Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Confluent Drawing Algorithms using Rectangular Dualization Gianluca Quercini1 1 Institute
2 Dipartimento
Massimo Ancona2
for Advanced Computer Studies University of Maryland
di Informatica e Scienze dell’Informazione University of Genoa
18th Symposium of Graph Drawing, 2010
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Problem Definition The Approach
Confluent Drawing Dickerson, Eppstein, Goodrich, Meng, Confluent Drawings: Visualizing Non-planar Diagrams in a Planar Way, GD 2004
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Problem Definition The Approach
Motivation Definition Clutter is the state in which excess items, or their representation or organization, lead to a degradation of performance at some task a . a
Rosenholtz, Ruth, Li, Mansfield, Jin. Feature Congestion: a Measure of Display Clutter, CHI 2005, pp. 761–770, 2005
In Graphs: Too many edges (possibly) intersecting. Solution: Confluent Drawing.
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Problem Definition The Approach
Rectangular Dual
Rectangular dual RD(G) = (Γ, f ) of a plane graph G = (V , E) with n nodes: Γ: set of n non-overlapping rectangles no four of which meet at the same point. f : V → Γ such that (u, v ) ∈ E ↔ f (u) and f (v ) share a common boundary. G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Problem Definition The Approach
Overview Dickerson, Eppstein, Goodrich, Meng, Confluent Drawings: Visualizing Nonplanar Diagrams in a Planar Way, GD 2004 clique gate
clique crossover node
biclique crossover node
clique gate G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Problem Definition The Approach
Overview Dickerson, Eppstein, Goodrich, Meng, Confluent Drawings: Visualizing Nonplanar Diagrams in a Planar Way, GD 2004 clique gate
clique crossover node
biclique crossover node
clique gate G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Problem Definition The Approach
Overview Dickerson, Eppstein, Goodrich, Meng, Confluent Drawings: Visualizing Nonplanar Diagrams in a Planar Way, GD 2004 clique gate
clique crossover node
biclique crossover node
clique gate G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Problem Definition The Approach
Overview Dickerson, Eppstein, Goodrich, Meng, Confluent Drawings: Visualizing Nonplanar Diagrams in a Planar Way, GD 2004 clique gate
clique crossover node
biclique crossover node
clique gate G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Problem Definition The Approach
Overview Dickerson, Eppstein, Goodrich, Meng, Confluent Drawings: Visualizing Nonplanar Diagrams in a Planar Way, GD 2004 biclique gate
clique crossover node
biclique crossover node
clique gate G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
Admissibility Conditions
A plane graph G admits a rectangular dual with four rectangles on the boundary if and only if 1 : There are exactly four nodes on the outer cycle of G. Every inner face of G is a 3−cycle. G has no separating triangles. A graph complies with these conditions is called Proper Triangular Planar (PTP).
1
Kant, He. Two Algorithms for Finding Rectangular Duals of Planar Graphs, LNCS vol. 790, pp. 396–410, 1994 G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
Enforcing the Admissibility Conditions Plane connected graph Biconnectivity augmentation New outer face Separating triangles search (Chiba et Nishizeki, 1985) Separating triangles breaking (Quercini, 2009) Triangulation (Biedl, Kant, Kaufmann, 1994) PTP GRAPH G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
Enforcing the Admissibility Conditions Plane connected graph Biconnectivity augmentation New outer face Separating triangles search (Chiba et Nishizeki, 1985) Separating triangles breaking (Quercini, 2009) Triangulation (Biedl, Kant, Kaufmann, 1994) PTP GRAPH G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
Enforcing the Admissibility Conditions Plane connected graph Biconnectivity augmentation New outer face Separating triangles search (Chiba et Nishizeki, 1985) Separating triangles breaking (Quercini, 2009) Triangulation (Biedl, Kant, Kaufmann, 1994) PTP GRAPH G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
Enforcing the Admissibility Conditions Plane connected graph Biconnectivity augmentation New outer face Separating triangles search (Chiba et Nishizeki, 1985) Separating triangles breaking (Quercini, 2009) Triangulation (Biedl, Kant, Kaufmann, 1994) PTP GRAPH G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
Enforcing the Admissibility Conditions Plane connected graph Biconnectivity augmentation New outer face Separating triangles search (Chiba et Nishizeki, 1985) Separating triangles breaking (Quercini, 2009) Triangulation (Biedl, Kant, Kaufmann, 1994) PTP GRAPH G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
Enforcing the Admissibility Conditions Plane connected graph Biconnectivity augmentation New outer face Separating triangles search (Chiba et Nishizeki, 1985) Separating triangles breaking (Quercini, 2009) Triangulation (Biedl, Kant, Kaufmann, 1994) PTP GRAPH G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
Enforcing the Admissibility Conditions Plane connected graph Biconnectivity augmentation New outer face Separating triangles search (Chiba et Nishizeki, 1985) Separating triangles breaking (Quercini, 2009) Triangulation (Biedl, Kant, Kaufmann, 1994) PTP GRAPH G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
