Journal of Computational Geometry
http://www.jocg.org/index.php/jocg
The Journal of Computational Geometry (JoCG) is an international open access journal devoted to publishing original research of the highest quality in all aspects of computational geometry.<p>JoCG articles and supplementary data are freely available for download and JoCG charges no publishing fees of any kind.</p><p>All JoCG issues and articles are assigned a DOI. JoCG's data and content are safeguarded through <a href="/index.php/jocg/pages/view/backup">several backup mechanisms</a>.</p><p>JoCG is a member of the <a href="http://freejournals.org/">Free Journal Network</a>.</p>Carleton Universityen-USJournal of Computational Geometry1920-180XAuthors who publish with this journal agree to the following terms:<br /><br /><ol type="a"><ol type="a"><li>Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a <a href="http://creativecommons.org/licenses/by/3.0/" target="_new">Creative Commons Attribution License</a> that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.</li></ol></ol><br /><ol type="a"><ol type="a"><li>Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.</li></ol></ol><br /><ol type="a"><li>Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See <a href="http://opcit.eprints.org/oacitation-biblio.html" target="_new">The Effect of Open Access</a>).</li></ol>Approximating minimum-area rectangular and convex containers for packing convex polygons
http://www.jocg.org/index.php/jocg/article/view/289
We investigate the problem of finding a minimum-area container for the disjoint packing of a set of convex polygons by translations. In particular, we consider axis-parallel rectangles or arbitrary convex sets as containers. For both optimization problems which are NP-hard we develop efficient constant factor approximation algorithms.Helmut AltMark de BergChristian Knauer2017-02-182017-02-188111010.20382/jocg.v8i1a1Towards plane spanners of degree 3
http://www.jocg.org/index.php/jocg/article/view/295
<p>Let $S$ be a finite set of points in the plane. In this paper we consider the problem of computing plane spanners of degree at most three for $S$.</p><ol><li>If $S$ is in convex position, then we present an algorithm that constructs a plane $\frac{3+4\pi}{3}$-spanner for $S$ whose vertex degree is at most 3. </li><li>If $S$ is the vertex set of a non-uniform rectangular lattice, then we present an algorithm that constructs a plane $3\sqrt{2}$-spanner for $S$ whose vertex degree is at most 3. </li><li>If $S$ is in general position, then we show how to compute plane degree-3 spanners for $S$ with a linear number of Steiner points.</li></ol>Ahmad BiniazProsenjit BoseJean-Lou De CarufelCyril GavoilleAnil MaheshwariMichiel Smid2017-03-132017-03-1381113110.20382/jocg.v8i1a2On interference among moving sensors and related problems
http://www.jocg.org/index.php/jocg/article/view/297
<p>We show that for any set of $n$ moving points in $\Re^d$ and any parameter $2 \le k \le n$, one can select a fixed non-empty subset of the points of size $O(k \log k)$, such that the Voronoi diagram of this subset is ``balanced'' at any given time (i.e., it contains $O(n/k)$ points per cell). We also show that the bound $O(k \log k)$ is near optimal even for the one dimensional case in which points move linearly in time. As an application, we show that one can assign communication radii to the sensors of a network of $n$ moving sensors so that at any given time, their interference is $O(\sqrt{n\log n})$. This is optimal up to an $O(\sqrt{\log n})$ factor. In order to obtain these results, we extend well-known results from $\varepsilon$-net theory to kinetic environments.</p>Jean-Lou De CarufelMatthew J. KatzMatias KormanAndré van RenssenMarcel RoeloffzenShakhar Smorodinsky2017-04-252017-04-2581324610.20382/jocg.v8i1a3Counting and enumerating crossing-free geometric graphs
http://www.jocg.org/index.php/jocg/article/view/280
