DOCSLIB.ORG

Shape of Shortest Paths in Random Spatial Networks Alexander Kartun-Giles, Marc Barthelemy, Carl Dettmann

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Shape of Shortest Paths in Random Spatial Networks Alexander Kartun-Giles, Marc Barthelemy, Carl Dettmann Shape of shortest paths in random spatial networks Alexander Kartun-Giles, Marc Barthelemy, Carl Dettmann To cite this version: Alexander Kartun-Giles, Marc Barthelemy, Carl Dettmann. Shape of shortest paths in random spatial networks. Physical Review E , American Physical Society (APS), 2019, 100 (3), pp.2315. 10.1103/PhysRevE.100.032315. cea-02564008 HAL Id: cea-02564008 https://hal-cea.archives-ouvertes.fr/cea-02564008 Submitted on 2 Sep 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The shape of shortest paths in random spatial networks Alexander P. Kartun-Giles,1 Marc Barthelemy,2 and Carl P. Dettmann3 1Max Plank Institute for Mathematics in the Sciences, Inselstr. 22, Leipzig, Germany 2Institut de Physique Th´eorique,CEA, CNRS-URA 2306, Gif-sur-Yvette, France 3School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance L, geodesics exhibit deviations from their mean length L that are of order Lχ, while the transversal fluctuations, known as wandering, grow as Lξ. We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents ξ = 3=5 and χ = 1=5, or ξ = 7=10 and χ = 2=5, depending only on coarse details of the specific connectivity laws used. Also, the travel time fluctuations are Gaussian, rather than Tracy-Widom, which is rarely seen in first passage models. The first class contains proximity graphs such as the hard and soft random geometric graph, and the k-nearest neighbour random geometric graphs, where via Monte Carlo simulations we find ξ = 0:60 ± 0:01 and χ = 0:20 ± 0:01, showing a theoretical minimal wandering. The second class contains graphs based on excluded regions such as β-skeletons and the Delaunay triangulation and are characterized by the values ξ = 0:70 ± 0:01 and χ = 0:40 ± 0:01, with a nearly theoretically maximal wandering exponent. We also show numerically that the KPZ relation χ = 2ξ − 1 is satisfied for all these models. These results shed some light on the Euclidean first passage process, but also raise some theoretical questions about the scaling laws and the derivation of the exponent values, and also whether a model can be constructed with maximal wandering, or non-Gaussian travel fluctuations, while embedded in space. I. INTRODUCTION [19], and key ideas of universality and scale invariance in the shape of shortest paths [23]. As an important inter- Many complex systems assume the form of a spatial section between probability and geometry, it is also part network [1, 2]. Transport networks, neural networks, of the mathematical aspects of a theory of gravity be- communication and wireless sensor networks, power and yond general relativity, and perhaps in the foundations of energy networks, and ecological interaction networks are quantum mechanics, since it displays fundamental links all important examples where the characteristics of a spa- to complexity, emergent phenomena, and randomness in tial network structure are key to understanding the cor- physics [24, 25]. responding emergent dynamics. A particular case of FPP is the topic of this article, Shortest paths form an important aspect of their study. known as Euclidean first passage percolation (EFPP). Consider for example the appearance of bottlenecks im- This is a probabilistic model of fluid flow between points peding traffic flow in a city [3, 4], the emergence of spa- of a d-dimensional Euclidean space, such as the surface of tial small worlds [5, 6], bounds on the diameter of spatial a hypersphere. One studies optimal routes from a source preferential attachment graphs [7{9], the random con- node to each possible destination node in a spatial net- nection model [10{13], or in spatial networks generally work built either randomly or deterministically on the [14, 15], as well as geometric effects on betweenness cen- points. Introduced by Howard and Newman much later trality measures in complex networks [11, 16], and navi- in 1997 [26] and originally a weighted complete graph, we gability [17]. adopt a different perspective by considering edge weights First passage percolation (FPP) [18] attempts to cap- given deterministically by the Euclidean distances be- ture these features with a probabilistic model. As with tween the spatial points themselves. This is in sharp percolation [19], the effect of spatial disorder is