Prosenjit Bose

Prosenjit Bose, Jean-Lou De Carufel, Darryl Hill et al.

Given a finite set $P\subset\mathbb{R}^2$, the directed Theta-6 graph, denoted $\vecΘ_6(P)$, is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The $\vecΘ_6(P)$-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 $\vecΘ_6(P)$-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 $\vecΘ_6(P)$-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 $\vecΘ_k(P)$-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.

Mark de Berg, Prosenjit Bose, Leonidas Theocharous

Many algorithmic problems can be solved (almost) as efficiently in metric spaces of bounded doubling dimension as in Euclidean space. Unfortunately, the metric space defined by points in a simple polygon equipped with the geodesic distance does not necessarily have bounded doubling dimension. We therefore study the doubling dimension of fat polygons, for two well-known fatness definitions. We prove that locally-fat simple polygons do not always have bounded doubling dimension, while any $(α,β)$-covered polygon does have bounded doubling dimension (even if it has holes). We also study the perimeter of geodesically convex sets in $(α,β)$-covered polygons (possibly with holes), and show that this perimeter is at most a constant times the Euclidean diameter of the set. Using these two results, we obtain new results for several problems on $(α,β)$-covered polygons, including an algorithm that computes the closest pair of a set of $m$ points in an $(α,β)$-covered polygon with $n$ vertices that runs in $O(n + m\log{n})$ expected time.

Prosenjit Bose, Guillermo Esteban, David Orden et al.

Let $ Π(n) $ be the largest number such that for every set $ S $ of $ n $ points in a polygon~$ P $, there always exist two points $ x, y \in S $, where every geodesic disk containing $ x $ and $ y $ contains $ Π(n) $ points of~$ S $. We establish upper and lower bounds for $ Π(n)$, and show that $ \left\lceil \frac{n}{5}\right\rceil+1 \leq Π(n) \leq \left\lceil \frac{n}{4} \right\rceil +1 $. We also show that there always exist two points $x, y\in S$ such that every geodesic disk with $x$ and $y$ on its boundary contains at least $ \frac{n}{7+\sqrt{37}} \approx \left\lceil \frac{n}{13.1} \right\rceil$ points both inside and outside the disk. For the special case where the points of $ S $ are restricted to be the vertices of a geodesically convex polygon we give a tight bound of $\left\lceil \frac{n}{3} \right\rceil + 1$. We provide the same tight bound when we only consider geodesic disks having $ x $ and $ y $ as diametral endpoints. We give upper and lower bounds of $\left\lceil \frac{n}{5} \right\rceil + 1 $ and $\frac{n}{6+\sqrt{26}} \approx \left\lceil \frac{n}{11.1} \right\rceil$, respectively, for the two-colored version of the problem. Finally, for the two-colored variant we show that there always exist two points $x, y\in S$ where $x$ and $y$ have different colors and every geodesic disk with $x$ and $y$ on its boundary contains at least $\left\lceil \frac{n}{27.1}\right\rceil+1$ points both inside and outside the disk.

Prosenjit Bose, Pilar Cano, Rodrigo I. Silveira

We study affine invariant 2D triangulation methods. That is, methods that produce the same triangulation for a point set $S$ for any (unknown) affine transformation of $S$. Our work is based on a method by Nielson [A characterization of an affine invariant triangulation. Geom. Mod, 191-210. Springer, 1993] that uses the inverse of the covariance matrix of $S$ to define an affine invariant norm, denoted $A_{S}$, and an affine invariant triangulation, denoted ${DT}_{A_{S}}[S]$. We revisit the $A_{S}$-norm from a geometric perspective, and show that ${DT}_{A_{S}}[S]$ can be seen as a standard Delaunay triangulation of a transformed point set based on $S$. We prove that it retains all of its well-known properties such as being 1-tough, containing a perfect matching, and being a constant spanner of the complete geometric graph of $S$. We show that the $A_{S}$-norm extends to a hierarchy of related geometric structures such as the minimum spanning tree, nearest neighbor graph, Gabriel graph, relative neighborhood graph, and higher order versions of these graphs. In addition, we provide different affine invariant sorting methods of a point set $S$ and of the vertices of a polygon $P$ that can be combined with known algorithms to obtain other affine invariant triangulation methods of $S$ and of $P$.

Prosenjit Bose, Jean-Lou De Carufel, Michael G. Dobbins et al.

A "shadow" of a subset $S$ of Euclidean space is an orthogonal projection of $S$ into one of the coordinate hyperplanes. In this paper we show that it is not possible for all three shadows of a cycle (i.e., a simple closed curve) in $\mathbb R^3$ to be paths (i.e., simple open curves). We also show two contrasting results: the three shadows of a path in $\mathbb R^3$ can all be cycles (although not all convex) and, for every $d\geq 1$, there exists a $d$-sphere embedded in $\mathbb R^{d+2}$ whose $d+2$ shadows have no holes (i.e., they deformation-retract onto a point).