We provide linear-time algorithms for geometric graphs with sublinearly many crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an iterated logarithmic factor. Specific problems we study include Voronoi diagrams and single-source shortest paths. Our algorithms all run in linear time in the standard comparison-based computational model; hence, we make no assumptions about the distribution or bit complexities of edge weights, nor do we utilize unusual bit-level operations on memory words. Instead, our algorithms are based on a planarization method that "zeroes in" on edge crossings, together with methods for extending planar separator decompositions to geometric graphs with sublinearly many crossings. Incidentally, our planarization algorithm also solves an open computational geometry problem of Chazelle for triangulating a self-intersecting polygonal chain having n segments and k crossings in linear time, for the case when k is sublinear in n by an iterated logarithmic factor.
A geometric graph is a graph whose vertices are points in general position in the plane and its edges are straight line segments joining these points. In this paper we give an $O(n^2 \log n)$ algorithm to compute the number of pairs of edges that cross in a geometric graph on $n$ points. For layered, and convex geometric graphs the algorithm takes $O(n^2)$ time.
The algorithm of Gutwenger et al. to insert an edge $e$ in linear time into a planar graph $G$ with a minimal number of crossings on $e$, is a helpful tool for designing heuristics that minimize edge crossings in drawings of general graphs. Unfortunately, some graphs do not have a geometric embedding $Γ$ such that $Γ+e$ has the same number of crossings as the embedding $G+e$. This motivates the study of the computational complexity of the following problem: Given a combinatorially embedded graph $G$, compute a geometric embedding $Γ$ that has the same combinatorial embedding as $G$ and that mi…
We consider the minimization of edge-crossings in geometric drawings of graphs $G=(V, E)$, i.e., in drawings where each edge is depicted as a line segment. The respective decision problem is NP-hard [Bienstock, '91]. In contrast to theory and the topological setting, the geometric setting did not receive a lot of attention in practice. Prior work [Radermacher et al., ALENEX'18] is limited to the crossing-minimization in geometric graphs with less than $200$ edges. The described heuristics base on the primitive operation of moving a single vertex $v$ to its crossing-minimal position, i.e., the …
Given $n$ points in the plane, we propose algorithms to compile connected crossing-free geometric graphs into directed acyclic graphs (DAGs). The DAGs allow efficient counting, enumeration, random sampling, and optimization. Our algorithms rely on Wettstein's framework to compile several crossing-free geometric graphs. One of the remarkable contributions of Wettstein is to allow dealing with geometric graphs with connectivity, since it is known to be difficult to efficiently represent geometric graphs with such global property. To achieve this, Wettstein proposed specialized techniques for cro…
We describe a framework for counting and enumerating various types of crossing-free geometric graphs on a planar point set. The framework generalizes ideas of Alvarez and Seidel, who used them to count triangulations in time $O(2^nn^2)$ where $n$ is the number of points. The main idea is to reduce the problem of counting geometric graphs to counting source-sink paths in a directed acyclic graph. The following new results will emerge. The number of all crossing-free geometric graphs can be computed in time $O(c^nn^4)$ for some $c < 2.83929$. The number of crossing-free convex partitions can be …
We prove crossing number inequalities for geometric graphs whose vertex sets are taken from a d-dimensional grid of volume N and give applications of these inequalities to counting the number of non-crossing geometric graphs that can be drawn on such grids. In particular, we show that any geometric graph with m >= 8N edges and with vertices on a 3D integer grid of volume N, has Ω((m^2/n)\log(m/n)) crossings. In d-dimensions, with d >= 4, this bound becomes Ω(m^2/n). We provide matching upper bounds for all d. Finally, for d >= 4 the upper bound implies that the maximum number of crossing-free …
Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-…
Let $S$ be a set of $n$ points in the plane in general position. Two line segments connecting pairs of points of $S$ cross if they have an interior point in common. Two vertex disjoint geometric graphs with vertices in $S$ cross if there are two edges, one from each graph, which cross. A set of vertex disjoint geometric graphs with vertices in $S$ is called mutually crossing if any two of them cross. We show that there exists a constant $c$ such that from any family of $n$ mutually crossing triangles, one can always obtain a family of at least $n^c$ mutually crossing $2$-paths (each of which i…
