DOCSLIB.ORG

Alaa AK Ismaeel

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Alaa AK Ismaeel Alaa A. K. Ismaeel Dynamic Hierarchical Graph Drawing Alaa A. K. Ismaeel Dynamic Hierarchical Graph Drawing Dynamic Hierarchical Graph Drawing by Alaa A. K. Ismaeel Dissertation, Karlsruher Instituts für Technologie (KIT), Fakultät für Wirtschaftswissenschaften, 2012 Tag der mündlichen Prüfung: 21. Mai 2012 Referent: Prof. Dr. Hartmut Schmeck, Institut für Angewandte Informatik und For- male Beschreibungsverfahren (AIFB), Karlsruher Instituts für Technologie (KIT), Karlsruhe, Germany. Korreferent: Prof. Dr. Jürgen Branke, Warwick Business School, The University of Warwick, Coventry, United Kingdom. Dynamic Hierarchical Graph Drawing Zur Erlangung des akademischen Grades eines Doktors der Wirtschaftswissenschaften (Dr. rer. pol.) von der Fakultät für Wirtschaftswissenschaften des Karlsruher Instituts für Technologie (KIT) genehmigte DISSERTATION von M.Sc. Alaa Aly Khalaf Ismaeel aus El-Minia, Ägypten Tag der mündlichen Prüfung: 21. Mai 2012 Referent: Prof. Dr. Hartmut Schmeck Korreferent: Prof. Dr. Jürgen Branke 2012 Karlsruhe Our Lord! Condemn us not if we forget, or miss the mark! ﺳﻮﺭﺓ ﺍﻟﺒﻘﺮﺓ – ﺁﻳﺔ ٢٨٦ Surah Al-Baqara – Verse 286 Acknowledgements First and foremost, my praise and deepest thanks are to God the Almighty for guiding me on the right path. Without HIS help and care, my efforts would have gone astray. Looking back, I feel immense gratitude to many people around me who have supported me while I was working on this Ph.D. thesis. The past four and a quarter years have given me many exciting, challenging and truly valuable experiences. This thesis is the result of a scholarship to carry out PhD in Computer Science at Karlsruhe Institute of Technology (KIT) in Germany. The scholarship was financially supported by the Egyptian Government (Egyptian Ministry of Higher Education) through the Egyptian Culture Bureau and Study Mission in Berlin. I would like to thank all of the administrative staff in Berlin and in the Culture Affairs & Mission Sector in Cairo. I am sincerely thankful to my supervisor, Prof. Dr. Hartmut Schmeck. Despite his busy schedule in an ever-expanding research group plus his many responsibilities at the KIT, he frequently had some time for valuable and effective discussions. I am really grateful for you, Prof. Schmeck, for accepting me as one of your PhD students and also for getting involved as a member in your Efficient Algorithms research group at the Institute AIFB. Furthermore, for your encouragement, guidance and support that enabled me to develop an understanding of the thesis. Similarly, I would also like to thank Prof. Dr. Jürgen Branke, from the University of Warwick, United Kingdom. He accepted, without hesitation, the request to serve as a second reviewer on the examination committee and also for his valuable comments. Before moving to the United Kingdom, Prof. Branke was working at the Institute AIFB, at the KIT. The preparation of the PhD proposal of this thesis was based on creative discussions with him. Moreover, I cannot forget how he helped me in my first few days in Karlsruhe. I am really very grateful for him. Furthermore, many thanks go to Prof. Dr. Karl-Heinz Waldmann and Prof. Dr. Ute Werner, both from the Karlsruhe Institute of Technology (KIT), who served as examiner and chairman, respectively, on the examination committee. At the AIFB intensive and collaborative cooperation has been carried out with my col- league Dr. Pradyumn Shukla during the last two years of the thesis. I am greatly indebted to you for your honestly, help and kind support. Furthermore, I am grateful to all of my colleagues with whom I have worked during the last few years. The current and former mem- bers of the research group "Efficient Algorithms" provided a pleasant and productive working atmosphere. I would like to express my thanks to Dr. Ingo Pänke, Felix Vogel, Lukas König, Dr. Sanaz Mostaghim, Dr. Holger Prothmann, Dr. Urban Richter, Dr. Andreas Kamper, xiii Acknowledgements Dr. Lei Liu, Dr. André Wiesner, Florian Allerding, Birger Becker, Frederic Toussaint, Nu- groho Fredivianus, Christian Hirsch, Sabrina Merkel, Marc Mültin, Daniel Pathmaperuma, Friederike Pfeiffer, Micaela Wünsche, Fabian Rigoll, Fredy Rios, and our secretary Mrs. In- geborg Götz. Special thanks also go to Felix, Sanaz, Lukas, and my office partner Nug. In the Graph Drawing community, it is an honour