Enforcing the Admissibility Conditions Plane connected graph Biconnectivity augmentation New outer face Separating triangles search (Chiba et Nishizeki, 1985) Separating triangles breaking (Quercini, 2009) Triangulation (Biedl, Kant, Kaufmann, 1994) PTP GRAPH G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
From a PTP Graph to a Rectangular Dual Kant, He, Two Algorithms for Finding Rectangular Duals of Planar Graphs, LNCS Vol. 790, pp. 396-410, 1994
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
From a PTP Graph to a Rectangular Dual Kant, He, Two Algorithms for Finding Rectangular Duals of Planar Graphs, LNCS Vol. 790, pp. 396-410, 1994
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Algorithms
From a PTP Graph to a Rectangular Dual Kant, He, Two Algorithms for Finding Rectangular Duals of Planar Graphs, LNCS Vol. 790, pp. 396-410, 1994
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
Methodology Two approaches: baseline (not efficient) and improved (efficient). The drawings are created on top of a rectangular dual. The resulting drawings are orthogonal-like with high angular resolution (≥ π/2). Strict guidelines to avoid edge intersections: 1 2
3
Nodes are drawn inside the corresponding rectangles. Edge (u, v ) must not cross the boundary of any rectangle, but f (u) and f (v ). Edges may run along the boundaries of any rectangle.
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
N W
E S
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
Discussion
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
Discussion
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
Discussion
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
How it Works
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
Discussion
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
Methodology
We compare the number of segments used to create the confluent drawing against the number of segments that would have been necessary to create the same drawing in a non-confluent fashion. 100 graphs from the AT&T data set (10-100 nodes). Kandinsky - Baseline - Improved.
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
Results 150
Kandinsky Baseline Improved
100
50
Bends
Segments G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Preliminaries Baseline Approach Improved Approach Experimental Results
Comparison
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Advantages Using rectangular dualization to create confluent drawings has several advantages: The layout of nodes and edges is immediate. The angular resolution of the resulting drawing is high (≥ π/2). Low number of segments and bends per edge 2 . The algorithms to draw plain graphs can be used to draw clustered graphs as well.
2
The number of bends can be minimized using a flow-based method such as in (Garg, Tamassia, 1997) G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Disadvantages
Using rectangular dualization to create confluent drawings has also drawbacks: The quality of resulting drawing is sometimes questionable and depends on the rectangular dual. The area of the resulting drawing is not optimal, especially when the rectangular dual contains gates. A c-rectangular dual sometimes can contain too many gates.
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Future Work
Minimization of the bends through a flow-based method like the one described in (Garg, Tamassia 1997). Assessment of the quality of the drawings with different rectangular duals. Thorough experiments with large graphs (thousand nodes and edges). Improvement of the algorithm for c-rectangular dualization.
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Thanks for your attention!
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Rectangular Dual for Clustered Graphs R-Confluent Drawing
Definition (C-Rectangular dual)
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Rectangular Dual for Clustered Graphs R-Confluent Drawing
Definition (C-Rectangular dual)
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Rectangular Dual for Clustered Graphs R-Confluent Drawing
Creation of the C-rectangular Dual
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Rectangular Dual for Clustered Graphs R-Confluent Drawing
Creation of the C-rectangular Dual
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Rectangular Dual for Clustered Graphs R-Confluent Drawing
Creation of the C-rectangular Dual
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Rectangular Dual for Clustered Graphs R-Confluent Drawing
Creation of the C-rectangular Dual
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Rectangular Dual for Clustered Graphs R-Confluent Drawing
Creation of the C-rectangular Dual
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Rectangular Dual for Clustered Graphs R-Confluent Drawing
Creation of the C-rectangular Dual
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Rectangular Dual for Clustered Graphs R-Confluent Drawing
From C-rectangular Dual to R-confluent Drawing
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
Introduction Rectangular Dualization Confluent Drawing Algorithms Conclusions and Future Work Confluent Drawing for Clustered Graphs
Rectangular Dual for Clustered Graphs R-Confluent Drawing
From C-rectangular Dual to R-confluent Drawing
G. Quercini, M. Ancona
Confluent Drawing Algorithms using Rectangular Dualization
