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>Computing maxmin edge length triangulations
http://www.jocg.org/index.php/jocg/article/view/319
<div class="page" title="Page 1"><div class="layoutArea"><div class="column">In 1991, Edelsbrunner and Tan gave an $O(n^2)$ algorithm for finding the MinMax Length triangulation of a set of points in the plane, but stated the complexity of finding a MaxMin Edge Length Triangulation (MELT) as a natural open problem. We resolve this long-standing problem by showing that computing a MELT is NP-complete. Moreover, we prove that (unless P=NP), there is no polynomial-time approximation algorithm that can approximate MELT within any polynomial factor. </div><div class="column"> </div><div class="column">While this may be taken as conclusive evidence from a theoretical point of view that the problem is hopelessly intractable, it still makes sense to consider powerful optimization methods, such as integer programming (IP), in order to obtain provably optimal solutions for intances of non-trivial size. A straightforward IP based on pairwise disjointness of the $\Theta(n^2)$ segments between the n points has $\Theta(n^4)$ constraints, making this IP hopelessly intractable from a practical point of view, even for relatively small $n$. The main algorithm engineering twist of this paper is to demonstrate how the combination of geometric insights with refined methods of combinatorial optimization can still help to put together an exact method capable of computing optimal MELT solutions for planar point sets up to $n = 200$. Our key idea is to exploit specific geometric properties in combination with more compact IP formulations, such that we are able to drastically reduce the IPs. On the practical side, we combine two of the most powerful software packages for the individual components: CGAL for carrying out the geometric computations, and CPLEX for solving the IPs. In addition, we discuss specific analytic aspects of the speedup for random point sets. </div></div></div>Sándor P. FeketeWinfried HellmannMichael HemmerArne SchmidtJulian Troegel2018-02-272018-02-279112610.20382/jocg.v9i1a1Scalable exact visualization of isocontours in road networks via minimum-link paths
http://www.jocg.org/index.php/jocg/article/view/313
Isocontours in road networks represent the area that is reachable from a source within a given resource limit. We study the problem of computing accurate isocontours in realistic, large-scale networks. We propose isocontours represented by polygons with minimum number of segments that separate reachable and unreachable components of the network. Since the resulting problem is not known to be solvable in polynomial time, we introduce several heuristics that run in (almost) linear time and are simple enough to be implemented in practice. A key ingredient is a new practical linear-time algorithm for minimum-link paths in simple polygons. Experiments in a challenging realistic setting show excellent performance of our algorithms in practice, computing near-optimal solutions in a few milliseconds on average, even for long ranges.Moritz BaumThomas BläsiusAndreas GemsaIgnaz RutterFranziska Wegner2018-02-272018-02-2791277310.20382/jocg.v9i1a2Flat foldings of plane graphs with prescribed angles and edge lengths
http://www.jocg.org/index.php/jocg/article/view/191
<p>When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ,<br />360^\circ\}$) be folded flat to lie in an infinitesimally thick line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.</p>Zachary AbelErik D. DemaineMartin L. DemaineDavid EppsteinAnna LubiwRyuhei Uehara2018-02-272018-02-2791749310.20382/jocg.v9i1a3Drawing planar graphs with many collinear vertices
http://www.jocg.org/index.php/jocg/article/view/326
<p>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 $\Omega(\sqrt{n})$ collinear vertices; for every $n$, there is an $n$-vertex planar graph whose every planar straight-line drawing has $O(n^\sigma)$ collinear vertices, where $\sigma<0.986$; every $n$-vertex planar graph of treewidth at most two has a planar straight-line drawing with $\Theta(n)$ 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<span>–</span>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.</p>Giordano Da LozzoVida DujmovićFabrizio FratiTamara MchedlidzeVincenzo Roselli2018-05-032018-05-03919413010.20382/jocg.v9i1a4On the geodesic centers of polygonal domains
http://www.jocg.org/index.php/jocg/article/view/290
<p>In this paper, we study the problem of computing Euclidean geodesic centers of a polygonal domain $P$ with a total of $n$ vertices. We discover many interesting observations. We give a necessary condition for a point being a geodesic center. We show that there is at most one geodesic center among all points of $P$ that have topologically-equivalent shortest path maps. This implies that the total number of geodesic centers is bounded by the combinatorial size of the shortest path map equivalence decomposition of $P$, which is known to be $O(n^{10})$. One key observation is a $\pi$-range property on shortest path lengths when points are moving. With these observations, we propose an algorithm that computes all geodesic centers in $O(n^{11} \log n)$ time. Previously, an algorithm of $O(n^{12+\epsilon})$ time was known for this problem, for any $\epsilon > 0$.</p>Haitao Wang2018-06-282018-06-289113119010.20382/jocg.v9i1a5Improved time-space trade-offs for computing Voronoi diagrams