for me to thank Christian Bachmaier and Andreas Gleißner (University of Passau, Germany) for our valuable discussions and the implementation of the Gravisto graph visualization toolkit. Furthermore, I am grateful to Prof. Perter Eades (University of Sydney, Australia), Prof. Roberto Tamassia (Brown University, USA), Prof. Ulrik Brandes (University of Konstanz, Germany) for our valuable discussions and the advice they gave during the 2010 and 2011 Graph Drawing conferences. Also, thanks are presented to Dr. Helen C. Purchase (University of Glasgow, Scotland), for her valuable comments. Moreover, for the non-scientific side of my thesis, I offer my sincerest regards and blessings to all of those who supported me in any capacity during the completion of this thesis. I am grateful to my family and friends in Egypt for their constant moral support. Special thanks go to my father, my brothers Ashraf and Osama, father-in-law, mother-in-law, and brothers and sisters in law, who always believed in me, supported me in every respect and for their constant encouragement and endless praying for me throughout the last few years. Last but not least, words fail to express my deepest thanks to my wife and life partner Mariam. I am so proud to dedicate this thesis to her, as she always enriches my life with her cheerful, spirited approach to life and countless and endless support. Really, she has been there for me every time I needed her even in stressful and chaotic times. Finally, I would like to show my gratitude to my children Alzhraa, Abdelmohsen, and Ashraqat for their innocent smiles, funny reactions and for making my Karlsruhe years so wonderful and unforgettable. Karlsruhe, May 2012 Alaa A. K. Ismaeel xiv Abstract Graph Drawing addresses the geometric representation of graphs. It is motivated by appli- cation fields in which it is crucial to visualize structural information as graphs. The majority of research into graph drawing has been in regard to efficient algorithms for working with static graphs in which the graph structure does not change. In dynamic graphs, changes in the graph structure and/or its drawing at use time of the graph are possible by adding, delet- ing, or modifying vertices and/or edges. Many graph drawing scenarios are dynamic since they involve a repeated updating of the graph, and consequently its drawing, after certain change is executed. One of the main objectives in dynamic graph drawing is preserving the user’s mental map, which is minimizes the effort the user has to spend in order to become re-familiarized with the drawing after some actions have been executed. The recommended approach for preserving the user’s mental map is to minimize the changes between the new and current drawings. Effectively, minimizing the changes involves computing a measure of similarity between the two drawings. Collectively, these measures are known as difference metrics. Some considered difference metrics are the Euclidean distance, relative distance, edge orthogonality, vertex width, and vertex ordering. This thesis is concerned with the problem of proposing and validating difference metrics of dynamic hierarchical graph drawings. The existing approaches in this direction focus on how to execute an action on a dynamic hierarchical graph, or on measuring the difference between drawings of a dynamic hierarchical graph using difference metrics. We introduce a new general framework on how to formulate difference metrics for hierarchical graphs. This framework is based on distinguishing between the vertices and edges that are inserted into or deleted from the graph and those that are still shared in both drawings. The general form of a difference metric is formulated by considering the topological and geometric characteristics of hierarchical graphs and have also been applied to existing metrics that are used with non- hierarchical graph drawings. The proposed formulation could be applied to any hierarchical graph of any size and could be easily extended to different graph types. Measuring the similarity between two different drawings should not be done with respect to some computations of similarity metrics only since the human judgement based on the perception of what appears aesthetic into account. So, an experimental user study is carried out in order to validate the introduced difference metrics based on user evaluations. Another problem for drawing hierarchical graphs aesthetically is minimizing