crucial. contrast with the usual FPP problem, where weights are Compare this to the elementary random graph [20]. In i.i.d. random variables. FPP one usually considers a deterministic lattice such Howard's model is defined on the complete graph con- as Zd with independent, identically distributed weights, structed on a point process. Long paths are discouraged known as local passage times, on the edges. With a fluid by taking powers of interpoint distances as edge weights. arXiv:1906.04314v2 [cond-mat.stat-mech] 13 Nov 2019 flowing outward from a point, the question is, what is the The variant of EFPP we study is instead defined on a minimum passage time over all possible routes between Poisson point process in an unbounded region (by def- this and another distant point, where routing is quicker inition, the number of points in a bounded region is a along lower weighted edges? Poisson random variable, see for example [27]), but with More than 50 years of intensive study of FPP has been links added between pairs of points according to given carried out [21]. This has lead to many results such as rules [28, 29], rather than the totality of the weighted the subadditive ergodic theorem, a key tool in probability complete graph. More precisely, the model we study in theory, but also a number of insights in crystal and in- this paper is defined as follows. We take a random spatial terface growth [22], the statistical physics of traffic jams network such as the random geometric graph constructed 2 a distribution. In particular, we find distinct exponents b from the KPZ class (see for example [39] and references c d therein) which has wandering and fluctuation exponents ξ = 2=3 and χ = 1=3, respectively. Importantly, we x y conjecture and numerically corroborate a Gaussian cen- tral limit theorem for the travel time fluctuations, on the scale t1=5 for the RGG and the other proximity graphs, 2=5 FIG. 1: Illustration of the problem on a small network. The and t for the Delaunay triangulation and other ex- network is constructed over a set of points denoted by circles cluded region graphs, which is also distinct from KPZ here and the edges are denoted by lines. For a pair of nodes where the Tracy-Widom distribution, and the scale t1=3, (x; y) we look for the shortest path (shown here by a dotted is the famous outcome. Finally, in Section V we present line) where the length of the path is given by the sum of all some analytic ideas which help explain the distinction be- edges length: d(x; y) = jx−aj+ja−bj+jb−cj+jc−dj+jd−yj. tween universality classes. We then conclude and discuss some open questions in Section VI. over a simple Poisson point process on a flat torus, and weight the edges with their Euclidean length (see Fig. 1). We then study the random length and transversal de- II. BACKGROUND: FPP AND EFPP viation of the shortest paths between two nodes in the network, denoted x and y, conditioned to lie at mutual In EFPP, we first construct a Poisson point process in Euclidean separation jx − yj, as a function of the point Rd which forms the basis of an undirected graph. A fluid process density and other parameters of the model used or current then flows outward from a single source at a (here and in the following jxj denotes the usual norm in constant speed with a travel time along an edge given euclidean space). The study of the scaling with jx − yj by a power α ≥ 1 of the Euclidean length of the edge of the length and the deviation allow to define the fluc- along which it travels [26]. See Fig. 2, where the model tuation and wandering exponents (see precise definitions is shown on six different random spatial network models. below). We will consider a variety of networks such as the Euclidean FPP on a large family of connected random random geometric graph with unit disk and Rayleigh fad- geometric graphs has been studied in detail by Hirsch, ing connection functions, the k-nearest neighbour graph, Neuh¨auser,Gloaguen and Schmidt in [32, 40, 41] and the the Delaunay triangulation, the relative neighbourhood closely related works [32, 40{45], and references therein. graph, the Gabriel graph, and the complete graph with Developing FPP in this setting, Santalla et al [46] re- (in this case only) the edge weights raised to the power cently studied the model on spatial networks, as we do α > 1. We describe these models in more detail in Sec- here. Instead of EFPP, they weight the edges of the De- tion III. launay triangulation, and also the square lattice, with To expand on two examples, the random geometric i.i.d.
