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Vida Dujmovic

Publications and source records attributed to Vida Dujmovic.

At least 19 recordsLinked to original sources
arXiv2019

Near-Optimal $O(k)$-Robust Geometric Spanners

For any constants $d\ge 1$, $ε>0$, $t>1$, and any $n$-point set $P\subset\mathbb{R}^d$, we show that there is a geometric graph $G=(P,E)$ having $O(n\log^2 n\log\log n)$ edges with the following property: For any $F\subseteq P$, there exists $F^+\supseteq F$, $|F^+| \le (1+ε)|F|$ such that, for any pair $p,q\in P\setminus F^+$, the graph $G-F$ contains a path from $p$ to $q$ whose (Euclidean) length is at most $t$ times the Euclidean distance between $p$ and $q$. In the terminology of robust spanners (Bose \et al, SICOMP, 42(4):1720--1736, 2013) the graph $G$ is a $(1+ε)k$-robust $t$-spanner of $P$. This construction is sparser than the recent constructions of Buchin, Olàh, and Har-Peled (arXiv:1811.06898) who prove the existence of $(1+ε)k$-robust $t$-spanners with $n\log^{O(d)} n$ edges.

cs.CG↗
arXiv2018

Pole Dancing: 3D Morphs for Tree Drawings

We study the question whether a crossing-free 3D morph between two straight-line drawings of an $n$-vertex tree can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with $O(\log n)$ steps, while for the latter $Θ(n)$ steps are always sufficient and sometimes necessary.

cs.CG↗
arXiv2018

Geodesic Obstacle Representation of Graphs

An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in the graph. The obstacle representation and its plane variant (in which the resulting representation is a plane straight-line embedding of the graph) have been extensively studied with the main objective of minimizing the number of obstacles. Recently, Biedl and Mehrabi (GD 2017) studied grid obstacle representations of graphs in which the vertices of the graph are mapped onto the points in the plane while the straight-line segments representing the adjacency between the vertices is replaced by the $L_1$ (Manhattan) shortest paths in the plane that avoid obstacles. In this paper, we introduce the notion of geodesic obstacle representations of graphs with the main goal of providing a generalized model, which comes naturally when viewing line segments as shortest paths in the Euclidean plane. To this end, we extend the definition of obstacle representation by allowing some obstacles-avoiding shortest path between the corresponding points in the underlying metric space whenever the vertices are adjacent in the graph. We consider both general and plane variants of geodesic obstacle representations (in a similar sense to obstacle representations) under any polyhedral distance function in $\mathbb{R}^d$ as well as shortest path distances in graphs. Our results generalize and unify the notions of obstacle representations, plane obstacle representations and grid obstacle representations, leading to a number of questions on such embeddings.

cs.CG↗
arXiv2017

EPG-representations with small grid-size

In an EPG-representation of a graph $G$ each vertex is represented by a path in the rectangular grid, and $(v,w)$ is an edge in $G$ if and only if the paths representing $v$ an $w$ share a grid-edge. Requiring paths representing edges to be x-monotone or, even stronger, both x- and y-monotone gives rise to three natural variants of EPG-representations, one where edges have no monotonicity requirements and two with the aforementioned monotonicity requirements. The focus of this paper is understanding how small a grid can be achieved for such EPG-representations with respect to various graph parameters. We show that there are $m$-edge graphs that require a grid of area $Ω(m)$ in any variant of EPG-representations. Similarly there are pathwidth-$k$ graphs that require height $Ω(k)$ and area $Ω(kn)$ in any variant of EPG-representations. We prove a matching upper bound of $O(kn)$ area for all pathwidth-$k$ graphs in the strongest model, the one where edges are required to be both x- and y-monotone. Thus in this strongest model, the result implies, for example, $O(n)$, $O(n \log n)$ and $O(n^{3/2})$ area bounds for bounded pathwidth graphs, bounded treewidth graphs and all classes of graphs that exclude a fixed minor, respectively. For the model with no restrictions on the monotonicity of the edges, stronger results can be achieved for some graph classes, for example an $O(n)$ area bound for bounded treewidth graphs and $O(n \log^2 n)$ bound for graphs of bounded genus.