crossings be- tween edges. Two approaches are considered in solving this problem. The first is the layer-by- layer sweep and the second is the global approach. In the layer-by-layer sweep approach, the xv Abstract problem is solved repeatedly by computing the order of the vertices in each layer according to their connectivity in the upper and lower layer, then the algorithm moves to the next two layers. In the global approach, the order of vertices in all layers is computed simultaneously. A new algorithm is introduced, named the efficient barycenter, which empirically produces better results than some recent global algorithms and traditional layer-by-layer algorithms, in both the number of crossings and the execution time. Finally, a new method for generating random hierarchical graphs is introduced. This method is based on a precise counting of the maximum number of edges in a hierarchical graph. Two generators for proper and generally non-proper hierarchical graph are introduced. Both generators control all the parameters of a hierarchical graph, like the number of layers, the number of vertices in each layer, the minimum number of outgoing edges of a vertex, the edge density and the ratio of the long edges in the graph. xvi Contents Figures List xix Tables List xxi Algorithms List xxiii 1 Introduction 1 1.1 Graph Drawing ..................................
Load more

Recommended publications
  • Space and Polynomial Time Algorithm for Reachability in Directed Layered Planar Graphs
    Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 16 (2015) An O(n) Space and Polynomial Time Algorithm for Reachability in Directed Layered Planar Graphs Diptarka Chakraborty∗and Raghunath Tewari y Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, Kanpur, India April 27, 2015 Abstract Given a graph G and two vertices s and t in it, graph reachability is the problem of checking whether there exists a path from s to t in G. We show that reachability in directed layered planar graphs can be decided in polynomial time and O(n ) space, for any > 0. Thep previous best known space bound for this problem with polynomial time was approximately O( n) space [INP+13]. Deciding graph reachability in SC is an important open question in complexity theory and in this paper we make progress towards resolving this question. 1 Introduction Given a graph and two vertices s and t in it, the problem of determining whether there is a path from s to t in the graph is known as the graph reachability problem. Graph reachability problem is an important question in complexity theory. Particularly in the domain of space bounded computa- tions, the reachability problem in various classes of graphs characterize the complexity of different complexity classes. The reachability problem in directed and undirected graphs, is complete for the classes non-deterministic log-space (NL) and deterministic log-space (L) respectively [LP82, Rei08]. The latter follows due to a famous result by Reingold who showed that undirected reachability is in L [Rei08].
    [Show full text]
  • Confluent Hasse Diagrams
    Confluent Hasse diagrams David Eppstein Joseph A. Simons Department of Computer Science, University of California, Irvine, USA. October 29, 2018 Abstract We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with O(n2) features, in an O(n) × O(n) grid in O(n2) time. For the digraphs representing series-parallel partial orders we show how to construct a drawing with O(n) fea- tures in an O(n) × O(n) grid in O(n) time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use. 1 Introduction One of the most important aspects of a graph drawing is that it should be readable: it should convey the structure of the graph in a clear and concise way. Ease of understanding is difficult to quantify, so various proxies for readability have been proposed; one of the most prominent is the number of edge crossings. That is, we should minimize the number of edge crossings in our drawing (a planar drawing, if possible, is ideal), since crossings make drawings harder to read. Another measure of readability is the total amount of ink required by the drawing [1]. This measure can be formulated in terms of Tufte’s “data-ink ratio” [22,35], according to which a large proportion of the ink on any infographic should be devoted to information. Thus given two different ways to present information, we should choose the more succinct and crossing-free presentation.