Load more

Recommended publications
  • Nonobtuse Triangulations of Pslgs
    NONOBTUSE TRIANGULATIONS OF PSLGS CHRISTOPHER J. BISHOP Abstract. We show that any planar PSLG with n vertices has a conforming triangulation by O(n2.5) nonobtuse triangles, answering the question of whether a polynomial bound exists. The triangles may be chosen to be all acute or all right. A nonobtuse triangulation is Delaunay, so this result improves a previous O(n3) bound of Eldesbrunner and Tan for conforming Delaunay triangulations. In the special case that the PSLG is the triangulation of a simple polygon, we will show that only O(n2) elements are needed, improving an O(n4) bound of Bern and Eppstein. We also show that for any ǫ> 0, every PSLG has a conforming triangulation with O(n2/ǫ2) elements and with all angles bounded above by 90◦ + ǫ. This improves a 3 ◦ 7 ◦ result of S. Mitchell when ǫ = 8 π = 67.5 and Tan when ǫ = 30 π = 42 . Date: March 1, 2011. 1991 Mathematics Subject Classification. Primary: 30C62 Secondary: Key words and phrases. nonobtuse, acute, triangulation, conforming triangulation, PSLG, De- launay, Gabriel, nearest-neighbor learning, quadrilateral mesh, beta-skeleton, thick-thin decompo- sition, polynomial complexity, The author is partially supported by NSF Grant DMS 10-06309. This is a very preliminary version and should not be distributed. Please contact me for the latest version if you wish to share this with someone. 1 NONOBTUSE TRIANGULATIONS OF PSLGS 1 1. Introduction A planar straight line graph (or PSLG from now on) is the union of a finite number (possibly none) of non-intersecting open line segments together with a disjoint finite set that includes all the endpoints of the line segments, but may include other points as well.
    [Show full text]
  • Hierarchical Correlation Clustering in Multiple 2D Scalar Fields
    Lawrence Berkeley National Laboratory Recent Work Title Hierarchical Correlation Clustering in Multiple 2D Scalar Fields Permalink https://escholarship.org/uc/item/09j2d6cg Journal Computer Graphics Forum, 37(3) ISSN 0167-7055 Authors Liebmann, T Weber, GH Scheuermann, G Publication Date 2018-06-01 DOI 10.1111/cgf.13396 Peer reviewed eScholarship.org Powered by the California Digital Library University of California Eurographics Conference on Visualization (EuroVis) 2018 Volume 37 (2018), Number 3 J. Heer, H. Leitte, and T. Ropinski (Guest Editors) Hierarchical Correlation Clustering in Multiple 2D Scalar Fields Tom Liebmann1, Gunther H. Weber2;3, and Gerik Scheuermann1 1Department of Computer Science, Leipzig University, Germany 2Lawrence Berkeley National Laboratory 3University of California, Davis Abstract Sets of multiple scalar fields can be used to model different types of varying data, such as uncertainty in measurements and simulations or time-dependent behavior of scalar quantities. Many structural properties of such fields can be explained by de- pendencies between different points in the scalar field. Although these dependencies can be of arbitrary complexity, correlation, i.e., the linear dependency, already provides significant structural information. Existing methods for correlation analysis are usually limited to positive correlation, handle only local dependencies, or use combinatorial approximations to this continuous problem. We present a new approach for computing and visualizing correlated regions in sets of 2-dimensional scalar fields. This work consists of three main parts: (i) An algorithm for hierarchical correlation clustering resulting in a dendrogram, (ii) adaption of topological landscapes for dendrogram visualization, and (iii) an optional extension of our clustering and visualization to handle negative correlation values.
