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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