    [Show full text]
  • Upward Planar Drawing of Single Source Acyclic Digraphs (Extended
    Upward Planar Drawing of Single Source Acyclic Digraphs (Extended Abstract) y z Michael D. Hutton Anna Lubiw UniversityofWaterlo o UniversityofWaterlo o convention, the edges in the diagrams in this pap er Abstract are directed upward unless sp eci cally stated otherwise, An upward plane drawing of a directed acyclic graph is a and direction arrows are omitted unless necessary.The plane drawing of the graph in which each directed edge is digraph on the left is upward planar: an upward plane represented as a curve monotone increasing in the vertical drawing is given. The digraph on the rightisnot direction. Thomassen [14 ] has given a non-algorithmic, upward planar|though it is planar, since placing v graph-theoretic characterization of those directed graphs inside the face f would eliminate crossings, at the with a single source that admit an upward plane drawing. cost of pro ducing a downward edge. Kelly [10] and We present an ecient algorithm to test whether a given single-source acyclic digraph has an upward plane drawing v and, if so, to nd a representation of one suchdrawing. The algorithm decomp oses the graph into biconnected and triconnected comp onents, and de nes conditions for merging the comp onents into an upward plane drawing of the original graph. To handle the triconnected comp onents f we provide a linear algorithm to test whether a given plane drawing admits an upward plane drawing with the same faces and outer face, which also gives a simpler, algorithmic Upward planar Non-upward-planar pro of of Thomassen's result. The entire testing algorithm (for general single-source directed acyclic graphs) op erates 2 in O (n ) time and O (n) space.
    [Show full text]
  • LNCS 7034, Pp
    Confluent Hasse Diagrams DavidEppsteinandJosephA.Simons Department of Computer Science, University of California, Irvine, USA Abstract. We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimen- sion at most two. In this case, we construct a confluent upward drawing with O(n2)features,inanO(n) × O(n)gridinO(n2)time.Forthe digraphs representing series-parallel partial orders we show how to con- struct a drawing with O(n)featuresinanO(n)×O(n)gridinO(n)time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use. 1 Introduction One of the most important aspects of a graph drawing is that it should be readable: it should convey the structure of the graph in a clear and concise way. Ease of understanding is difficult to quantify, so various proxies for it have been proposed, including the number of crossings and the total amount of ink required by the drawing [1,18]. Thus given two different ways to present information, we should choose the more succinct and crossing-free presentation. Confluent drawing [7,8,9,15,16] is a style of graph drawing in which multiple edges are combined into shared tracks, and two vertices are considered to be adjacent if a smooth path connects them in these tracks (Figure 1). This style was introduced to re- duce crossings, and in many cases it will also Fig. 1. Conventional and confluent improve the ink requirement by represent- drawings of K5,5 ing dense subgraphs concisely.
    [Show full text]
  • Upward Planar Drawings with Three Slopes
    Upward Planar Drawings with Three and More Slopes Jonathan Klawitter and Johannes Zink Universit¨atW¨urzburg,W¨urzburg,Germany Abstract. We study upward planar straight-line drawings that use only a constant number of slopes. In particular, we are interested in whether a given directed graph with maximum in- and outdegree at most k admits such a drawing with k slopes. We show that this is in general NP-hard to decide for outerplanar graphs (k = 3) and planar graphs (k ≥ 3). On the positive side, for cactus graphs deciding and constructing a drawing can be done in polynomial time. Furthermore, we can determine the minimum number of slopes required for a given tree in linear time and compute the corresponding drawing efficiently. Keywords: upward planar · slope number · NP-hardness 1 Introduction One of the main goals in graph drawing is to generate clear drawings. For visualizations of directed graphs (or digraphs for short) that model hierarchical relations, this could mean that we explicitly represent edge directions by letting each edge point upward. We may also require a planar drawing and, if possible, we would thus get an upward planar drawing. For schematic drawings, we try to keep the visual complexity low, for example by using only few different geometric primitives [28] { in our case few slopes for edges. If we allow two different slopes we get orthogonal drawings [14], with three or four slopes we get hexalinear and octilinear drawings [36], respectively. Here, we combine these requirements and study upward planar straight-line drawings that use only few slopes.