    [Show full text]
  • Spatial Networks and Probability
    Spatial Networks and Probability David Aldous April 17, 2014 This talk has a different style. Not start with a probability model for a random network and then study its mathematical properties instead start with some data, then try to find some simplified math model that illuminates the data. Advice to young researchers: the Socratic question: if you wish to make an impact with new theory, should you (i) choose a field with extensive theory and no data (ii) choose a field with extensive data and no theory? Also: I teach an undergraduate \probability in the real world" course { 20 lectures on different topics, each \anchored" by (ideally) new real-world data that students could obtain themselves. (22) Psychology of probability: predictable irrationality (18) Global economic risks (17) Everyday perception of chance (16) Luck (16) Science fiction meets science (14) Risk to individuals: perception and reality (13) Probability and algorithms. (13) Game theory (13) Coincidences, near misses and paradoxes. (11) So what do I do in my own research? (spatial networks) (10) Stock Market investment, as gambling on a favorable game (10) Mixing and sorting (9) Tipping points and phase transitions (9) Size-biasing, regression effect and dust-to-dust phenomena (6) Prediction markets, fair games and martingales (6) Branching processes, advantageous mutations and epidemics (5) Toy models of social networks (4) The local uniformity principle (2) Coding and entropy (-5) From neutral alleles to diversity statistics A 2012 paper \Evolution of subway networks" concludes subway systems of very large cities consist of a highly-connected core with branches radiating outwards.
    [Show full text]
  • Minimum-Weight Triangulation Is NP-Hard
    Minimum-Weight Triangulation is NP-hard WOLFGANG MULZER Princeton University and GUNTER¨ ROTE Freie Universit¨at Berlin A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard, using a reduction from PLANAR 1-IN-3-SAT. The correct working of the gadgets is established with computer assistance, using dynamic programming on polygonal faces, as well as the β-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation. Categories and Subject Descriptors: F.2.2 [Nonnumerical Algorithms and Problems]: Geo- metrical problems and computations; G.2.2 [Graph Theory]: Graph algorithms General Terms: Algorithms, Theory Additional Key Words and Phrases: Optimal triangulations, PLANAR 1-IN-3-SAT 1. INTRODUCTION Given a set S of points in the Euclidean plane, a triangulation T of S is a maximal plane straight-line graph with vertex set S. The weight of T is defined as the total Euclidean length of all edges in T . A triangulation that achieves the minimum weight is called a minimum-weight triangulation (MWT) of S. The problem of computing a triangulation for a given planar point set arises nat- urally in many applications such as stock cutting, finite element analysis, terrain modeling, and numerical approximation. The minimum-weight triangulation has attracted the attention of many researchers, mainly due to its natural definition of optimality, and not so much because it would be important for the mentioned ap- plications.
    [Show full text]
  • Algorithmic and Geometric Aspects of Data Depth with Focus on Β-Skeleton Depth
    Algorithmic and Geometric Aspects of Data Depth With Focus on β-skeleton Depth by Rasoul Shahsavarifar Master of Science, University of Tabriz, Iran, 2008 Bachelor of Science, Razi University, Iran, 2006 A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy In the Graduate Academic Unit of Computer Science Supervisor(s): David Bremner, Ph.D, Computer Science Examining Board: Patricia Evans, Ph.D, Computer Science, Chair Suprio Ray, Ph.D, Computer Science Jeffrey Picka, Ph.D Mathematics and Statistics External Examiner: Pat Morin, Ph.D, Computer Science, Carleton University This dissertation is accepted by Dean of Graduate Studies THE UNIVERSITY OF NEW BRUNSWICK April, 2019 c Rasoul Shahsavarifar, 2019 Abstract The statistical rank tests play important roles in univariate non-parametric data analysis. If one attempts to generalize the rank tests to a multivariate case, the problem of defining a multivariate order will occur. It is not clear how to define a multivariate order or statistical rank in a meaningful way. One approach to overcome this problem is to use the notion of data depth which measures the centrality of a point with respect to a given data set. In other words, a data depth can be applied to indicate how deep a point is lo- cated with respect to a given data set. Using data depth, a multivariate order can be defined by ordering the data points according to their depth values. Various notions of data depth have been introduced over the last decades. In this thesis, we discuss three depth functions: two well-known depth functions halfspace depth and simplicial depth, and one recently defined depth function named as β-skeleton depth, β ≥ 1.
