with 7 cones is decreased from at most 7.5625 to at most 3.5132. These spanning proofs also imply improved upper bounds on the competitiveness of the θ-routing algorithm. In particular, we show that the θ-routing algorithm is (1 + 2 sin(θ/2)/(cos(θ/2)− sin(θ/2)))-competitive on θ-graphs with 4k + 4 cones and that this ratio is tight. Finally, we present improved lower bounds on the spanning ratio of these Research supported in part by FQRNT, NSERC, and Carleton University’s President’s 2010 Doctoral Fellowship. ∗School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, K1S 5B6, ON, Canada, Tel.: +1-613-520-2600 x4336 Fax: +1-613-520-2600 x4334 Email addresses: [email protected] (Prosenjit Bose), [email protected] (Jean-Lou De Carufel), [email protected] (Pat Morin), [email protected] (André van Renssen), [email protected] (Sander Verdonschot) Preprint submitted to Theoretical Computer Science April 11, 2014 graphs. Using these bounds, we provide a partial order on these families of θ-graphs. In particular, we show that θ-graphs with 4k + 4 cones have spanning ratio at least 1 + 2 tan(θ/2) + 2 tan(θ/2), where θ is 2π/(4k + 4). This is somewhat surprising since, for equal values of k, the spanning ratio of θ-graphs with 4k + 4 cones is greater than that of θ-graphs with 4k + 2 cones, showing that increasing the number of cones can make the spanning ratio worse.