    [Show full text]
  • Algorithms for Incremental Planar Graph Drawing and Two-Page Book Embeddings
    Algorithms for Incremental Planar Graph Drawing and Two-page Book Embeddings Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨at zu Koln¨ vorgelegt von Martin Gronemann aus Unna Koln¨ 2015 Berichterstatter (Gutachter): Prof. Dr. Michael Junger¨ Prof. Dr. Markus Chimani Prof. Dr. Bettina Speckmann Tag der m¨undlichenPr¨ufung: 23. Juni 2015 Zusammenfassung Diese Arbeit besch¨aftigt sich mit zwei Problemen bei denen es um Knoten- ordnungen in planaren Graphen geht. Hierbei werden als erstes Ordnungen betrachtet, die als Grundlage fur¨ inkrementelle Zeichenalgorithmen dienen. Solche Algorithmen erweitern in der Regel eine vorhandene Zeichnung durch schrittweises Hinzufugen¨ von Knoten in der durch die Ordnung gegebene Rei- henfolge. Zu diesem Zweck kommen im Gebiet des Graphenzeichnens verschie- dene Ordnungstypen zum Einsatz. Einigen dieser Ordnungen fehlen allerdings gewunschte¨ oder sogar fur¨ einige Algorithmen notwendige Eigenschaften. Diese Eigenschaften werden genauer untersucht und dabei ein neuer Typ von Ord- nung entwickelt, die sogenannte bitonische st-Ordnung, eine Ordnung, welche Eigenschaften kanonischer Ordnungen mit der Flexibilit¨at herk¨ommlicher st- Ordnungen kombiniert. Die zus¨atzliche Eigenschaft bitonisch zu sein erm¨oglicht es, eine st-Ordnung wie eine kanonische Ordnung zu verwenden. Es wird gezeigt, dass fur¨ jeden zwei-zusammenh¨angenden planaren Graphen eine bitonische st-Ordnung in linearer Zeit berechnet werden kann. Im Ge- gensatz zu kanonischen Ordnungen, k¨onnen st-Ordnungen naturgem¨aß auch fur¨ gerichtete Graphen verwendet werden. Diese F¨ahigkeit ist fur¨ das Zeichnen von aufw¨artsplanaren Graphen von besonderem Interesse, da eine bitonische st-Ordnung unter Umst¨anden es erlauben wurde,¨ vorhandene ungerichtete Zei- chenverfahren fur¨ den gerichteten Fall anzupassen.
    [Show full text]
  • On Layered Drawings of Planar Graphs
    On Layered Drawings of Planar Graphs Bachelor Thesis of Sarah Lutteropp At the Department of Informatics Institute of Theoretical Computer Science Reviewers: Prof. Dr. Maria Axenovich Prof. Dr. Dorothea Wagner Advisors: Dipl.-Inform. Thomas Bläsius Dr. Tamara Mchedlidze Dr. Torsten Ueckerdt Time Period: 28th January 2014 – 28th May 2014 KIT – University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association www.kit.edu Acknowledgements I would like to thank my advisors Thomas Bläsius, Dr. Tamara Mchedlidze and Dr. Torsten Ueckerdt for their great support in form of many hours of discussion, useful ideas and extensive commenting on iterative versions. It was not trivial to find a topic and place for a combined mathematics and computer science thesis and I am grateful to Prof. Dr. Dorothea Wagner, who allowed me to write my thesis at her institute. I would also like to thank Prof. Dr. Dorothea Wagner and Prof. Dr. Maria Axenovich for grading my thesis. Last but not least, I would like to thank all my proofreaders (supervisors included) for giving last-minute comments on my thesis. Statement of Authorship I hereby declare that this document has been composed by myself and describes my own work, unless otherwise acknowledged in the text. Karlsruhe, 28th May 2014 iii Abstract A graph is k-level planar if it admits a planar drawing in which each vertex is mapped to one of k horizontal parallel lines and the edges are drawn as non-crossing y-monotone line segments between these lines. It is not known whether the decision problem of a graph being k-level planar is solvable in polynomial time complexity (in P) or not.
