Computational Geometry-Theory and Applications最新文献

We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each “hole” is bounded by three pairwise tangent discs are called triangulated. There are 164 pairs $(r,s)$, $1>r>s$, allowing triangulated packings by discs of radii 1, r and s. In this paper, we enhance existing methods of dealing with maximal-density packings in order to find ternary triangulated packings which maximize the density among all the packings with the same disc radii. We showed for 16 pairs that the density is maximized by a triangulated ternary packing; for 16 other pairs, we proved the density to be maximized by a triangulated packing using only two sizes of discs; for 45 pairs, we found non-triangulated packings strictly denser than any triangulated one; finally, we classified the remaining cases where our methods are not applicable.

Let P be a set of n points in $R^d$ where each point $p\in P$ carries a weight drawn from a commutative monoid $(M,+,0)$. Given a d-rectangle $r_{\mathrm{upd}}$ (i.e., an orthogonal rectangle in $R^d$) and a value $\Delta \in M$, a range update adds Δ to the weight of every point $p\in P\cap r_{\mathrm{upd}}$; given a d-rectangle $r_{\mathrm{qry}}$, a range sum query returns the total weight of the points in $P\cap r_{\mathrm{qry}}$. The goal is to store P in a structure to support updates and queries with attractive performance guarantees. We describe a structure of $\tilde{O}\left(n\right)$ space that handles an update in $\tilde{O}\left(T_{\mathrm{upd}}\right)$ time and a query in $\tilde{O}\left(T_{\mathrm{qry}}\right)$ time for arbitrary functions $T_{\mathrm{upd}}\left(n\right)$ and $T_{\mathrm{qry}}\left(n\right)$ satisfying $T_{\mathrm{upd}}\cdot T_{\mathrm{qry}}=n$. The result holds for any fixed dimensionality $d\ge 2$. Our query-update tradeoff is tight up to a polylog factor subject to the OMv-conjecture.

We characterize the triples of interior angles that are possible in non-self-crossing triangles with circular-arc sides, and we prove that a given cyclic sequence of angles can be realized by a non-self-crossing polygon with circular-arc sides whenever all angles are ≤π. As a consequence of these results, we prove that every cactus has a planar Lombardi drawing (a drawing with edges depicted as circular arcs, meeting at equal angles at each vertex) for its natural embedding in which every cycle of the cactus is a face of the drawing. However, there exist planar embeddings of cacti that do not have planar Lombardi drawings.

We present a randomized algorithm that with high probability finds a bottleneck matching in a set of $n=2\ell$ points in the plane. The algorithm's running time is $O(n^{\omega /2}\log n)$, where $\omega >2$ is a constant such that any two $n\times n$ matrices can be multiplied in time $O\left(n^{\omega }\right)$. The state of the art in fast matrix multiplication allows us to set $\omega =2.3728596$.

We study the priority set cover problem for simple geometric set systems in the plane. For pseudo-halfspaces in the plane we obtain a PTAS via local search by showing that the corresponding set system admits a planar support. We show that the problem is APX-hard even for unit disks in the plane and argue that in this case the standard local search algorithm can output a solution that is arbitrarily bad compared to the optimal solution. We then present an LP-relative constant factor approximation algorithm (which also works in the weighted setting) for unit disks via quasi-uniform sampling. As a consequence we obtain a constant factor approximation for the capacitated set cover problem with unit disks. For arbitrary size disks, we show that the problem is at least as hard as the vertex cover problem in general graphs even when the disks have nearly equal sizes. We also present a few simple results for unit squares and orthants in the plane.

In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with $\Delta >0$ unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of n axis-parallel affine subspaces of dimension Δ that yields an $(1+ϵ)$-approximate solution in $O\left(nd\right)$ time. The constants hidden behind $O(\cdot )$ depend only on $\Delta ,ϵ$ and k. This improves the $O\left(n^{2}d\right)$-time algorithm by Eiben et al. (2021) [7] by a factor of n.

We study problems related to colouring families of bottomless rectangles in the plane, in an attempt to improve the polychromatic k-colouring number $m_k^{⁎}$. This number is the smallest m such that any collection of bottomless rectangles can be k-coloured so that any m-fold covered point is covered by all k colours. We show that for many families of bottomless rectangles, such as unit-width bottomless rectangles, or bottomless rectangles whose left corners lie on a line, $m_k^{⁎}$ is linear in k. We present the lower bound $m_k^{⁎}\ge 2k-1$ for general families.

We also investigate semi-online colouring algorithms, which need not colour each vertex immediately, but must maintain a proper colouring. We prove that for many sweeping orders, for any positive integers $m,k$, there is no semi-online algorithm that can k-colour bottomless rectangles presented in that order, so that any m-fold covered point is covered by at least two colours. This holds even for translates of quadrants, and is a corollary of a stronger result for arborescence colourings: Any semi-online colouring algorithm that colours an arborescence presented in post-order may produce arbitrarily long monochromatic paths.

We study how to obtain partial matchings using the block function $M_f$, induced by a morphism f between persistence modules. $M_f$ is defined algebraically and is linear with respect to direct sums of morphisms. We study some interesting properties of $M_f$, and provide a way of obtaining $M_f$ using matrix operations.

Let P be a polygon and $C$ a set of shortcuts, where each shortcut is a directed straight-line segment connecting two vertices of P. A shortcut hull of P is another polygon that encloses P and whose oriented boundary is composed of elements from $C$. We require P and the output shortcut hull to be weakly simple polygons, which we define as a generalization of simple polygons. Shortcut hulls find their application in cartography, where a common task is to compute simplified representations of area features. We aim at a shortcut hull that has a small area and a small perimeter. Our optimization objective is to minimize a convex combination of these two criteria. If no holes in the shortcut hull are allowed, the problem admits a straight-forward solution via computation of shortest paths. For the more challenging case in which the shortcut hull may contain holes, we present a polynomial-time algorithm that is based on computing a constrained, weighted triangulation of the input polygon's exterior. We use this problem as a starting point for investigating further variants, e.g., restricting the number of edges or bends. We demonstrate that shortcut hulls can be used for the schematization of polygons.