A Framework for Drawing Planar Graphs with Curves and Polylines

We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well...

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Published in:Journal of algorithms Vol. 37; no. 2; pp. 399 - 421
Main Authors: Goodrich, Michael T., Wagner, Christopher G.
Format: Journal Article
Language:English
Published: San Diego, CA Elsevier Inc 01.11.2000
Elsevier
Subjects:
Combinatorics
Combinatorics. Ordered structures
Exact sciences and technology
Graph theory
Mathematics
Sciences and techniques of general use
Graph construction
Planar graph
Worst case method
Asymptotic behavior
Curve
Asymptotic optimality
Graph theory
Aspect ratio
Algorithm
Complexity
Bézier curve
Optimal solution
Aesthetics
ISSN:0196-6774, 1090-2678
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Abstract We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any n-vertex planar graph in an O(n)×O(n) grid using polylines that have at most two bends per edge and asymptotically-optimal worst-case angular resolution. More significantly, we show how to adapt this algorithm to draw any n-vertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n)×O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. All of our algorithms run in O(n) time.
AbstractList We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any n-vertex planar graph in an O(n)×O(n) grid using polylines that have at most two bends per edge and asymptotically-optimal worst-case angular resolution. More significantly, we show how to adapt this algorithm to draw any n-vertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n)×O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. All of our algorithms run in O(n) time.
Author Goodrich, Michael T.
Wagner, Christopher G.
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  organization: Department of Computer Science, The Johns Hopkins University, Baltimore, Maryland, 21218, f1goodrich@cs.jhu.eduf1
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  givenname: Christopher G.
  surname: Wagner
  fullname: Wagner, Christopher G.
  organization: Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland, 21218, f2wagner@brutus.mts.jhu.eduf2
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Issue 2
Keywords Graph construction
Planar graph
Worst case method
Asymptotic behavior
Curve
Asymptotic optimality
Graph theory
Aspect ratio
Algorithm
Complexity
Bézier curve
Optimal solution
Aesthetics
Language English
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References Chrobak, Payne (RF2) 1995; 54
Tamassia (RF14) 1997
de Fraysseix, Pach, Pollack (RF3) 1990; 10
Tamassia (RF13) 1987; 16
Schnyder (RF12) 1990
Chan, Goodrich, Kosaraju, Tamassia (RF1) 1997; 1190
G. Kant, Drawing planar graphs using the canonical ordering, Algorithmica161996, 4–32. 1996. [Special issue on Graph Drawing, edited by G. Di Battista and R. Tamassia]
Garg, Tamassia (RF7) 1994; 855
Hopcroft, Tarjan (RF8) 1974; 21
Malitz, Papakostas (RF11) 1994; 7
Formann, Hagerup, Haralambides, Kaufmann, Leighton, Simvonis, Welzl, Woeginger (RF5) 1993; 22
Di Battista, Eades, Tamassia, Tollis (RF4) 1994; 4
Gansner, Koutsofios, North, Vo (RF6) 1993; 19
G. Kant, Algorithms for Drawing Planar Graphs, Ph.D. thesis, Dept. Comput. Sci. Univ. Utrecht, Utrecht, Netherlands, 1993.
Formann (10.1006/jagm.2000.1115_RF5) 1993; 22
10.1006/jagm.2000.1115_RF9
Chan (10.1006/jagm.2000.1115_RF1) 1997; 1190
Garg (10.1006/jagm.2000.1115_RF7) 1994; 855
Chrobak (10.1006/jagm.2000.1115_RF2) 1995; 54
de Fraysseix (10.1006/jagm.2000.1115_RF3) 1990; 10
Di Battista (10.1006/jagm.2000.1115_RF4) 1994; 4
Hopcroft (10.1006/jagm.2000.1115_RF8) 1974; 21
10.1006/jagm.2000.1115_RF10
Tamassia (10.1006/jagm.2000.1115_RF13) 1987; 16
Tamassia (10.1006/jagm.2000.1115_RF14) 1997
Malitz (10.1006/jagm.2000.1115_RF11) 1994; 7
Schnyder (10.1006/jagm.2000.1115_RF12) 1990
Gansner (10.1006/jagm.2000.1115_RF6) 1993; 19
References_xml – reference: G. Kant, Algorithms for Drawing Planar Graphs, Ph.D. thesis, Dept. Comput. Sci. Univ. Utrecht, Utrecht, Netherlands, 1993.
– volume: 7
  start-page: 172
  year: 1994
  end-page: 183
  ident: RF11
  article-title: On the angular resolution of planar graphs
  publication-title: SIAM J. Discrete Math.