cs.CG↗
arXiv2016

Drawing Planar Graphs with Many Collinear Vertices

Consider the following problem: Given a planar graph $G$, what is the maximum number $p$ such that $G$ has a planar straight-line drawing with $p$ collinear vertices? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known for it: Every $n$-vertex planar graph has a planar straight-line drawing with $Ω(\sqrt{n})$ collinear vertices; for every $n$, there is an $n$-vertex planar graph whose every planar straight-line drawing has $O(n^σ)$ collinear vertices, where $σ<0.986$; every $n$-vertex planar graph of treewidth at most two has a planar straight-line drawing with $Θ(n)$ collinear vertices. We extend the linear bound to planar graphs of treewidth at most three and to triconnected cubic planar graphs. This (partially) answers two open problems posed by Ravsky and Verbitsky [WG 2011:295--306]. Similar results are not possible for all bounded treewidth planar graphs or for all bounded degree planar graphs. For planar graphs of treewidth at most three, our results also imply asymptotically tight bounds for all of the other above mentioned graph drawing problems.

cs.CG↗
arXiv2013

Average Stretch Factor: How Low Does It Go?

In a geometric graph, $G$, the \emph{stretch factor} between two vertices, $u$ and $w$, is the ratio between the Euclidean length of the shortest path from $u$ to $w$ in $G$ and the Euclidean distance between $u$ and $w$. The \emph{average stretch factor} of $G$ is the average stretch factor taken over all pairs of vertices in $G$. We show that, for any constant dimension, $d$, and any set, $V$, of $n$ points in $\mathbb{R}^d$, there exists a geometric graph with vertex set $V$, that has $O(n)$ edges, and that has average stretch factor $1+ o_n(1)$. More precisely, the average stretch factor of this graph is $1+O((\log n/n)^{1/(2d+1)})$. We complement this upper-bound with a lower bound: There exist $n$-point sets in $\mathbb{R}^2$ for which any graph with $O(n)$ edges has average stretch factor $1+Ω(1/\sqrt{n})$. Bounds of this type are not possible for the more commonly studied worst-case stretch factor. In particular, there exists point sets, $V$, such that any graph with worst-case stretch factor $1+o_n(1)$ has a superlinear number of edges.

cs.CG↗
arXiv2013

Robust Geometric Spanners

Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in addition, geometric spanners. We define a property of spanners called robustness. Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices. We show that robust spanners must have a superlinear number of edges, even in one dimension. On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges.

cs.CG↗
arXiv2013

Graph Layouts via Layered Separators

A k-queue layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets such that no two edges that are in the same set are nested with respect to the vertex ordering. A k-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The queue-number (track-number) of a graph G, is the minimum k such that G has a k-queue (k-track) layout. This paper proves that every n-vertex planar graph has track number and queue number at most O(log n). This improves the result of Di Battista, Frati and Pach [Foundations of Computer Science, (FOCS '10), pp. 365--374] who proved the first sub-polynomial bounds on the queue number and track number of planar graphs. Specifically, they obtained O(log^2 n) queue number and O(log^8 n) track number bounds for planar graphs. The result also implies that every planar graph has a 3D crossing-free grid drawing in O(n log n) volume. The proof uses a non-standard type of graph separators.

cs.CG↗
arXiv2013

Crossings in Grid Drawings

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 geometric graphs with vertices on some d-dimensional grid of volume N is n^Θ(n). In 3 dimensions it remains open to improve the trivial bounds, namely, the 2^Ω(n) lower bound and the n^O(n) upper bound.

math.CO↗
arXiv2012

A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l such that in both line sets, for both halfplanes delimited by l, there are n^{1/2} lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are (n/3)^{1/2} of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to Graph Drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labelled lines that are universal for all n-vertex labelled planar graphs. As a side note, we prove that every set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar graphs.

cs.CG↗
arXiv2012

An Affine Invariant $k$-Nearest Neighbor Regression Estimate

We design a data-dependent metric in $\mathbb R^d$ and use it to define the $k$-nearest neighbors of a given point. Our metric is invariant under all affine transformations. We show that, with this metric, the standard $k$-nearest neighbor regression estimate is asymptotically consistent under the usual conditions on $k$, and minimal requirements on the input data.

math.ST↗
arXiv2010

Notes on large angle crossing graphs

A graph G is an a-angle crossing (aAC) graph if every pair of crossing edges in G intersect at an angle of at least a. The concept of right angle crossing (RAC) graphs (a=Pi/2) was recently introduced by Didimo et. al. It was shown that any RAC graph with n vertices has at most 4n-10 edges and that there are infinitely many values of n for which there exists a RAC graph with n vertices and 4n-10 edges. In this paper, we give upper and lower bounds for the number of edges in aAC graphs for all 0 < a < Pi/2.