http://www.jocg.org/index.php/jocg/article/view/345
<p>Let $P$ be a planar set of $n$ sites in general position. For $k \in \{1, \dots, n-1\}$, the Voronoi diagram of order $k$ for $P$ is obtained by subdividing the plane into cells such that points in the same cell have the same set of nearest $k$ neighbors in $P$. The (nearest site) Voronoi diagram (NVD) and the farthest site Voronoi diagram (FVD) are the particular cases of $k=1$ and $k=n-1$, respectively. For any given $K \in \{1, \dots, n-1\}$, the family of all higher-order Voronoi diagrams of order $k = 1, \dots, K$ for $P$ can be computed in total time $O(nK^2+ n \log n)$ using $O(K^2(n-K))$ space [Aggarwal et al., DCG'89; Lee, TC'82]. Moreover, NVD and FVD for $P$ can be computed in $O(n\log n)$ time using $O(n)$ space [Preparata, Shamos, Springer'85].</p><p>For $s \in \{1, \dots, n\}$, an $s$-workspace algorithm has random access to a read-only array with the sites of $P$ in arbitrary order. Additionally, the algorithm may use $O(s)$ words, of $\Theta(\log n)$ bits each, for reading and writing intermediate data. The output can be written only once and cannot be accessed or modified afterwards.</p><p>We describe a deterministic $s$-workspace algorithm for computing NVD and FVD for $P$ that runs in $O((n^2/s)\log s)$ time. Moreover, we generalize our $s$-workspace algorithm so that for any given $K \in O(\sqrt{s})$, we compute the family of all higher-order Voronoi diagrams of order $k = 1, \dots, K$ for $P$ in total expected time $O\bigl(\frac{n^2 K^5}{s}(\log s + K \, 2^{O(\log^* K)}) \bigr)$ or in total deterministic time $O\bigl(\frac{n^2 K^5}{s}(\log s + K \log K) \bigr)$. Previously, for Voronoi diagrams, the only known $s$-workspace algorithm runs in expected time $O\bigl((n^2/s) \log s + n \log s \log^*s\bigr)$ [Korman et al., WADS'15] and only works for NVD (i.e., $k=1$). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques.</p><pre><!--EndFragment--></pre>Bahareh BanyassadyMatias KormanWolfgang MulzerAndré van RenssenMarcel RoeloffzenPaul SeiferthYannik Stein2018-06-292018-06-299119121210.20382/jocg.v9i1a6Topological drawings of complete bipartite graphs
http://www.jocg.org/index.php/jocg/article/view/347
<p>Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been studied extensively in the context of crossing number problems. We consider a natural class of simple topological drawings of {\em complete bipartite} graphs, in which we require that one side of the vertex set bipartition lies on the outer boundary of the drawing.</p><p>We investigate the combinatorics of such drawings. For this purpose, we define combinatorial encodings of the drawings by enumerating the distinct drawings of subgraphs isomorphic to $K_{2,2}$ and $K_{3,2}$, and investigate the constraints they must satisfy. We prove that a drawing of $K_{k,n}$ exists if and only if some simple local conditions are satisfied by the encodings. This directly yields a polynomial-time algorithm for deciding the existence of such a drawing given the encoding. We show the encoding is equivalent to specifying which pairs of edges cross, yielding a similar polynomial-time algorithm for the realizability of abstract topological graphs.</p><p>We also completely characterize and enumerate such drawings of $K_{k,n}$ in which the order of the edges around each vertex is the same for vertices on the same side of the bipartition. Finally, we investigate drawings of $K_{k,n}$ using straight lines and pseudolines, and consider the complexity of the corresponding realizability problems.</p>Jean CardinalStefan Felsner2018-07-182018-07-189121324610.20382/jocg.v9i1a7Array-based compact data structures for triangulations: Practical solutions with theoretical guarantees
http://www.jocg.org/index.php/jocg/article/view/332
We consider the problem of designing space efficient solutions for representing triangle meshes. Our main result is a new explicit data structure for compactly representing planar triangulations: if one is allowed to permute input vertices, then a triangulation with $n$ vertices requires at most $4n$ references ($5n$ references if vertex permutations are not allowed). Our solution combines existing techniques from mesh encoding with a novel use of maximal Schnyder woods. Our approach extends to higher genus triangulations and could be applied to other familiesof meshes (such as quadrangular or polygonal meshes). As far as we know, our solution provides the most parsimonious data structures for triangulations, allowing constant time navigation. Our data structures require linear construction time, and are fast decodable from a standard compressed format without using additional memory allocation. All bounds, concerning storage requirements and navigation performances, hold in the worst case. We have implemented and tested our results, and experiments confirm the practical interest of compact data structures.Luca Castelli AleardiOlivier Devillers2018-07-182018-07-189124728910.20382/jocg.v9i1a8Drawing planar graphs with prescribed face areas
