On Embeddability of Buses in Point Sets

Set membership of points in the plane can be visualized by connecting corresponding points via graphical features, like paths, trees, polygons, ellipses. In this paper we study the bus embeddability problem BEP: given a set of colored points we ask whether there exists a planar realization with one horizontal straight-line segment per color, called bus, such that all points with the same color are connected with vertical line segments to their bus. We present an ILP and an FPT algorithm for the general problem. For restricted versions of this problem, such as when the relative order of buses is predefined, or when a bus must be placed above all its points, we provide efficient algorithms. We show that another restricted version of the problem can be solved using 2-stack pushall sorting. On the negative side we prove the NP-completeness of a special case of BEP.

[1]  Sue Whitesides,et al.  On Bus Graph Realizability , 2006, CCCG.

[2]  Joseph L. Ganley,et al.  Computing Optimal Rectilinear Steiner Trees: A Survey and Experimental Evaluation , 1999, Discret. Appl. Math..

[3]  Brett Stevens,et al.  The Directed Anti-Oberwolfach Solution: Pancyclic 2-Factorizations of Complete Directed Graphs of Odd Order , 2002, Electron. J. Comb..

[4]  Tim Dwyer,et al.  Untangling Euler Diagrams , 2010, IEEE Transactions on Visualization and Computer Graphics.

[5]  Bettina Speckmann,et al.  On Planar Supports for Hypergraphs , 2009, J. Graph Algorithms Appl..

[6]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[7]  Stefan Felsner,et al.  On the Characterization of Plane Bus Graphs , 2013, CIAC.

[8]  C. Thomborson,et al.  A Complexity Theory for VLSI , 1980 .

[9]  David Eppstein,et al.  Journal of Graph Algorithms and Applications Confluent Drawings: Visualizing Non-planar Diagrams in a Planar Way , 2022 .

[10]  Bettina Speckmann,et al.  KelpFusion: A Hybrid Set Visualization Technique , 2013, IEEE Transactions on Visualization and Computer Graphics.

[11]  Daniel W. Archambault,et al.  Fully Automatic Visualisation of Overlapping Sets , 2009, Comput. Graph. Forum.

[12]  Yifan Hu,et al.  MapSets: Visualizing Embedded and Clustered Graphs , 2014, J. Graph Algorithms Appl..

[13]  Aaron D. Jaggard.  Prefix Exchanging and Pattern Avoidance by Involutions , 2002, Electron. J. Comb..

[14]  Sergio Cabello.  Planar embeddability of the vertices of a graph using a fixed point set is NP-hard , 2006, J. Graph Algorithms Appl..

[15]  Dominique Rossin,et al.  2-Stack Sorting is Polynomial , 2013, Theory of Computing Systems.

[16]  Roberto Tamassia,et al.  A unified approach to visibility representations of planar graphs , 1986, Discret. Comput. Geom..

[17]  Yehuda Koren,et al.  Improved Circular Layouts , 2006, GD.

[18]  Michael A. Bekos,et al.  Many-to-One Boundary Labeling with Backbones , 2015, J. Graph Algorithms Appl..

[19]  Martin Nöllenburg,et al.  Minimum Tree Supports for Hypergraphs and Low-Concurrency Euler Diagrams , 2014, SWAT.

[20]  Donald E. Knuth.  The art of computer programming: fundamental algorithms , 1969 .

[21]  Xin He,et al.  On Finding the Rectangular Duals of Planar Triangular Graphs , 1993, SIAM J. Comput..

[22]  Ioannis G. Tollis,et al.  Graph Drawing , 1994, Lecture Notes in Computer Science.

[23]  Mary Czerwinski,et al.  Design Study of LineSets, a Novel Set Visualization Technique , 2011, IEEE Transactions on Visualization and Computer Graphics.

[24]  Miklós Bóna,et al.  A Survey of Stack-Sorting Disciplines , 2003, Electron. J. Comb..

[25]  Ulrik Brandes,et al.  Path-based supports for hypergraphs , 2012, J. Discrete Algorithms.

[26]  Robert E. Tarjan,et al.  A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas , 1979, Inf. Process. Lett..

[27]  David Lichtenstein,et al.  Planar Formulae and Their Uses , 1982, SIAM J. Comput..

[28]  David S. Johnson,et al.  The Rectilinear Steiner Tree Problem is NP Complete , 1977, SIAM Journal of Applied Mathematics.

[29]  Alexander Wolff,et al.  Manhattan-Geodesic Embedding of Planar Graphs , 2009, Graph Drawing.

[30]  Jan van Leeuwen,et al.  Handbook of Theoretical Computer Science, Vol. A: Algorithms and Complexity , 1994 .

[31]  Donald E. Knuth,et al.  The Art of Computer Programming, Volume I: Fundamental Algorithms, 2nd Edition , 1997 .

[32]  Sergey Bereg,et al.  Edge Routing with Ordered Bundles , 2011, Graph Drawing.

[33]  Feng Zhou,et al.  Refined single trunk tree: a rectilinear steiner tree generator for interconnect prediction , 2002, SLIP '02.

[34]  David S. Johnson,et al.  The Complexity of Computing Steiner Minimal Trees , 1977 .

[35]  M. Sheelagh T. Carpendale,et al.  Bubble Sets: Revealing Set Relations with Isocontours over Existing Visualizations , 2009, IEEE Transactions on Visualization and Computer Graphics.

[36]  Thomas Lengauer.  VLSI Theory , 1990, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[37]  F. Hwang,et al.  The Steiner Tree Problem , 2012 .

[38]  Dominique Rossin,et al.  2-stack Pushall Sortable Permutations , 2013, ArXiv.