Computational Geometry-Theory and Applications最新文献

Many papers study the natural problem of drawing nonplanar graphs with few crossings per edge. In particular, a graph is 1-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time and several efficient algorithms have been described in the literature, deciding whether a graph is 1-planar is NP-complete, even for restricted classes of graphs. Despite some polynomial-time algorithms are known for recognizing specific subfamilies of 1-planar graphs, there is still a lack of practical 1-planarity testing algorithms and no implementation is available for general graphs. This paper investigates the feasibility of a 1-planarity testing and embedding algorithm based on a backtracking strategy. Our contribution provides initial indications that have the potential to stimulate further research on the design of practical approaches for the 1-planarity testing problem. On the one hand, our experiments show that a backtracking strategy can be successfully applied to graphs with up to 30 vertices. On the other hand, our study suggests that alternative techniques are needed to attack larger graphs.

In 2012 Driemel et al. introduced the concept of c-packed curves as a realistic input model. In the case when c is a constant they gave a near linear time $(1+\epsilon )$-approximation algorithm for computing the Fréchet distance between two c-packed polygonal curves. Since then a number of papers have used the model.

In this paper we consider the problem of computing the smallest c for which a given polygonal curve in $R^d$ is c-packed. We present two approximation algorithms. The first algorithm is a 2-approximation algorithm and runs in $O(d{n}^2\log n)$ time. In the case $d=2$ we develop a faster algorithm that returns a $(6+\epsilon )$-approximation and runs in $O\left({(n/{\epsilon }^3)}^{4/3}\mathrm{polylog}\right(n/\epsilon \left)\right))$ time.

We also implemented the first algorithm and computed the approximate packedness-value for 16 sets of real-world trajectories. The experiments indicate that the notion of c-packedness is a useful realistic input model for many curves and trajectories.

We consider the problem of enumerating integer tetrahedra of fixed perimeter (sum of side-lengths) and/or diameter (maximum side-length), up to congruence. As we will see, this problem is considerably more difficult than the corresponding problem for triangles, which has long been solved. We expect there are no closed-form solutions to the tetrahedron enumeration problems, but we explore the extent to which they can be approached via classical methods, such as orbit enumeration. We also discuss algorithms for computing the numbers, and present several tables and figures that can be used to visualise the data. Several intriguing patterns seem to emerge, leading to a number of natural conjectures. The central conjecture is that the number of integer tetrahedra of perimeter n, up to congruence, is asymptotic to $n^5/C$ for some constant $C\approx 229000$.

Bounded-angle (minimum) spanning trees were first introduced in the context of wireless networks with directional antennas. They are reminiscent of bounded-degree (minimum) spanning trees, which have received significant attention. Let P be a set of n points in the plane, and let $0<\alpha <2\pi$ be an angle. An α-spanning tree (α-ST) of P is a spanning tree of the complete Euclidean graph over P, with the following property: For each vertex $p_i\in P$, the (smallest) angle that is spanned by all the edges incident to $p_i$ is at most α. An α-minimum spanning tree (α-MST) is an α-ST of P of minimum weight, where the weight of an α-ST is the sum of the lengths of its edges. In this paper, we consider the problem of computing an α-MST for the case where $\alpha =\frac{2\pi }{3}$. We present a 4-approximation algorithm, thus improving upon the previous results of Aschner and Katz and Biniaz et al., who presented algorithms with approximation ratios 6 and $\frac{16}{3}$, respectively.

To obtain this result, we devise an $O\left(n\right)$-time algorithm that, given any Hamiltonian path Π of P, constructs a $\frac{2\pi }{3}$-ST $T$ of P, such that $T$'s weight is at most twice that of Π and, moreover, $T$ is a 3-hop spanner of Π. This latter result is optimal (with respect to $T$'s weight), since for any $\epsilon >0$ there exists a polygonal path for which every $\frac{2\pi }{3}$-ST (of the corresponding set of points) has weight greater than $2-\epsilon$ times the weight of the path.

We study polyiamonds (polygons arising from the triangular grid) that fold into the smallest yet unstudied platonic solid – the octahedron. We show a number of results. Firstly, we characterize foldable polyiamonds containing a hole of positive area, namely each but one polyiamond is foldable. Secondly, we show that a convex polyiamond folds into the octahedron if and only if it contains one of five polyiamonds. We thirdly present a sharp size bound: While there exist unfoldable polyiamonds of size 14, every polyiamond of size at least 15 folds into the octahedron. This clearly implies that one can test in polynomial time whether a given polyiamond folds into the octahedron. Lastly, we show that for any assignment of positive integers to the faces, there exists a polyiamond that folds into the octahedron such that the number of triangles covering a face is equal to the assigned number.