<p>We describe a framework for constructing data structures which allow fast counting and enumeration of 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 represent geometric graphs as source-sink paths in a directed acyclic graph.</p><p>The following 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 computed in time $O(2^nn^4)$. The number of crossing-free perfect matchings can be computed in time $O(2^nn^4)$. The number of convex subdivisions can be computed in time $O(2^nn^4)$. The number of crossing-free spanning trees can be computed in time $O(c^nn^4)$ for some $c < 7.04313$. The number of crossing-free spanning cycles can be computed in time $O(c^nn^4)$ for some $c < 5.61804$.</p><p>Moreover, after a preprocessing phase with the same time bounds as above, we can enumerate the respective classes efficiently. For example, after $O(2^nn^4)$ time of preprocessing we can enumerate the set of all crossing-free perfect matchings using polynomial time per enumerated object. For crossing-free perfect matchings and convex partitions we further obtain enumeration algorithms where the time delay for each (in particular, the first) output is bounded by a polynomial in $n$.</p><p>All described algorithms are comparatively simple, both in terms of their analysis and implementation.</p>Manuel Wettstein2017-04-252017-04-2581477710.20382/jocg.v8i1a4The projection median as a weighted average
http://www.jocg.org/index.php/jocg/article/view/244
The projection median of a set $P$ of $n$ points in $\mathbb{R}^d$ is a robust geometric generalization of the notion of univariate median to higher dimensions. In its original definition, the projection median is expressed as a normalized integral of the medians of the projections of $P$ onto all lines through the origin. We introduce a new definition in which the projection median is expressed as a weighted mean of $P$, and show the equivalence of the two definitions. In addition to providing a definition whose form is more consistent with those of traditional statistical estimators of location, this new definition for the projection median allows many of its geometric properties to be established more easily, as well as enabling new randomized algorithms that compute approximations of the projection median with increased accuracy and efficiency, reducing computation time from $O(n^{d+\epsilon})$ to $O(mnd)$, where $m$ denotes the number of random projections sampled. Selecting $m \in \Theta(\epsilon^{-2} d^2 \log n)$ or $m \in \Theta(\min ( d + \epsilon^{-2} \log n, \epsilon^{-2} n))$, suffices for our algorithms to return a point within relative distance $\epsilon$ of the true projection median with high probability, resulting in running times $O(d^3 n \log n)$ and $O(\min(d^2 n, d n^2))$ respectively, for any fixed $\epsilon$.Stephane DurocherAlexandre LeblancMatthew Skala2017-05-012017-05-01817810410.20382/jocg.v8i1a5Time-space trade-offs for triangulating a simple polygon
http://www.jocg.org/index.php/jocg/article/view/307
<p>An $s$-workspace algorithm is an algorithm that has read-only access to the values of the input, write-only access to the output, and only uses $O(s)$ additional words of space. We present a randomized $s$-workspace algorithm for triangulating a simple polygon $P$ of $n$ vertices that runs in $O(n^2/s+n \log n \log^{5} (n/s))$ expected time using $O(s)$ variables, for any $s \leq n$. In particular, when $s \leq \frac{n}{\log n\log^{5}\log n}$ the algorithm runs in $O(n^2/s)$ expected time.</p>Boris AronovMatias KormanSimon PrattAndré van RenssenMarcel Roeloffzen2017-05-012017-05-018110512410.20382/jocg.v8i1a6Competitive local routing with constraints
http://www.jocg.org/index.php/jocg/article/view/288
Let $P$ be a set of $n$ vertices in the plane and $S$ a set of non-crossing line segments between vertices in $P$, called constraints. Two vertices are visible if the straight line segment connecting them does not properly intersect any constraints. The constrained $\Theta_m$-graph is constructed by partitioning the plane around each vertex into $m$ disjoint cones, each with aperture $\theta = 2 \pi/m$, and adding an edge to the `closest' visible vertex in each cone. We consider how to route on the constrained $\Theta_6$-graph. We first show that no deterministic 1-local routing algorithm is $o(\sqrt{n})$-competitive on all pairs of vertices of the constrained $\Theta_6$-graph. After that, we show how to route between any two visible vertices of the constrained $\Theta_6$-graph using only 1-local information. Our routing algorithm guarantees that the returned path has length at most 2 times the Euclidean distance between the source and destination. Additionally, we provide a 1-local 18-competitive routing algorithm for visible vertices in the constrained half-$\Theta_6$-graph, a subgraph of the constrained $\Theta_6$-graph that is equivalent to the Delaunay graph where the empty region is an equilateral triangle. To the best of our knowledge, these are the first local routing algorithms in the constrained setting with guarantees on the length of the returned path.Prosenjit BoseRolf FagerbergAndré van RenssenSander Verdonschot2017-05-152017-05-158112515210.20382/jocg.v8i1a7A new drawing for simple Venn diagrams based on algebraic construction