cs.DS↗
arXiv2010

Odds-On Trees

Let R^d -> A be a query problem over R^d for which there exists a data structure S that can compute P(q) in O(log n) time for any query point q in R^d. Let D be a probability measure over R^d representing a distribution of queries. We describe a data structure called the odds-on tree, of size O(n^ε) that can be used as a filter that quickly computes P(q) for some query values q in R^d and relies on S for the remaining queries. With an odds-on tree, the expected query time for a point drawn according to D is O(H*+1), where H* is a lower-bound on the expected cost of any linear decision tree that solves P. Odds-on trees have a number of applications, including distribution-sensitive data structures for point location in 2-d, point-in-polytope testing in d dimensions, ray shooting in simple polygons, ray shooting in polytopes, nearest-neighbour queries in R^d, point-location in arrangements of hyperplanes in R^d, and many other geometric searching problems that can be solved in the linear-decision tree model. A standard lifting technique extends these results to algebraic decision trees of constant degree. A slightly different version of odds-on trees yields similar results for orthogonal searching problems that can be solved in the comparison tree model.

cs.CG↗
arXiv2010

Point Location in Disconnected Planar Subdivisions

Let $G$ be a (possibly disconnected) planar subdivision and let $D$ be a probability measure over $\R^2$. The current paper shows how to preprocess $(G,D)$ into an O(n) size data structure that can answer planar point location queries over $G$. The expected query time of this data structure, for a query point drawn according to $D$, is $O(H+1)$, where $H$ is a lower bound on the expected query time of any linear decision tree for point location in $G$. This extends the results of Collette et al (2008, 2009) from connected planar subdivisions to disconnected planar subdivisions. A version of this structure, when combined with existing results on succinct point location, provides a succinct distribution-sensitive point location structure.

cs.CG↗
arXiv2009

Memoryless Routing in Convex Subdivisions: Random Walks are Optimal

A memoryless routing algorithm is one in which the decision about the next edge on the route to a vertex t for a packet currently located at vertex v is made based only on the coordinates of v, t, and the neighbourhood, N(v), of v. The current paper explores the limitations of such algorithms by showing that, for any (randomized) memoryless routing algorithm A, there exists a convex subdivision on which A takes Omega(n^2) expected time to route a message between some pair of vertices. Since this lower bound is matched by a random walk, this result implies that the geometric information available in convex subdivisions is not helpful for this class of routing algorithms. The current paper also shows the existence of triangulations for which the Random-Compass algorithm proposed by Bose etal (2002,2004) requires 2^{Ω(n)} time to route between some pair of vertices.

cs.CG↗
arXiv2009

Minimum feature size preserving decompositions

The minimum feature size of a crossing-free straight line drawing is the minimum distance between a vertex and a non-incident edge. This quantity measures the resolution needed to display a figure or the tool size needed to mill the figure. The spread is the ratio of the diameter to the minimum feature size. While many algorithms (particularly in meshing) depend on the spread of the input, none explicitly consider finding a mesh whose spread is similar to the input. When a polygon is partitioned into smaller regions, such as triangles or quadrangles, the degradation is the ratio of original to final spread (the final spread is always greater). Here we present an algorithm to quadrangulate a simple n-gon, while achieving constant degradation. Note that although all faces have a quadrangular shape, the number of edges bounding each face may be larger. This method uses Theta(n) Steiner points and produces Theta(n) quadrangles. In fact to obtain constant degradation, Omega(n) Steiner points are required by any algorithm. We also show that, for some polygons, a constant factor cannot be achieved by any triangulation, even with an unbounded number of Steiner points. The specific lower bounds depend on whether Steiner vertices are used or not.

cs.CG↗
arXiv2009

Entropy, Triangulation, and Point Location in Planar Subdivisions

A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of point-line comparisons performed by this data structure, when the queries are distributed according to D, is H + O(H^{2/3}+1) where H=H(G,D) is a lower bound on the expected number of point-line comparisons performed by any linear decision tree for point location in G under the query distribution D. The preprocessing algorithm runs in O(n log n) time and produces a data structure of size O(n). These results are obtained by creating a Steiner triangulation of G that has near-minimum entropy.

cs.CG↗