Drawing Ordered (k - 1)-Ary Trees on k-Grids

We explore the complexity of drawing ordered (k - 1)-ary trees on grids with k directions for k ∈ {4, 6, 8} and within a given area. This includes, e. g., ternary trees drawn on the orthogonal grid. For aesthetically pleasing tree drawings on these grids, we additionally present various restrictions similar to the common hierarchical case. First, we generalize the NP-hardness of minimal width in hierarchical drawings of ordered trees to (k - 1)-ary trees on k-grids and then we generalize the Reingold and Tilford algorithm to k-grids.

[1]  John Q. Walker,et al.  A node‐positioning algorithm for general trees , 1990, Softw. Pract. Exp..

[2]  Goos Kant.  Hexagonal Grid Drawings , 1992, WG.

[3]  Giuseppe Di Battista,et al.  A Note on Optimal Area Algorithms for Upward Drawings of Binary Trees , 1992, Comput. Geom..

[4]  Roberto Tamassia,et al.  On Embedding a Graph in the Grid with the Minimum Number of Bends , 1987, SIAM J. Comput..

[5]  Tao Lin,et al.  Minimum Size h-v Drawings , 1992, Advanced Visual Interfaces.

[6]  Edward M. Reingold,et al.  Tidier Drawings of Trees , 1981, IEEE Transactions on Software Engineering.

[7]  Stavros S. Cosmadakis,et al.  The Complexity of Minimizing Wire Lengths in VLSI Layouts , 1987, Inf. Process. Lett..

[8]  Peter Eades,et al.  The Logic Engine and the Realization Problem for Nearest Neighbor Graphs , 1996, Theor. Comput. Sci..

[9]  Timothy M. Chan,et al.  Optimizing area and aspect ration in straight-line orthogonal tree drawings , 1996, Comput. Geom..

[10]  Michael Jünger,et al.  On the complexity of drawing trees nicely: corrigendum , 2004, Acta Informatica.

[11]  Christian Bachmaier,et al.  Tree Drawings on the Hexagonal Grid , 2009, GD.

[12]  Anne Mulkers,et al.  Live Data Structures in Logic Programs , 1993, Lecture Notes in Computer Science.

[13]  Kyung-Yong Chwa,et al.  Area-efficient algorithms for straight-line tree drawings , 2000, Comput. Geom..

[14]  Edward M. Reingold,et al.  The complexity of drawing trees nicely , 2004, Acta Informatica.

[15]  Therese C. Biedl,et al.  Hexagonal Grid Drawings: Algorithms and Lower Bounds , 2004, Graph Drawing.

[16]  Ashim Garg,et al.  Straight-line Drawings of Binary Trees with Linear Area and Arbitrary Aspect Ratio , 2002, J. Graph Algorithms Appl..

[17]  Michael Forster,et al.  Gravisto: Graph Visualization Toolkit , 2004, GD.

[18]  Peter J. Stuckey,et al.  NP-Completeness of Minimal Width Unordered Tree Layout , 2004, J. Graph Algorithms Appl..

[19]  J. van Leeuwen,et al.  Drawing Graphs , 2001, Lecture Notes in Computer Science.

[20]  Fabrizio Frati,et al.  Straight-Line Orthogonal Drawings of Binary and Ternary Trees , 2007, GD.