http://www.jocg.org/index.php/jocg/article/view/271
Venn diagrams are used to display all relations between a finite number of sets. Recent researches in this domain concern the mathematical aspects of these constructions, but are not directed towards the readability of the diagram. This article presents a new way to draw easy-to-read Venn diagrams, in which each region tends to be drawn with the same size when the number of sets grows, and tends to draw a grid. Finally, using linear algebra, we prove that this construction gives a simple Venn diagram for any number of sets.Arnaud BannierNicolas Bodin2017-05-292017-05-298115317310.20382/jocg.v8i1a8Classifying unavoidable Tverberg partitions
http://www.jocg.org/index.php/jocg/article/view/308
<div>Let $T(d,r) = (r-1)(d+1)+1$ be the parameter in Tverberg's theorem, and call a partition $\mathcal I$ of $\{1,2,\ldots,T(d,r)\}$ into $r$ parts a <em>Tverberg type</em>. We say that $\mathcal I$ o<em>ccurs</em> in an ordered point sequence $P$ if $P$ contains a subsequence $P'$ of $T(d,r)$ points such that the partition of $P'$ that is order-isomorphic to $\mathcal I$ is a Tverberg partition. We say that $\mathcal I$ is <em>unavoidable</em> if it occurs in every sufficiently long point sequence.<br /><br />In this paper we study the problem of determining which Tverberg types are unavoidable. We conjecture a complete characterization of the unavoidable Tverberg types, and we prove some cases of our conjecture for $d\le 4$. Along the way, we study the avoidability of many other geometric predicates.<br /><br />Our techniques also yield a large family of $T(d,r)$-point sets for which the number of Tverberg partitions is exactly $(r-1)!^d$. This lends further support for Sierksma's conjecture on the number of Tverberg partitions.</div><div> </div>Boris BukhPo-Shen LohGabriel Nivasch2017-07-042017-07-048117420510.20382/jocg.v8i1a9How many three-dimensional Hilbert curves are there?
http://www.jocg.org/index.php/jocg/article/view/298
<p>Hilbert's two-dimensional space-filling curve is appreciated for its good locality-preserving properties and easy implementation for many applications. However, Hilbert did not describe how to generalize his construction to higher dimensions. In fact, the number of ways in which this may be done ranges from zero to infinite, depending on what properties of the Hilbert curve one considers to be essential.<br /><br />In this work we take the point of view that a Hilbert curve should at least be self-similar and traverse cubes octant by octant. We organize and explore the space of possible three-dimensional Hilbert curves and the potentially useful properties which they may have. We discuss a notation system that allows us to distinguish the curves from one another and enumerate them. This system has been implemented in a software prototype, available from the author's website.</p><p>Several examples of possible three-dimensional Hilbert curves are presented, including a curve that visits the points on most sides of the unit cube in the order of the two-dimensional Hilbert curve; curves of which not only the eight octants are similar to each other, but also the four quarters; a curve with excellent locality-preserving properties and endpoints that are not vertices of the cube; a curve in which all but two octants are each other's images with respect to reflections in axis-parallel planes; and curves that can be sketched on a grid without using vertical line segments. In addition, we discuss several four-dimensional Hilbert curves.<br /><br /></p>Herman Haverkort2017-09-122017-09-128120628110.20382/jocg.v8i1a10Qualitative symbolic perturbation: two applications of a new geometry-based perturbation framework
http://www.jocg.org/index.php/jocg/article/view/253
<br />In a classical <em>Symbolic Perturbation</em> scheme, degeneracies are handled by substituting some polynomials in $\varepsilon$ for the inputs of a predicate. Instead of a single perturbation, we propose to use a sequence of (simpler) perturbations. Moreover, we look at their effects geometrically instead of algebraically; this allows us to tackle cases that were not tractable with the classical algebraic approach.Olivier DevillersMenelaos KaravelasMonique Teillaud2017-09-122017-09-128128231510.20382/jocg.v8i1a11Maximizing the sum of radii of disjoint balls or disks