    [Show full text]
  • Upward Planar Drawing of Single Source Acyclic Digraphs
    Upward Planar Drawing of Single Source Acyclic Digraphs by Michael D. Hutton A thesis presented to the UniversityofWaterlo o in ful lmentofthe thesis requirement for the degree of Master of Mathematics in Computer Science Waterlo o, Ontario, Canada, 1990 c Michael D. Hutton 1990 ii I hereby declare that I am the sole author of this thesis. I authorize the UniversityofWaterlo o to lend this thesis to other institutions or individuals for the purp ose of scholarly research. I further authorize the UniversityofWaterlo o to repro duce this thesis by pho- to copying or by other means, in total or in part, at the request of other institutions or individuals for the purp ose of scholarly research. ii The UniversityofWaterlo o requires the signatures of all p ersons using or pho- to copying this thesis. Please sign b elow, and give address and date. iii Acknowledgements The results of this thesis arise from joint researchwithmy sup ervisor, Anna Lubiw. Anna also deserves credit for my initial interest in Algorithms and Graph Theory, for her extra e ort to help me makemy deadlines, and for nancial supp ort from her NSERC Op erating Grant. Thanks also to NSERC for nancial supp ort, and to my readers Kelly Bo oth and DerickWo o d for their valuable comments and suggestions. iv Dedication This thesis is dedicated to my parents, David and Barbara on the o ccasion of their 26'th wedding anniversary. Without their years of hard work, love, supp ort and endless encouragement, it would not exist. Michael David Hutton August 29, 1990 v Abstract Aupward plane drawing of a directed acyclic graph is a straightlinedrawing in the Euclidean plane such that all directed arcs p ointupwards.
    [Show full text]
  • Drawing of Level Planar Graphs with Fixed Slopes
    Drawing of Level Planar Graphs with Fixed Slopes Bachelor Thesis of Nadine Davina Krisam At the Department of Informatics Institute of Theoretical Computer Science Reviewers: Prof. Dr. Dorothea Wagner Prof. Dr. rer. nat. Peter Sanders Advisors: Guido Brückner, M.Sc. Dr. Tamara Mchedlidze Time Period: May 1st, 2018 – August 31th, 2018 KIT – University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association www.kit.edu Statement of Authorship I hereby declare that this document has been composed by myself and describes my own work, unless otherwise acknowledged in the text. I also declare that I have read the Satzung zur Sicherung guter wissenschaftlicher Praxis am Karlsruher Institut für Technologie (KIT). Karlsruhe, August 30, 2018 iii Abstract A drawing of a directed acyclic planar graph is called upward planar if all edges are represented by monotonically increasing non-intersecting curves. A directed acyclic planar graph is called level planar if it has an upward planar drawing such that all vertices that are assigned to the same level have the same y-coordinate. In this thesis we study level planar drawings with two fixed slopes of the edges and call them LP2-drawings. We present an algorithm that, given a level planar graph, decides whether it has an LP2-drawing with an additional requirement of rectangular inner faces. After that, we drop the requirement of rectangular inner faces and provide an algorithm that decides whether a general LP2-drawing exists. Both algorithms have polynomial running time. In the conclusion of the thesis, we give an outlook how to extend the latter algorithm to work on multiple fixed slopes.
