arXiv subjects

Darryl Hill

Publications and source records attributed to Darryl Hill.

6 recordsLinked to original sources

arXiv2026

The Spanning Ratio of the Directed $\Theta_6$-Graph is 5

Given a finite set $P\subset\mathbb{R}^2$, the directed Theta-6 graph, denoted $\vec{\Theta}_6(P)$, is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The $\vec{\Theta}_6(P)$-graph is defined as follows: the plane around each point $u\in P$ is partitioned into $6$ equiangular cones with apex $u$, and in each cone, $u$ is joined to the point whose projection on the bisector of the cone is closest. Equivalently, the $\vec{\Theta}_6(P)$-graph contains an edge from $u$ to $v$ exactly when the interior of $\nabla_u^v$ is disjoint from $P$, where $\nabla_u^v$ is the unique equilateral triangle containing $u$ on a corner, $v$ on the opposite side, and whose sides are parallel to the cone boundaries. It was previously shown that the spanning ratio of the $\vec{\Theta}_6(P)$-graph is between $4$ and $7$ in the worst case (Akitaya, Biniaz, and Bose \emph{Comput. Geom.}, 105-106:101881, 2022). We close this gap by showing a tight spanning ratio of 5. This is the first tight bound proven for the spanning ratio of any $\vec{\Theta}_k(P)$-graph. Our lower bound models a long path by mapping it to a converging series. Our upper bound proof uses techniques novel to the area of spanners. We use linear programming to prove that among several candidate paths, there exists a path satisfying our bound.

cs.CG↗

arXiv2021

On the Spanning and Routing Ratio of Directed Theta-Four

We present a routing algorithm for the directed $Θ_4$-graph, here denoted as the $\overrightarrow{Θ_4}}$-graph, that computes a path between any two vertices $s$ and $t$ having length at most $17$ times the Euclidean distance between $s$ and $t$. To compute this path, at each step, the algorithm only uses knowledge of the location of the current vertex, its (at most four) outgoing edges, the destination vertex, and one additional bit of information in order to determine the next edge to follow. This provides the first known online, local, competitive routing algorithm with constant routing ratio for the $Θ_4$-graph, as well as improving the best known upper bound on the spanning ratio of these graphs from $237$ to $17$. We also show that without this additional bit of information, the routing ratio increases to $\sqrt{290} \approx 17.03$.

cs.CG↗

arXiv2021

Constant Delay Lattice Train Schedules

The following geometric vehicle scheduling problem has been considered: given continuous curves $f_1, \ldots, f_n : \mathbb{R} \rightarrow \mathbb{R}^2$, find non-negative delays $t_1, \ldots, t_n$ minimizing $\max \{ t_1, \ldots, t_n \}$ such that, for every distinct $i$ {and $j$} and every time $t$, $| f_j (t - t_j) - f_i (t - t_i) | > \ell$, where~$\ell$ is a given safety distance. We study a variant of this problem where we consider trains (rods) of fixed length $\ell$ that move at constant speed and sets of train lines (tracks), each of which consisting of an axis-parallel line-segment with endpoints in the integer lattice $\mathbb{Z}^d$ and of a direction of movement (towards $\infty$ {or $- \infty$}). We are interested in upper bounds on the maximum delay we need to introduce on any line to avoid collisions, but more specifically on universal upper bounds that apply no matter the set of train lines. We show small universal constant upper bounds for $d = 2$ and any given $\ell$ and also for $d = 3$ and $\ell = 1$. Through clique searching, we are also able to show that several of these upper bounds are tight.

cs.CG↗

arXiv2021

Improved Spanning on Theta-5

We show an upper bound of $\frac{ \sin\left(\frac{3π}{10}\right) }{ \sin\left(\frac{2π}{5}\right)-\sin\left(\frac{3π}{10}\right) } <5.70$ on the spanning ratio of $Θ_5$-graphs, improving on the previous best known upper bound of $9.96$ [Bose, Morin, van Renssen, and Verdonschot. The Theta-5-graph is a spanner. Computational Geometry, 2015.]

cs.CG↗

arXiv2016

On the Stretch Factor of Convex Polyhedra whose Vertices are (Almost) on a Sphere

Let $P$ be a convex polyhedron in $\mathbb{R}^3$. The skeleton of $P$ is the graph whose vertices and edges are the vertices and edges of $P$, respectively. We prove that, if these vertices are on the unit-sphere, the skeleton is a $(0.999 \cdot π)$-spanner. If the vertices are very close to this sphere, then the skeleton is not necessarily a spanner. For the case when the boundary of $P$ is between two concentric spheres of radii $1$ and $R>1$, and the angles in all faces are at least $θ$, we prove that the skeleton is a $t$-spanner, where $t$ depends only on $R$ and $θ$. One of the ingredients in the proof is a tight upper bound on the geometric dilation of a convex cycle that is contained in an annulus.

cs.CG↗

arXiv2015

Improved Spanning Ratio for Low Degree Plane Spanners

We describe an algorithm that builds a plane spanner with a maximum degree of 8 and a spanning ratio of approximately 4.414 with respect to the complete graph. This is the best currently known spanning ratio for a plane spanner with a maximum degree of less than 14.

cs.CG↗