    [Show full text]
  • The Shape of Shortest Paths in Random Spatial Networks
    The shape of shortest paths in random spatial networks Alexander P. Kartun-Giles,1 Marc Barthelemy,2 and Carl P. Dettmann3 1Max Plank Institute for Mathematics in the Sciences, Inselstr. 22, Leipzig, Germany 2Institut de Physique Th´eorique,CEA, CNRS-URA 2306, Gif-sur-Yvette, France 3School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance L, geodesics exhibit deviations from their mean length L that are of order Lχ, while the transversal fluctuations, known as wandering, grow as Lξ. We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents ξ = 3=5 and χ = 1=5, or ξ = 7=10 and χ = 2=5, depending only on coarse details of the specific connectivity laws used. Also, the travel time fluctuations are Gaussian, rather than Tracy-Widom, which is rarely seen in first passage models. The first class contains proximity graphs such as the hard and soft random geometric graph, and the k-nearest neighbour random geometric graphs, where via Monte Carlo simulations we find ξ = 0:60 ± 0:01 and χ = 0:20 ± 0:01, showing a theoretical minimal wandering. The second class contains graphs based on excluded regions such as β-skeletons and the Delaunay triangulation and are characterized by the values ξ = 0:70 ± 0:01 and χ = 0:40 ± 0:01, with a nearly theoretically maximal wandering exponent. We also show numerically that the KPZ relation χ = 2ξ − 1 is satisfied for all these models.
    [Show full text]
  • 2D Points Curve Reconstruction Survey and Benchmark
    EUROGRAPHICS ’0x/ N.N. and N.N. STAR – State of The Art Report 2D Points Curve Reconstruction Survey and Benchmark S. Ohrhallinger1 and J. Peethambaran2 and A. D. Parakkat3 and T. K. Dey4 and R. Muthuganapathy5 1Institute of Visual Computing and Human-Centered Technology, TU Wien, Austria 2Department of Math and Computing Science, Saint Mary’s University, Halifax, Canada 3Department of Computer Science and Engineering, Indian Institute of Technology Guwahati, India 4Department of Computer Science, Purdue University, Indiana, USA 5Department of Engineering Design, Indian Institute of Technology Madras, India Figure 1: We survey 36 curve reconstruction algorithms and compare 14 of these, with quantitative and qualitative analysis. As inputs we take unorganized points, samples on the boundary of binary images or smooth curves, and evaluate with ground truth. Abstract Curve reconstruction from unstructured points in a plane is a fundamental problem with many applications that has generated research interest for decades. Involved aspects like handling open, sharp, multiple and non- manifold outlines, run-time and provability as well as potential extension to 3D for surface reconstruction have led to many different algorithms. We survey the literature on 2D curve reconstruction and then present an open- sourced benchmark for the experimental study. Our unprecedented evaluation on a selected set of planar curve reconstruction algorithms aims to give an overview of both quantitative analysis and qualitative aspects for helping arXiv:2103.09583v1 [cs.GR] 17 Mar 2021 users to select the right algorithm for specific problems in the field. Our benchmark framework is available online to permit reproducing the results, and easy integration of new algorithms.
