The Price of Order

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 $k \geq 1$, 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 $k \geq 2$, 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

Spanning Properties of Theta-Theta Graphs

We study the spanning properties of Theta-Theta graphs. Similar in spirit with the Yao-Yao graphs, Theta-Theta graphs partition the space around each vertex into a set of k cones, for some fixed integer k > 1, and select at most one edge per cone. The difference is in the way edges are selected. Yao-Yao graphs select an edge of minimum length, whereas Theta-Theta graphs select an edge of minimum orthogonal projection onto the cone bisector. It has been established that the Yao-Yao graphs with parameter k = 6k' have spanning ratio 11.67, for k' >= 6. In this paper we establish a first spanning ratio of $7.82$ for Theta-Theta graphs, for the same values of $k$. We also extend the class of Theta-Theta spanners with parameter 6k', and establish a spanning ratio of $16.76$ for k' >= 5. We surmise that these stronger results are mainly due to a tighter analysis in this paper, rather than Theta-Theta being superior to Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta graphs decreases to 4.64 as k' increases to 8. These are the first results on the spanning properties of Theta-Theta graphs.Comment: 20 pages, 6 figures, 3 table

Routing on the Visibility Graph

We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let $P$ be a set of $n$ points in the plane and let $S$ be a set of non-crossing line segments whose endpoints are in $P$. We present two deterministic 1-local $O(1)$-memory routing algorithms that are guaranteed to find a path of at most linear size between any pair of vertices of the \emph{visibility graph} of $P$ with respect to a set of constraints $S$ (i.e., the algorithms never look beyond the direct neighbours of the current location and store only a constant amount of additional information). Contrary to {\em all} existing deterministic local routing algorithms, our routing algorithms do not route on a plane subgraph of the visibility graph. Additionally, we provide lower bounds on the routing ratio of any deterministic local routing algorithm on the visibility graph.Comment: An extended abstract of this paper appeared in the proceedings of the 28th International Symposium on Algorithms and Computation (ISAAC 2017). Final version appeared in the Journal of Computational Geometr

Expected Complexity of Routing in Θ6\Theta_6 and Half-Θ6\Theta_6 Graphs

We study online routing algorithms on the Θ6-graph and the half-Θ6-graph (which is equivalent to a variant of the Delaunay triangulation). Given a source vertex s and a target vertex t in the Θ6-graph (resp. half-Θ6-graph), there exists a deterministic online routing algorithm that finds a path from s to t whose length is at most 2 st (resp. 2.89 st) which is optimal in the worst case [Bose et al., siam J. on Computing, 44(6)]. We propose alternative, slightly simpler routing algorithms that are optimal in the worst case and for which we provide an analysis of the average routing ratio for the Θ6-graph and half-Θ6-graph defined on a Poisson point process. For the Θ6-graph, our online routing algorithm has an expected routing ratio of 1.161 (when s and t random) and a maximum expected routing ratio of 1.22 (maximum for fixed s and t where all other points are random), much better than the worst-case routing ratio of 2. For the half-Θ6-graph, our memoryless online routing algorithm has an expected routing ratio of 1.43 and a maximum expected routing ratio of 1.58. Our online routing algorithm that uses a constant amount of additional memory has an expected routing ratio of 1.34 and a maximum expected routing ratio of 1.40. The additional memory is only used to remember the coordinates of the starting point of the route. Both of these algorithms have an expected routing ratio that is much better than their worst-case routing ratio of 2.89