Computational Geometry-Theory and Applications最新文献

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. However, there are 1-planar graphs that do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a two-coloring of the edges such that edges of the same color do not cross. Thus 1-planar graphs have geometric thickness two. In addition, the drawing is nearly 1-planar, that is, it is 1-planar if all fan-crossed edges are removed. An edge is fan-crossed if it is crossed by edges with a common vertex if it is crossed more than twice. The drawing algorithm uses high precision arithmetic with numbers with $O(n\log n)$ digits and computes the straight-line drawing from a 1-planar drawing in linear time on a real RAM.

A 3-prismatoid is the convex hull of two convex polygons A and B which lie in parallel planes $H_A,H_B\subset R^3$. Let $\tilde{A}$ be the orthogonal projection of A onto $H_B$. A 3-prismatoid is called nested if $\tilde{A}$ is properly contained in B, or vice versa. We show that every nested 3-prismatoid has an edge-unfolding to a non-overlapping polygon in the plane.

The Gromov-Hausdorff distance $\left(d_{GH}\right)$ proves to be a useful distance measure between shapes. In order to approximate $d_{GH}$ for $X,Y\subset R^d$, we look into its relationship with $d_{H,iso}$, the infimum Hausdorff distance under Euclidean isometries. As already known for dimension $d\ge 2$, $d_{H,iso}$ cannot be bounded above by a constant factor times $d_{GH}$. For $d=1$, however, we prove that $d_{H,iso}\le \frac{5}{4}{d}_{GH}$. We also show that the bound is tight. In effect, for $X,Y\subset R$ with at most n points, this gives rise to an $O(n\log n)$-time algorithm to approximate $d_{GH}(X,Y)$ with an approximation factor of $(1+\frac{1}{4})$.

In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. We show how to lift this combinatorial characterization of barcodes to an analogous combinatorialization of merge trees. As result of this study, we provide the first clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees: generic phylogenetic trees on $n+1$ leaf nodes fall into $(2n-1)!!$ distinct equivalence classes, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., $(n+1)!n!2^{-n}$. The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group. We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation of our distribution in terms of Dirichlet convolution. This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data, which opens the door to doing more precise statistics and science.

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.

This paper addresses a family of geometric half-plane retrieval queries of points in the plane in the presence of geometric uncertainty. The problems include exact and uncertain point sets and half-plane queries defined by an exact or uncertain line whose location uncertainties are independent or dependent and are defined by k real-valued parameters. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), an expressive and computationally efficient worst-case, first order linear approximation of geometric uncertainty that supports parametric dependencies between point locations. We present an efficient $O\left(k^2\right)$ time and space algorithm for computing the envelope of the LPGUM line that defines the half-plane query. For an exact line and an LPGUM n points set, we present an $O(\log nk+mk)$ time query and $O\left(nk\right)$ space algorithm, where m is the number of LPGUM points on or above the half-plane line. For a LPGUM line and an exact points set, we present a $O(k^2+\frac{(k\log n\log \log n)}{\epsilon }+m)$ time and $O(\frac{n^2}{\log n}+\frac{k}{\epsilon })$ space approximation algorithm, where $0<\epsilon \le 1$ is the desired approximation error. For a LPGUM line and an LPGUM points set, we present two $O(k^2+\frac{(k\log nk\log \log nk)}{\epsilon }+mk)$ and $O(m{k}^2+\frac{(k\log nk\log \log nk)}{\epsilon })$ time query and