arXiv subjects
Publications and source records attributed to Anna Lubiw.
We consider the conjecture by Aichholzer, Aurenhammer, Hurtado, and Krasser that any two points sets with the same cardinality and the same size convex hull can be triangulated in the "same" way, more precisely via \emph{compatible triangulations}. We show counterexamples to various strengthened versions of this conjecture.
For a matroid with an ordered (or "labelled") basis, a basis exchange step removes one element with label $l$ and replaces it by a new element that results in a new basis, and with the new element assigned label $l$. We prove that one labelled basis can be reconfigured to another if and only if for every label, the initial and final elements with that label lie in the same connected component of the matroid. Furthermore, we prove that when the reconfiguration is possible, the number of basis exchange steps required is $O(r^{1.5})$ for a rank $r$ matroid. For a graphic matroid we improve the bound to $O(r \log r)$.
A geometric graph is angle-monotone if every pair of vertices has a path between them that---after some rotation---is $x$- and $y$-monotone. Angle-monotone graphs are $\sqrt 2$-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone---specifically, we prove that the half-$θ_6$-graph is generalized angle-monotone. We give a local routing algorithm for Gabriel triangulations that finds a path from any vertex $s$ to any vertex $t$ whose length is within $1 + \sqrt 2$ times the Euclidean distance from $s$ to $t$. Finally, we prove some lower bounds and limits on local routing algorithms on Gabriel triangulations.
Given an $n$-vertex graph and two straight-line planar drawings of the graph that have the same faces and the same outer face, we show that there is a morph (i.e., a continuous transformation) between the two drawings that preserves straight-line planarity and consists of $O(n)$ steps, which we prove is optimal in the worst case. Each step is a unidirectional linear morph, which means that every vertex moves at constant speed along a straight line, and the lines are parallel although the vertex speeds may differ. Thus we provide an efficient version of Cairns' 1944 proof of the existence of straight-line planarity-preserving morphs for triangulated graphs, which required an exponential number of steps.
A rectifier network is a directed acyclic graph with distinguished sources and sinks; it is said to compute a Boolean matrix $M$ that has a $1$ in the entry $(i,j)$ iff there is a path from the $j$th source to the $i$th sink. The smallest number of edges in a rectifier network computing $M$ is a classic complexity measure on matrices, which has been studied for more than half a century. We explore two well-known techniques that have hitherto found little to no applications in this theory. Both of them build on a basic fact that depth-$2$ rectifier networks are essentially weighted coverings of Boolean matrices with rectangles. We obtain new results by using fractional and greedy coverings (defined in the standard way). First, we show that all fractional coverings of the so-called full triangular matrix have cost at least $n\log n$. This provides (a fortiori) a new proof of the tight lower bound on its depth-$2$ complexity (the exact value has been known since 1965, but previous proofs are based on different arguments). Second, we show that the greedy heuristic is instrumental in tightening the upper bound on the depth-$2$ complexity of the Kneser-Sierpiński (disjointness) matrix. The previous upper bound is $O(n^{1.28})$, and we improve it to $O(n^{1.17})$, while the best known lower bound is $Ω(n^{1.16})$. Third, using fractional coverings, we obtain a form of direct product theorem that gives a lower bound on unbounded-depth complexity of Kronecker (tensor) products of matrices. In this case, the greedy heuristic shows (by an argument due to Lovász) that our result is only a logarithmic factor away from the "full" direct product theorem. Our second and third results constitute progress on open problem 7.3 and resolve, up to a logarithmic factor, open problem 7.5 from a recent book by Jukna and Sergeev (in Foundations and Trends in Theoretical Computer Science (2013)).
Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique labels to the triangulation edges. We introduce the concept of the orbit of an edge $e$, which is the set of all edges reachable from $e$ via flips. We establish the first upper and lower bounds on the diameter of the flip graph in this setting. Specifically, we prove tight $Θ(n \log n)$ bounds for edge-labelled triangulations of $n$-vertex convex polygons and combinatorial triangulations, contrasting with the $Θ(n)$ bounds in their respective unlabelled settings. The $Ω(n \log n)$ lower bound for the convex polygon setting might be of independent interest, as it generalizes lower bounds on certain sorting models. When simultaneous flips are allowed, the upper bound for convex polygons decreases to $O(\log^2 n)$, although we no longer have a matching lower bound. Moving beyond convex polygons, we show that edge-labelled triangulated polygons with a single reflex vertex can have a disconnected flip graph. This is in sharp contrast with the unlabelled case, where the flip graph is connected for any triangulated polygon. For spiral polygons, we provide a complete characterization of the orbits. This allows us to decide connectivity of the flip graph of a spiral polygon in linear time. We also prove an upper bound of $O(n^2)$ on the diameter of each connected component, which is optimal in the worst case. We conclude with an example of a non-spiral polygon whose flip graph has diameter $Ω(n^3)$.
We study versions of cop and robber pursuit-evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit-evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs.
We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case.
In this paper, we study how to fold a specified origami crease pattern in order to minimize the impact of paper thickness. Specifically, origami designs are often expressed by a mountain-valley pattern (plane graph of creases with relative fold orientations), but in general this specification is consistent with exponentially many possible folded states. We analyze the complexity of finding the best consistent folded state according to two metrics: minimizing the total number of layers in the folded state (so that a "flat folding" is indeed close to flat), and minimizing the total amount of paper required to execute the folding (where "thicker" creases consume more paper). We prove both problems strongly NP-complete even for 1D folding. On the other hand, we prove the first problem fixed-parameter tractable in 1D with respect to the number of layers.