We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits Ω(8.65^n) different triangulations. This improves the bound Ω(8.…
Inspired by the increasingly popular research on extending partial graph drawings, we propose a new perspective on the traditional and arguably most important geometric graph parameter, the crossing number. Specifically, we define the partially predrawn crossing number to be the smallest number of crossings in any drawing of a graph, part of which is prescribed on the input (not counting the prescribed crossings). Our main result - an FPT-algorithm to compute the partially predrawn crossing number - combines advanced ideas from research on the classical crossing number and so called partial pl…
Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is one in which each graph is drawn planar, there are no edge overlaps and the crossings between the two graphs form right angles. The geometric version restricts the problem to straight-line drawings. It is known, however, that there exists a wheel and a matching which do not admit a geometric RAC simultaneous drawing. In order to enlarge the class of graphs that admit RAC simultaneous drawings, we allow bends in the resulting drawings. We prove that two planar graphs always admit a RAC simultaneo…
Let $P$ be a set of points in general position in the plane. Join all pairs of points in $P$ with straight line segments. The number of segment-crossings in such a drawing, denoted by $\crg(P)$, is the \emph{rectilinear crossing number} of $P$. A \emph{halving line} of $P$ is a line passing though two points of $P$ that divides the rest of the points of $P$ in (almost) half. The number of halving lines of $P$ is denoted by $h(P)$. Similarly, a $k$\emph{-edge}, $0\leq k\leq n/2-1$, is a line passing through two points of $P$ and leaving exactly $k$ points of $P$ on one side. The number of $(\le…
Beyond-planarity focuses on the study of geometric and topological graphs that are in some sense nearly-planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations. Early research dates back to the 1960s (e.g., Avital and Hanani 1966) for extremal problems on geometric graphs, but is also related to graph drawing problems where visual clutter by edge crossings should be minimized (e.g., Huang et al. 2008) that could negatively affect the readability of the drawing. Different types of forbidden crossing configurations give…
Let $X_{2k}$ be a set of $2k$ labeled points in convex position in the plane. We consider geometric non-intersecting straight-line perfect matchings of $X_{2k}$. Two such matchings, $M$ and $M'$, are disjoint compatible if they do not have common edges, and no edge of $M$ crosses an edge of $M'$. Denote by $\mathrm{DCM}_k$ the graph whose vertices correspond to such matchings, and two vertices are adjacent if and only if the corresponding matchings are disjoint compatible. We show that for each $k \geq 9$, the connected components of $\mathrm{DCM}_k$ form exactly three isomorphism classes -- n…
Khoury College of Computer Sciences, Northeastern University（东北大学科里计算机科学学院）；School of Computer Science, University of Windsor（温莎大学计算机学院）；School of Computer Science, Carleton University（卡尔顿大学计算机学院）；School of Electrical Engineering and Computer Science, University of Ottawa（渥太华大学电气与计算机工程学院）；Computer Science Department, University of California, Irvine（加州大学欧文分校计算机科学系）；Department of Mathematics, California State University Northridge（北岭加利福尼亚州立大学数学系）；Department of Computer Science, Tufts University（塔夫茨大学计算机科学系）；Department of Applied Mathematics, Charles University（查理大学应用数学系）
We use production matrices to count several classes of geometric graphs. We present novel production matrices for non-crossing partitions, connected geometric graphs, and k-angulations, which provide another way of counting the number of such objects. Counting geometric graphs is then equivalent to calculating the powers of a production matrix. Applying the technique of Riordan Arrays to these production matrices, we establish new formulas for the numbers of geometric graphs as well as combinatorial identities derived from the production matrices. Further, we obtain the characteristic polynomi…
A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which i…
Using Reinforcement Learning to Optimize the Global and Local Crossing Number
Technical University of Munich, Heilbronn, Germany（慕尼黑技术大学（海因斯贝格））；John Cabot University, Rome, Italy（约翰·卡博特大学）；Technical University of Munich, Garching, Germany（慕尼黑技术大学（戈林根））