Schelling's classical segregation model gives a coherent explanation for the wide-spread phenomenon of residential segregation. We introduce an agent-based saturated open-city variant, the Flip Schelling Process (FSP), in which agents, placed on a graph, have one out of two types and, based on the predominant type in their neighborhood, decide whether to change their types; similar to a new agent arriving as soon as another agent leaves the vertex.

We investigate the probability that an edge $\{u,v\}$ is monochrome, i.e., that both vertices u and v have the same type in the FSP, and we provide a general framework for analyzing the influence of the underlying graph topology on residential segregation. In particular, for two adjacent vertices, we show that a highly decisive common neighborhood, i.e., a common neighborhood where the absolute value of the difference between the number of vertices with different types is high, supports segregation and, moreover, that large common neighborhoods are more decisive.

As an application, we study the expected behavior of the FSP on two common random graph models with and without geometry: (1) For random geometric graphs, we show that the existence of an edge $\{u,v\}$ makes a highly decisive common neighborhood for u and v more likely. Based on this, we prove the existence of a constant $c>0$ such that the expected fraction of monochrome edges after the FSP is at least $1/2+c$. (2) For Erdős–Rényi graphs we show that large common neighborhoods are unlikely and that the expected fraction of monochrome edges after the FSP is at most $1/2+o\left(1\right)$. Our results indicate that the cluster structure of the underlying graph has a significant impact on the obtained segregation strength.

We consider a minimizing variant of the well-known No-Three-In-Line Problem, the Geometric Dominating Set Problem: What is the smallest number of points in an $n\times n$ grid such that every grid point lies on a common line with two of the points in the set? We show a lower bound of $\Omega \left(n^{2/3}\right)$ points and provide a constructive upper bound of size $2\lceil n/2\rceil$. If the points of the dominating sets are required to be in general position we provide optimal solutions for grids of size up to $12\times 12$. For arbitrary n the currently best upper bound for points in general position remains the obvious 2n. Finally, we discuss the problem on the discrete torus where we prove an upper bound of $O\left({(n\log n)}^{1/2}\right)$. For n even or a multiple of 3, we can even show a constant upper bound of 4. We also mention a number of open questions and some further variations of the problem.

A graph is 1-planar, if it can be drawn in the plane such that there is at most one crossing on every edge. It is known, that 1-planar graphs have at most $4n-8$ edges.

We prove the following odd-even generalization. If a graph can be drawn in the plane such that every edge is crossed by at most one other edge an odd number of times, then it is called 1-odd-planar and it has at most $5n-9$ edges. As a consequence, we improve the constant in the Crossing Lemma for the odd-crossing number, if adjacent edges cross an even number of times. We also give upper bound for the number of edges of k-odd-planar graphs.

In this paper, we investigate the way of unfolding a given tetramonohedron, which is a tetrahedron that consists of four congruent triangles. Our aim is finding a way that achieves the minimum cut length to develop it. We first show the rigorous way to unfold any given tetramonohedron with minimum cut length. Next, we focus on a family of tetramonohedra that consist of four congruent isosceles triangles. For this family, we apply our result and investigate their behavior.

A polyomino is a shape best described as a connected set of cells in the square lattice. As part of recreational mathematics, polyominoes have seen active research since the 1950s. Simultaneously, polyominoes have been investigated in statistical physics under the name “lattice animals,” mainly in regards to percolation problems. One of the main points of interest is to solve the yet unanswered question of how many different polyominoes exist. Most of the focus, so far, went to estimating the number of different polyominoes that can be made with a given fixed number of cells. Recently, there are increased efforts to discover the number of polyominoes with not only a given area, but with a given perimeter size or perimeter defect as well.

Roughly speaking, the perimeter defect is a number that measures how many twists a polyomino has. Formally, the defect of a polyomino P is defined as the deviation of the perimeter size of P from the maximum possible perimeter size taken over all polyominoes of the same area as P. Interestingly, a polyomino might contain a column or row which can be “cut” out of the polyomino, then the remaining parts of the polyomino are “glued” back together along the cut, and result in a smaller polyomino with equal perimeter defect to the original. By repeating such “cut-and-glue” operations on a polyomino until all matching columns and rows have been removed, one obtains a so-called “reduced polyomino.”

We expand on the efforts regarding perimeter defect and reduced polyominoes, in two directions. First, given a fixed perimeter defect, we demonstrate and prove upper and lower bounds on the width and height, as well as an upper bound on the area, of any reduced polyomino with a given perimeter defect. Second, we present an algorithm for enumerating all reduced polyominoes with a given perimeter defect, and calculate their combined generating function. From the generating function, we can extract a formula $A_{\left(k\right)}\left(n\right)$ for the number of polyominoes that have the fixed perimeter defect k and area n, for any n. Using our new algorithm, and in the case of $k=5$ some additional manual calculations, we provide closed formulae of $A_{\left(k\right)}\left(n\right)$, for up to $k=5$, as well as the generating functions for up to $k=5$. This is an improvement over the previously known formulae, which were known only up to $k=3$.