    [Show full text]
  • 55 GRAPH DRAWING Emilio Di Giacomo, Giuseppe Liotta, Roberto Tamassia
    55 GRAPH DRAWING Emilio Di Giacomo, Giuseppe Liotta, Roberto Tamassia INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as soft- ware engineering, database systems, visual interfaces, and computer-aided design. Research on graph drawing has been conducted within several diverse areas, includ- ing discrete mathematics (topological graph theory, geometric graph theory, order theory), algorithmics (graph algorithms, data structures, computational geometry, vlsi), and human-computer interaction (visual languages, graphical user interfaces, information visualization). This chapter overviews aspects of graph drawing that are especially relevant to computational geometry. Basic definitions on drawings and their properties are given in Section 55.1. Bounds on geometric and topological properties of drawings (e.g., area and crossings) are presented in Section 55.2. Sec- tion 55.3 deals with the time complexity of fundamental graph drawing problems. An example of a drawing algorithm is given in Section 55.4. Techniques for drawing general graphs are surveyed in Section 55.5. 55.1 DRAWINGS AND THEIR PROPERTIES TYPES OF GRAPHS First, we define some terminology on graphs pertinent to graph drawing. Through- out this chapter let n and m be the number of graph vertices and edges respectively, and d the maximum vertex degree (i.e., number of edges incident to a vertex). GLOSSARY Degree-k graph: Graph with maximum degree d k. ≤ Digraph: Directed graph, i.e., graph with directed edges. Acyclic digraph: Digraph without directed cycles. Transitive edge: Edge (u, v) of a digraph is transitive if there is a directed path from u to v not containing edge (u, v).
    [Show full text]
  • Upward Planar Drawing of Single Source Acyclic
    UPWARD PLANAR DRAWING OF SINGLE SOURCE ACYCLIC DIGRAPHS y z MICHAEL D. HUTTON AND ANNA LUBIW Abstract. An upward plane drawing of a directed acyclic graph is a plane drawing of the digraph in which each directed edge is represented as a curve monotone increasing in the vertical direction. Thomassen [24]hasgiven a non-algorithmic, graph-theoretic characterization of those directed graphs with a single source that admit an upward plane drawing. We present an ecient algorithm to test whether a given single-source acyclic digraph has an upward plane drawing and, if so, to nd a representation of one suchdrawing. This result is made more signi cant in lightofthe recent pro of, by Garg and Tamassia, that the problem is NP-complete for general digraphs [12]. The algorithm decomp oses the digraph into biconnected and triconnected comp onents, and de- nes conditions for merging the comp onents into an upward plane drawing of the original digraph. To handle the triconnected comp onentsweprovide a linear algorithm to test whether a given plane drawing of a single source digraph admits an upward plane drawing with the same faces and outer face, which also gives a simpler, algorithmic pro of of Thomassen's result. The entire testing algo- 2 rithm (for general single-source directed acyclic graphs) op erates in O (n )timeandO (n) space (n b eing the number of vertices in the input digraph) and represents the rst p olynomial time solution to the problem. Key words. algorithms, upward planar, graph drawing, graph emb edding, graph decomp osition, graph recognition, planar graph, directed graph AMS sub ject classi cations.
    [Show full text]
  • Technical Foundations
    Technical Foundations Michael J¨unger1 and Petra Mutzel2 1 University of Cologne, Department of Computer Science, Pohligstraße 1, D-50969 K¨oln, Germany 2 Vienna University of Technology, Institute of Computer Graphics and Algorithms, Favoritenstraße 9–11, A-1040 Wien, Austria 1 Introduction Graph drawing software relies on a variety of mathematical results, mainly in graph theory, topology,andgeometry, as well as computer science techniques, mainly in the areas algorithms and data structures, software engineering, and user interfaces. Many of the core techniques used in automatic graph drawing come from the intersection of mathematics and computer science in combinatorial and continuous optimization. Even though automatic graph drawing is a relatively young scientific field, a few generic approaches have emerged in the graph drawing community. They allow a classification of layout methods so that most software packages implement variations of such approaches. The purpose of this chapter is to lay the foundations for all subsequent chapters so that they can be read independently from each other while re- ferring back to the common material presented here. This chapter has been written based on the requirements and the contributions of all authors in this book. This chapter is not an introduction to automatic graph drawing, because it is neither complete nor balanced. In order to avoid repetitions, we only explain subjects that are basic or used in at least two subsequent chapters. The following chapters contain a lot of additional material. For introductions into the field of automatic graph drawing we recommend the books “Graph Drawing” by Di Battista, Eades, Tamassia, and Tollis [23] and “Drawing Graphs” edited by Kaufmann and Wagner [52].
    [Show full text]