    [Show full text]
  • Stable Delaunay Graphs
    Discrete Comput Geom (2015) 54:905–929 DOI 10.1007/s00454-015-9730-x Stable Delaunay Graphs Pankaj K. Agarwal1 · Jie Gao2 · Leonidas J. Guibas3 · Haim Kaplan4 · Natan Rubin5 · Micha Sharir6,7 Received: 29 April 2014 / Revised: 19 June 2015 / Accepted: 10 August 2015 / Published online: 3 September 2015 © Springer Science+Business Media New York 2015 Abstract Let P be a set of n points in R2, and let DT(P) denote its Euclidean Delaunay triangulation. We introduce the notion of the stability of edges of DT(P). Editor in Charge: János Pach An earlier version [4] of this paper appeared in Proceedings of the 26th Annual Symposium on Computational Geometry, 2010, 127–136. Pankaj K. Agarwal [email protected] Jie Gao [email protected] Leonidas J. Guibas [email protected] Haim Kaplan [email protected] Natan Rubin [email protected] Micha Sharir [email protected] 1 Department of Computer Science, Duke University, Durham, NC 27708-0129, USA 2 Department of Computer Science, Stony Brook University, Stony Brook, NY 11794, USA 3 Department of Computer Science, Stanford University, Stanford, CA 94305, USA 4 School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel 5 Department of Computer Science, Ben-Gurion University of the Negev, Beersheva 84105, Israel 6 School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel 7 Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA 123 906 Discrete Comput Geom (2015) 54:905–929 Specifically, defined in terms of a parameter α>0, a Delaunay edge pq is called α-stable, if the (equal) angles at which p and q see the corresponding Voronoi edge epq are at least α.
    [Show full text]
  • Models for Connected Networks Over Random Points and a Route-Length Statistic
    Models for Connected Networks over Random Points and a Route-Length Statistic David J. Aldous∗ Julian Shun July 8, 2008 Abstract We review mathematically tractable models for connected networks on random points in the plane, emphasising the little-studied class of proximity graphs and introducing a new model called the Hammersley network. We introduce and motivate a particular statistic R measur- ing shortness of routes in a network. We show (via Monte Carlo, in part) the trade-off between normalized network length and R in a one- parameter family of proximity graphs. How close this family comes to the optimal trade-off over all possible networks remains an intriguing open question. xxx Almost finished draft. MSC 2000 subject classifications. 60D05, 90B15. Key words and phrases. Proximity graph, random graph, spatial net- work, geometric graph. Short title: Connected Networks over Random Points ∗Department of Statistics, 367 Evans Hall # 3860, U.C. Berkeley CA 94720; al- [email protected]; www.stat.berkeley.edu/users/aldous. Aldous's research sup- ported by N.S.F Grant DMS-0704159. 1 1 Introduction This is intended as the least technical in a series of papers [4, 7, 5] on spatial networks. For concreteness, visualize a road network linking a given set of cities. Our main purpose is to introduce a specific statistic R for measuring the effectiveness of a network in providing short routes (\route-length effi- ciency"). The definition of R and its motivation are given in section 3.2. A second purpose, perhaps of broader interest (hence presented first, in section 2), is to review models for connected networks on deterministic or random points in the plane.
    [Show full text]
  • Efficiently Computing All Delaunay Triangles Occurring Over
    Efficiently Computing All Delaunay Triangles Occurring over All Contiguous Subsequences Stefan Funke Universität Stuttgart, Germany [email protected] Felix Weitbrecht Universität Stuttgart, Germany [email protected] Abstract Given an ordered sequence of points P = {p1, p2, . , pn}, we are interested in computing T , the set of distinct triangles occurring over all Delaunay triangulations of contiguous subsequences within P . We present a deterministic algorithm for this purpose with near-optimal time complexity O(|T | log n). Additionally, we prove that for an arbitrary point set in random order, the expected number of Delaunay triangles occurring over all contiguous subsequences is Θ(n log n). 2012 ACM Subject Classification Theory of computation → Computational geometry Keywords and phrases Computational Geometry, Delaunay Triangulation, Randomized Analysis Digital Object Identifier 10.4230/LIPIcs.ISAAC.2020.28 1 Introduction For an ordered sequence of points P = {p1, p2, . , pn}, we consider for 1 ≤ i < j ≤ n the contiguous subsequences Pi,j := {pi, pi+1, . , pj} and their Delaunay triangulations S Ti,j := DT (Pi,j). We are interested in the set T := i<j{t | t ∈ Ti,j} of distinct Delaunay triangles occurring over all contiguous subsequences. Figure 1 shows a sequence of points 2 n p1, . , pn where |T | = Ω(n ) (collinearities could be perturbed away). For j > 2 , any point p will be connected to all points {p , . , p n } in T , so any such T contains Θ(n) j 1 2 1,j 1,j 0 2 Delaunay triangles not contained in any T1,j0 with j < j, hence |T | = Ω(n ).