– volume: 855
  year: 1994
  ident: RF7
  article-title: Planar drawings and angular resolution: Algorithms and bounds
  publication-title: Proc. 2nd Annu. European Sympos. Algorithms
– volume: 1190
  start-page: 63
  year: 1997
  end-page: 75
  ident: RF1
  article-title: Optimizing area and aspect ratio in straight-line orthogonal tree drawings
  publication-title: Graph Drawing (Proc. GD '96)
– year: 1990
  ident: RF12
  article-title: Embedding planar graphs on the grid
  publication-title: Proc. 1st ACM-SIAM Sympos. Discrete Algorithms
– reference: G. Kant, Drawing planar graphs using the canonical ordering, Algorithmica161996, 4–32. 1996. [Special issue on Graph Drawing, edited by G. Di Battista and R. Tamassia]
– volume: 19
  start-page: 214
  year: 1993
  end-page: 230
  ident: RF6
  article-title: A technique for drawing directed graphs
  publication-title: IEEE Trans. Software Eng.
– volume: 22
  start-page: 1035
  year: 1993
  end-page: 1052
  ident: RF5
  article-title: Drawing graphs in the plane with high resolution
  publication-title: SIAM J. Comput.
– volume: 16
  start-page: 421
  year: 1987
  end-page: 444
  ident: RF13
  article-title: On embedding a graph in the grid with the minimum number of bends
  publication-title: SIAM J. Comput.
– volume: 4
  start-page: 235
  year: 1994
  end-page: 282
  ident: RF4
  article-title: Algorithms for drawing graphs: an annotated bibliography
  publication-title: Comput. Geom. Theory Appl.
– volume: 21
  start-page: 549
  year: 1974
  end-page: 568
  ident: RF8
  article-title: Efficient planarity testing
  publication-title: J. Assoc. Comput. Mach.
– volume: 54
  start-page: 241
  year: 1995
  end-page: 246
  ident: RF2
  article-title: A linear-time algorithm for drawing planar graphs
  publication-title: Inform. Process. Lett.
– volume: 10
  start-page: 41
  year: 1990
  end-page: 51
  ident: RF3
  article-title: How to draw a planar graph on a grid
  publication-title: Combinatorica
– start-page: 815
  year: 1997
  end-page: 832
  ident: RF14
  article-title: Graph drawing
  publication-title: Handbook of Discrete and Computational Geometry
– volume: 22
  start-page: 1035
  year: 1993
  ident: 10.1006/jagm.2000.1115_RF5
  article-title: Drawing graphs in the plane with high resolution
  publication-title: SIAM J. Comput.
  doi: 10.1137/0222063
– volume: 21
  start-page: 549
  year: 1974
  ident: 10.1006/jagm.2000.1115_RF8
  article-title: Efficient planarity testing
  publication-title: J. Assoc. Comput. Mach.
  doi: 10.1145/321850.321852
– volume: 855
  year: 1994
  ident: 10.1006/jagm.2000.1115_RF7
  article-title: Planar drawings and angular resolution: Algorithms and bounds
– year: 1990
  ident: 10.1006/jagm.2000.1115_RF12
  article-title: Embedding planar graphs on the grid
– ident: 10.1006/jagm.2000.1115_RF9
– volume: 54
  start-page: 241
  year: 1995
  ident: 10.1006/jagm.2000.1115_RF2
  article-title: A linear-time algorithm for drawing planar graphs
  publication-title: Inform. Process. Lett.
  doi: 10.1016/0020-0190(95)00020-D
– volume: 19
  start-page: 214
  year: 1993
  ident: 10.1006/jagm.2000.1115_RF6
  article-title: A technique for drawing directed graphs
  publication-title: IEEE Trans. Software Eng.
  doi: 10.1109/32.221135
– volume: 7
  start-page: 172
  year: 1994
  ident: 10.1006/jagm.2000.1115_RF11
  article-title: On the angular resolution of planar graphs
  publication-title: SIAM J. Discrete Math.
  doi: 10.1137/S0895480193242931
– volume: 16
  start-page: 421
  year: 1987
  ident: 10.1006/jagm.2000.1115_RF13
  article-title: On embedding a graph in the grid with the minimum number of bends
  publication-title: SIAM J. Comput.
  doi: 10.1137/0216030
– start-page: 815
  year: 1997
  ident: 10.1006/jagm.2000.1115_RF14
  article-title: Graph drawing
– volume: 4
  start-page: 235
  year: 1994
  ident: 10.1006/jagm.2000.1115_RF4
  article-title: Algorithms for drawing graphs: an annotated bibliography
  publication-title: Comput. Geom. Theory Appl.
  doi: 10.1016/0925-7721(94)00014-X
– volume: 1190
  start-page: 63
  year: 1997
  ident: 10.1006/jagm.2000.1115_RF1
  article-title: Optimizing area and aspect ratio in straight-line orthogonal tree drawings
– ident: 10.1006/jagm.2000.1115_RF10
  doi: 10.1007/BF02086606
– volume: 10
  start-page: 41
  year: 1990
  ident: 10.1006/jagm.2000.1115_RF3
  article-title: How to draw a planar graph on a grid
  publication-title: Combinatorica
  doi: 10.1007/BF02122694
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SubjectTerms Combinatorics
Combinatorics. Ordered structures
Exact sciences and technology
Graph theory
Mathematics
Sciences and techniques of general use
Title A Framework for Drawing Planar Graphs with Curves and Polylines
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