Alamdari et al. showed that given two straight-line planar drawings of a graph, there is a morph between them that preserves planarity and consists of a polynomial number of steps where each step is a \emph{linear morph} that moves each vertex at constant speed along a straight line. An important step in their proof consists of converting a \emph{pseudo-morph} (in which contractions are allowed) to a true morph. Here we introduce the notion of \emph{unidirectional morphing} step, where the vertices move along lines that all have the same direction. Our main result is to show that any planarity preserving pseudo-morph consisting of unidirectional steps and contraction of low degree vertices can be turned into a true morph without increasing the number of steps. Using this, we strengthen Alamdari et al.'s result to use only unidirectional morphs, and in the process we simplify the proof.
We consider the problem of morphing between two planar drawings of the same triangulated graph, maintaining straight-line planarity. A paper in SODA 2013 gave a morph that consists of $O(n^2)$ steps where each step is a linear morph that moves each of the $n$ vertices in a straight line at uniform speed. However, their method imitates edge contractions so the grid size of the intermediate drawings is not bounded and the morphs are not good for visualization purposes. Using Schnyder embeddings, we are able to morph in $O(n^2)$ linear morphing steps and improve the grid size to $O(n)\times O(n)$ for a significant class of drawings of triangulations, namely the class of weighted Schnyder drawings. The morphs are visually attractive. Our method involves implementing the basic "flip" operations of Schnyder woods as linear morphs.
We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem---to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph---and the simultaneous planarity problem---to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required planar drawings exist, then there are planar drawings with a linear number of bends per edge and, in the case of simultaneous planarity, a constant number of crossings between every pair of edges. Our proofs provide efficient algorithms if the combinatorial embedding of the drawing is given. Our result on partially embedded graph drawing generalizes a classic result by Pach and Wenger which shows that any planar graph can be drawn with a linear number of bends per edge if the location of each vertex is fixed.
Can folding a piece of paper flat make it larger? We explore whether a shape $S$ must be scaled to cover a flat-folded copy of itself. We consider both single folds and arbitrary folds (continuous piecewise isometries $S\rightarrow R^2$). The underlying problem is motivated by computational origami, and is related to other covering and fixturing problems, such as Lebesgue's universal cover problem and force closure grasps. In addition to considering special shapes (squares, equilateral triangles, polygons and disks), we give upper and lower bounds on scale factors for single folds of convex objects and arbitrary folds of simply connected objects.
Consider an orthogonal polyhedron, i.e., a polyhedron where (at least after a suitable rotation) all faces are perpendicular to a coordinate axis, and hence all edges are parallel to a coordinate axis. Clearly, any facial angle and any dihedral angle is a multiple of $π/2$. In this note we explore the converse: if the facial and/or dihedral angles are all multiples of $π/2$, is the polyhedron necessarily orthogonal? The case of facial angles was answered previously. In this note we show that if both the facial and dihedral angles are multiples of $π/2$ then the polyhedron is orthogonal (presuming connectivity), and we give examples to show that the condition for dihedral angles alone does not suffice.
We study a geometric representation problem, where we are given a set $\cal R$ of axis-aligned rectangles with fixed dimensions and a graph with vertex set $\cal R$. The task is to place the rectangles without overlap such that two rectangles touch if and only if the graph contains an edge between them. We call this problem Contact Representation of Word Networks (CROWN). It formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. Here, we represent words by rectangles and semantic relationships by edges. We show that CROWN is strongly NP-hard even restricted trees and weakly NP-hard if restricted stars. We consider the optimization problem Max-CROWN where each adjacency induces a certain profit and the task is to maximize the sum of the profits. For this problem, we present constant-factor approximations for several graph classes, namely stars, trees, planar graphs, and graphs of bounded degree. Finally, we evaluate the algorithms experimentally and show that our best method improves upon the best existing heuristic by 45%.
In this paper we introduce self-approaching graph drawings. A straight-line drawing of a graph is self-approaching if, for any origin vertex s and any destination vertex t, there is an st-path in the graph such that, for any point q on the path, as a point p moves continuously along the path from the origin to q, the Euclidean distance from p to q is always decreasing. This is a more stringent condition than a greedy drawing (where only the distance between vertices on the path and the destination vertex must decrease), and guarantees that the drawing is a 5.33-spanner. We study three topics: (1) recognizing self-approaching drawings; (2) constructing self-approaching drawings of a given graph; (3) constructing a self-approaching Steiner network connecting a given set of points. We show that: (1) there are efficient algorithms to test if a polygonal path is self-approaching in R^2 and R^3, but it is NP-hard to test if a given graph drawing in R^3 has a self-approaching uv-path; (2) we can characterize the trees that have self-approaching drawings; (3) for any given set of terminal points in the plane, we can find a linear sized network that has a self-approaching path between any ordered pair of terminals.
Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem. It is not known whether the problem is in P or NP-complete. We prove that two natural generalizations of the problem are NP-complete, namely computing the minimum number of flips between two triangulations of (1) a polygon with holes; (2) a set of points in the plane.
Two planar graphs G1 and G2 sharing some vertices and edges are `simultaneously planar' if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a linear-time algorithm to test simultaneous planarity when the two graphs share a 2-connected subgraph. Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them.