    [Show full text]
  • Graph-Theoretic Scagnostics∗
    Graph-Theoretic Scagnostics∗ Leland Wilkinson† Anushka Anand‡ Robert Grossman§ SPSS Inc. University of Illinois at Chicago University of Illinois at Chicago Northwestern University ABSTRACT 2 TUKEY AND TUKEY SCAGNOSTICS We introduce Tukey and Tukey scagnostics and develop graph- A scatterplot matrix, variously called a SPLOM or casement plot theoretic methods for implementing their procedure on large or draftman’s plot, is a (usually) symmetric matrix of pairwise scat- datasets. terplots. An easy way to conceptualize a symmetric SPLOM is CR Categories: H.5.2 [User Interfaces]: Graphical to think of a covariance matrix of p variables and imagine that User Interfaces—Visualization; I.3.6 [Computing Methodologies]: each off-diagonal cell consists of a scatterplot of n cases rather Computer Graphics—Methodology and Techniques; than a scalar number representing a single covariance. This display was first published by John Hartigan [19] and was popularized by Keywords: visualization, statistical graphics Tukey and his associates at Bell Laboratories [9]. Figure 1 shows a SPLOM of measurements on abalones using data from [27]. Off the 1 INTRODUCTION diagonal are the pairwise scatterplots of nine variables. The vari- ables are sex (indeterminate, male, female), shell length, shell di- Around 20 years ago, John and Paul Tukey developed an ex- ameter, shell height, whole weight, shucked weight, viscera weight, ploratory visualization method called scagnostics. While they shell weight, and number of rings in shell. briefly mentioned their invention in [42], the specifics of the method were never published. Paul Tukey did offer more detail at an IMA visualization workshop a few years later, but he did not include the talk in the workshop volume he and Andreas Buja edited [7].
    [Show full text]
  • Minimum Weight Triangulation Is NP-Hard
    Minimum Weight Triangulation is NP-hard Wolfgang Mulzer, G¨unter Rote B 05-23 (revised) February 2008 Minimum-Weight Triangulation is NP-hard WOLFGANG MULZER Princeton University and GUNTER¨ ROTE Freie Universit¨at Berlin A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard, using a reduction from PLANAR 1-IN-3-SAT. The correct working of the gadgets is established with computer assistance, using dynamic programming on polygonal faces, as well as the β-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation. Categories and Subject Descriptors: F.2.2 [Nonnumerical Algorithms and Problems]: Geo- metrical problems and computations; G.2.2 [Graph Theory]: Graph algorithms General Terms: Algorithms, Theory Additional Key Words and Phrases: Optimal triangulations, PLANAR 1-IN-3-SAT 1. INTRODUCTION Given a set S of points in the Euclidean plane, a triangulation T of S is a maximal plane straight-line graph with vertex set S. The weight of T is defined as the total Euclidean length of all edges in T . A triangulation that achieves the minimum weight is called a minimum-weight triangulation (MWT) of S. The problem of computing a triangulation for a given planar point set arises nat- urally in many applications such as stock cutting, finite element analysis, terrain modeling, and numerical approximation.
    [Show full text]