Computational Geometry-Theory and Applications最新文献

We show that several classes of polyhedra are joined by a sequence of $O\left(1\right)$ refolding steps, where each refolding step unfolds the current polyhedron (allowing cuts anywhere on the surface and allowing overlap) and folds that unfolding into exactly the next polyhedron; in other words, a polyhedron is refoldable into another polyhedron if they share a common unfolding. Specifically, assuming equal surface area, we prove that (1) any two tetramonohedra are refoldable to each other, (2) any doubly covered triangle is refoldable to a tetramonohedron, (3) any (augmented) regular prismatoid and doubly covered regular polygon is refoldable to a tetramonohedron, (4) any tetrahedron has a 3-step refolding sequence to a tetramonohedron, and (5) the regular dodecahedron has a 4-step refolding sequence to a tetramonohedron. In particular, we obtain a ≤6-step refolding sequence between any pair of Platonic solids, applying (5) for the dodecahedron and (1) and/or (2) for all other Platonic solids. As far as the authors know, this is the first result about common unfolding involving the regular dodecahedron.

In this paper, we study two classic optimization problems: minimum geometric dominating set and set cover. In the dominating-set problem, for a given set of objects in the plane as input, the objective is to choose a minimum number of input objects such that every input object is dominated by the chosen set of objects. Here, we say that one object is dominated by another if their intersection is nonempty. For the second problem, for a given set of points and objects in the plane, the objective is to choose a minimum number of objects to cover all the points. This is a particular version of the set-cover problem.

Both problems have been well-studied, subject to various restrictions on the input objects. These problems are $\mathrm{APX}$-hard for object sets consisting of axis-parallel rectangles, ellipses, α-fat objects of constant description complexity, and convex polygons. On the other hand, $\mathrm{PTAS}$s (polynomial time approximation schemes) are known for object sets consisting of disks or unit squares. Surprisingly, a $\mathrm{PTAS}$ was unknown even for arbitrary squares. For both problems obtaining a $\mathrm{PTAS}$ remains open for a large class of objects.

For the dominating-set problem, we prove that a popular local-search algorithm leads to a $(1+\epsilon )$ approximation for a family of homothets of a convex object (which includes arbitrary squares, k-regular polygons, translated and scaled copies of a convex set, etc.) in $n^{O(1/{\epsilon }^2)}$ time. On the other hand, the same approach leads to a $\mathrm{PTAS}$ for the geometric covering problem when the objects are convex pseudodisks (which include disks, unit height rectangles, homothetic convex objects, etc.). Consequently, we obtain an easy-to-implement approximation algorithm for both problems for a large class of objects, significantly improving the best-known approximation guarantees.

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.

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.