http://www.jocg.org/index.php/jocg/article/view/351
We study drawings of planar graphs where every inner face has a prescribed area. A plane graph is 'area-universal' if for every area assignment on the inner faces, there exists a straight-line drawing realizing the assigned areas. The only non-area-universal graph known so far is the octahedron graph.<br /><br />We give a simple counting argument that allows to prove non-area-universality for a large class of triangulations, namely Eulerian triangulations. Adding some geometric arguments, the concept allows to prove non-area universality for other graphs including the icosahedron.<br /><br />Relaxing the straight-line property by allowing the edges to bend, we show that one bend per edge is sufficient to realize any face area assignment of every plane graph. For plane bipartite graphs, it even suffices that half of the edges have a bend.Linda Kleist2018-07-272018-07-279129031110.20382/jocg.v9i1a9Placing your coins on a shelf
http://www.jocg.org/index.php/jocg/article/view/361
We consider the problem of packing a family of disks ``on a shelf,'' that is, such that each disk touches the <em>x</em>-axis from above and such that no two disks overlap. We study the problem of minimizing the distance between the leftmost point and the rightmost point of any disk in such a packing. We show how to approximate this problem within a factor of <em>4/3</em> in <em>O(n log n)</em> time. We further provide an <em>O(n log n)</em>-time exact algorithm for a special case which includes inputs where the ratio between the largest radius and the smallest radius is less than four. On the negative side, we prove that the problem is NP-hard even when the ratio between the largest radius and the smallest radius is at most 36.Helmut AltKevin BuchinSteven ChaplickOtfried CheongPhilipp KindermannChristian KnauerFabian Stehn2018-09-052018-09-059131232710.20382/jocg.v9i1a10Planar and poly-arc Lombardi drawings
http://www.jocg.org/index.php/jocg/article/view/322
<p>In Lombardi drawings of graphs, edges are represented as circular arcs and the edges incident on vertices have perfect angular resolution. It is known that not every planar graph has a planar Lombardi drawing. We give an example of a planar 3-tree that has no planar Lombardi drawing and we show that all outerpaths do have a planar Lombardi drawing. Further, we show that there are graphs that do not even have any Lombardi drawing at all. With this in mind, we generalize the notion of Lombardi drawings to that of (smooth) $k$-Lombardi drawings, in which each edge may be drawn as a (differentiable) sequence of $k$ circular arcs; we show that every graph has a smooth $2$-Lombardi drawing and every planar graph has a smooth planar $3$-Lombardi drawing. We further investigate related topics connecting planarity and Lombardi drawings.</p>Christian A. DuncanDavid EppsteinMichael T. GoodrichStephen G. KobourovMaarten LöfflerMartin Nöllenburg2018-09-072018-09-079132835510.20382/jocg.v9i1a11Thickness and antithickness of graphs
http://www.jocg.org/index.php/jocg/article/view/348
This paper studies questions about duality between crossings and non-crossings in graph drawings via the notions of thickness and antithickness. The <em>thickness</em> of a graph $G$ is the minimum integer $k$ such that in some drawing of $G$, the edges can be partitioned into $k$ noncrossing subgraphs. The <em>antithickness</em> of a graph $G$ is the minimum integer $k$ such that in some drawing of $G$, the edges can be partitioned into $k$ thrackles, where a <em>thrackle</em> is a set of edges, each pair of which intersect exactly once. So thickness is a measure of how close a graph is to being planar, whereas antithickness is a measure of how close a graph is to being a thrackle. This paper explores the relationship between the thickness and antithickness of a graph, under various graph drawing models, with an emphasis on extremal questions.Vida DujmovićDavid R. Wood2018-09-252018-09-259135638610.20382/jocg.v9i1a12A stability theorem on cube tessellations
http://www.jocg.org/index.php/jocg/article/view/398
<p>It is shown that if a $d$-dimensional cube is decomposed into $n$ cubes, the side lengths<br />of which belong to the interval $(1 − \frac{n}{1/d 1 +1} , 1]$, then $n$ is a perfect $d$-th power and all<br />cubes are of the same size. This result is essentially tight.</p>János PachPeter Frankl2018-10-102018-10-109138739010.20382/jocg.v9i1a13