Given a set $P$ of red and blue points in the plane, a planar bichromatic spanning tree of $P$ is a geometric spanning tree of $P$, such that each edge connects a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree $T$, such that the length of the longest edge (i.e., bottleneck) in $T$ is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time $(8\sqrt{2})$-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck $\lambda$ can be converted to a planar bichromatic spanning tree of bottleneck at most $8\sqrt{2}\lambda$.

We study range-searching for colored objects, where one has to count (approxi- mately) the number of colors present in a query range. The problems studied mostly involve orthogonal range-searching in two and three dimensions, and the dual setting of rectangle stabbing by points. We present optimal and near-optimal solutions for these problems. Most of the results are obtained via reductions to the approximate uncolored version, and improved data-structures for them. An additional contribution of this work is the introduction of nested shallow cuttings.

The well-separated pair decomposition (WSPD) of the complete Euclidean graph defined on points in $R^2$, introduced by Callahan and Kosaraju [JACM, 42 (1): 67-90, 1995], is a technique for partitioning the edges of the complete graph based on length into a linear number of sets. Among the many different applications of WSPDs, Callahan and Kosaraju proved that the sparse subgraph that results by selecting an arbitrary edge from each set (called WSPD-spanner) is a $1 + 8/(s − 4)$-spanner, where $s > 4$ is the separation ratio used for partitioning the edges.Although competitive local-routing strategies exist for various spanners such as Yao-graphs, $\Theta$-graphs, and variants of Delaunay graphs, few local-routing strategies are known for any WSPD-spanner. Our main contribution is a local-routing algorithm with a near-optimal competitive routing ratio of $1 + O(1/s)$ on a WSPD-spanner.Specifically, using Callahan and Kosaraju’s fair split-tree, we show how to build a WSPD-spanner with spanning ratio $1 + 4/s + 4/(s − 2)$ which is a slight improvement over $1 + 8/(s − 4)$. We then present a 2-local and a 1-local routing algorithm on this spanner with competitive routing ratios of $1 + 6/(s − 2) + 4/s$ and $1 + 8/(s − 2) + 4/s + 8/s^2$, respectively. Moreover, we prove that there exists a point set for which our WSPD-spanner has a spanning ratio of at least $1 + 8/s$, thereby proving the near-optimality of its spanning ratio and the near-optimality of the routing ratio of both our routing algorithms.

In the recent study of crossing numbers, drawings of graphs that can be extended to an arrangement of pseudolines (pseudolinear drawings) have played an important role as they are a natural combinatorial extension of rectilinear (or straight-line) drawings. A characterization of the pseudolinear drawings of $K_n$ was found recently. We extend this characterization to all graphs, by describing the set of minimal forbidden subdrawings for pseudolinear drawings. Our characterization also leads to a polynomial-time algorithm to recognize pseudolinear drawings and construct the pseudolines when it is possible.

$\DeclareMathOperator{\poly}{poly}$In the classic polyline simplification problem, given a polygonal curve~$P$ consisting of $n$ vertices and an error threshold $\delta \geq 0$, we want to replace $P$ by a subsequence~$Q$ of minimal size such that the distance between the polygonal curves $P$ and $Q$ is at most $\delta$. The distance between curves is usually measured using the Hausdorff or continuous Frechet distance. These distance measures can be applied globally, i.e., to the whole curves $P$ and $Q$, or locally, i.e., to each simplified subcurve and the line segment that it was replaced with separately (and then taking the maximum). This gives rise to four problem variants: Global-Hausdorff simplification (known to be NP-hard), Local-Hausdorff simplification (can be solved in time $O(n^3)$), Global-Frechet simplification (can be solved in time $O(k n^5)$, where $k$ is the size of the optimum simplification), and Local-Fr\'{e}chet simplification (can be solved in time $O(n^3)$).Our contribution is as follows:Cubic time for all variants:\ For Global-Frechet simplification, we design an algorithm running in time $O(n^3)$. This shows that all three problems (Local-Hausdorff, Local-Frechet, and Global-Frechet) can be solved in cubic time. All these algorithms work over a general metric space such as $(\mathbb{R}^d,L_p)$, but the hidden constant depends on $p$ and (linearly) on~$d$.Cubic conditional lower bound: We provide evidence that in high dimensions, cubic time is essentially optimal for all three problems (Local-Hausdorff, Local-Frechet, and Global-Frechet). Specifically, improving the cubic time to $O(n^{3-\epsilon} \poly(d))$ for polyline simplification over $(\mathbb{R}^d,L_p)$ for $p = 1$ would violate plausible conjectures. We obtain similar results for all $p \in [1,\infty), p \ne 2$.In total, in high dimensions and over general $L_p$-norms we resolve the complexity of polyline simplification with respect to Local-Hausdorff, Local-Frechet, and Global-Frechet, by a providing new algorithm and conditional lower bounds.

This note corrects an error in the paper Approximating minimum-area rectangular and convex containers for packing convex polygons, which appeared in JoCG, Vol.~8(1), pages~1--10.

We consider a range-search variant of the closest-pair problem. Let $\varGamma$ be a fixed shape in the plane. We are interested in storing a given set of $n$ points in the plane in some data structure such that for any specified translate of $\varGamma$, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when $\varGamma$ is a polygon (possibly with holes) and when $\varGamma$ is a general convex body whose boundary is smooth. When $\varGamma$ is a polygon, we present a data structure using $O(n)$ space and $O(\log n)$ query time, which is asymptotically optimal. When $\varGamma$ is a general convex body with a smooth boundary, we give a near-optimal data structure using $O(n \log n)$ space and $O(\log^2 n)$ query time. Our results settle some open questions posed by Xue et al. [SoCG 2018].

We present an explicit PL-embedding of the flat square torus $\mathbb{T}^2=\mathbb{E}^2/\mathbb{Z}^2$ into $\mathbb{E}^3$, with 40 vertices and 80 faces.

Let $D$ be a set of $n$ pairwise disjoint unit balls in $R^d$ and $P$ the set of their centers. A hyperplane $H$ is an $m$-separator for $D$ if every closed halfspace bounded by $H$ contains at least $m$ points from $P$. This generalizes the notion of halving hyperplanes, which correspond to $n/2$-separators. The analogous notion for point sets is well studied. Separators have various applications, for instance, in divide-and-conquer schemes. In such a scheme, any ball that is intersected by the separating hyperplane may still interact with both sides of the partition. Therefore it is desirable that the separating hyperplane intersects a small number of balls only. We present three deterministic algorithms to bisect a given set of pairwise disjoint unit balls by a hyperplane. Firstly, we present a simple linear-time algorithm to construct an $\alpha n$-separator for balls in $R^d$, for any $0<\alpha<1/2$, that intersects at most $cn^{(d-1)/d}$ balls, for some constant $c$ that depends on $d$ and $\alpha$. The number of intersected balls is best possible up to the constant $c$. Secondly, we present a near-linear-time algorithm to construct an $(n/2-o(n))$-separator in $R^d$ that intersects $o(n)$ balls. Finally, we give a linear-time algorithm to construct a halving line in $\mathbb{R}^2$ for $P$ that intersects $O(n^{(2/3)+\epsilon})$ disks.We also point out how the above theorems can be generalized to more general classes of shapes, possibly with some overlap, and what are the limits of those generalizations.Our results improve the runtime of a disk sliding algorithm by Bereg, Dumitrescu and Pach. In addition, our results simplify and derandomize an algorithm to construct a space decomposition used by Loffler and Mulzer to construct an onion (convex layer) decomposition for imprecise points (any point resides at an unknown location within a given disk).