http://www.jocg.org/index.php/jocg/article/view/286
<p>Finding nonoverlapping balls with given centers in any metric space, maximizing the sum of radii of the balls, can be expressed as a linear program. Its dual linear program expresses the problem of finding a minimum-weight set of cycles (allowing 2-cycles) covering all vertices in a complete geometric graph. For points in a Euclidean space of any finite dimension $d$, with any convex distance function on this space, this graph can be replaced by a sparse subgraph obeying a separator theorem. This graph structure leads to an algorithm for finding the optimum set of balls in time $O(n^{2-1/d})$, improving the $O(n^3)$ time of a naive cycle cover algorithm. As a subroutine, we provide an algorithm for weighted bipartite matching in graphs with separators, which speeds up the best previous algorithm for this problem on planar bipartite graphs from $O(n^{3/2}\log n)$ to $O(n^{3/2})$ time.</p>David Eppstein2017-10-062017-10-068131633910.20382/jocg.v8i1a12Computing nonsimple polygons of minimum perimeter
http://www.jocg.org/index.php/jocg/article/view/303
<div class="page" title="Page 1"><div class="layoutArea"><div class="column">We consider the Minimum Perimeter Polygon Problem (MP3): for a given set V of points in the plane, find a polygon P with holes that has vertex set V , such that the total boundary length is smallest possible. The MP3 can be considered a natural geometric generalization of the Traveling Salesman Problem (TSP), which asks for a simple polygon with minimum perimeter. Just like the TSP, the MP3 occurs naturally in the context of curve reconstruction.</div><div class="column"> </div><div class="column">Even though the closely related problem of finding a minimum cycle cover is polynomially solvable by matching techniques, we prove how the topological structure of a polygon leads to NP-hardness of the MP3. On the positive side, we provide constant-factor approximation algorithms.</div><div class="column"> </div><div class="column">In addition to algorithms with theoretical worst-case guarantess, we provide practical methods for computing provably optimal solutions for relatively large instances, based on integer programming. An additional difficulty compared to the TSP is the fact that only a subset of subtour constraints is valid, depending not on combinatorics, but on geometry. We overcome this difficulty by establishing and exploiting geometric properties. This allows us to reliably solve a wide range of benchmark instances with up to 600 vertices within reasonable time on a standard machine. We also show that restricting the set of connections between points to edges of the Delaunay triangulation yields results that are on average within 0.5% of the optimum for large classes of benchmark instances. </div></div></div>Sándor P. FeketeAndreas HaasMichael HemmerMichael HoffmannIrina KostitsynaDominik KrupkeFlorian MaurerJoseph S. B. MitchellArne SchmidtChristiane SchmidtJulian Troegel2017-10-062017-10-068134036510.20382/jocg.v8i1a13Central trajectories
http://www.jocg.org/index.php/jocg/article/view/302
$\newcommand{\c}{\mathcal{C}}\newcommand{\R}{\mathbb{R}}$An important task in trajectory analysis is clustering. The results of a clustering are often summarized by a single representative trajectory and an associated size of each cluster. We study the problem of computing a suitable representative of a set of similar trajectories. To this end we define a <em>central trajectory</em> $\c$, which consists of pieces of the input trajectories, switches from one entity to another only if they are within a small distance of each other, and such that at any time $t$, the point $\c(t)$ is as central as possible. We measure centrality in terms of the radius of the smallest disk centered at $\c(t)$ enclosing all entities at time $t$, and discuss how the techniques can be adapted to other measures of centrality. We first study the problem in $\R^1$, where we show that an optimal central trajectory $\c$ representing $n$ trajectories, each consisting of $\tau$ edges, has complexity $\Theta(\tau n^2)$ and can be computed in $O(\tau n^2 \log n)$ time. We then consider trajectories in $\R^d$ with $d\geq 2$, and show that the complexity of $\c$ is at most $O(\tau n^{5/2})$ and can be computed in $O(\tau n^3)$ time.<br /><br />Marc van KreveldMaarten LöfflerFrank Staals2017-10-162017-10-168136638610.20382/jocg.v8i1a14