On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the...

On the Existence and Optimality of some

Scuola Dottorale di IngegneriaSezione di Informatica e Automazione

PATRIZIO ANGELINIXXII C ICLO

ADVISOR: PROF. G. DI BATTISTA

On the Existence and Optimality of some

Planar Graph Embeddings

Graphs

� A graph is a mathematical representation of a set of objectsand of a relationship among them� A vertex represents an object

� An edge between two vertices represents the fact that the two objects are related

Patrizio Angelini - PhD Dissertation

Graphs

� Areas of application:

� Computer Networks

� Social Networks

� Interpersonal Relationships

� Geographical and


� Geographical and Transportation Maps

� Biological Networks

� Knowledge Representation

� …

Graphs


Graph Drawing

� Graph Drawing is the reserach field dealing with the visualization of graphs

� Cross between Graph Theory, Graph Algorithmics, andComputational Geometry

� The graphical representation should be nice and readable


� The graphical representation should be nice and readable

Graph Drawing

� Some aesthetic criteria have to be defined� Planarity

� Small Area

� Few number of bends

� Convex faces

� Straight-line edges

� …


Planarity

� If a graph admits a planar drawing, then it admits a planar drawing in which the edges are straight-linesegments

� Fary’s Theorem, 1948


Planarity

Planarity can be studied in terms of topology

Embedding of a graph: rotation scheme of each vertex

1


3

1 2

5

6

42

56

3

3

5

6

14

2 5

6

3

Topology/Geometry

Some of the asthetic criteria depend on the topological features of the graph (embedding),


topological features of the graph (embedding), while other criteria also depend on

its geometrical realization (drawing).

Connectivity

� Simply connected graphs

cut-vertex


Connectivity

� Biconnected graphs

Separation pair


Connectivity

� Triconnected graphs

Separating triplet


Connectivity - Embedding

� A triconnected graph admits only 2 planar embeddings, which differ for a flip of the whole graph


SPQR-Trees

Data structure describing the embeddings of a biconnected graph

� Di Battista and Tamassia, 1996

S P Q R


skeleton

Planarity Testing


� Planarity can be tested in linear time

� Hopcroft and Tarjan, 1974

� Boyer and Myrvold, 1999

� Boyer, Cortese, Patrignani, and Di Battista, 2003

� de Fraisseix, de Mendez, and Rosenstiehl, 2006

Extending a Planar Embedding

What happens to planarity if the input graph is


What happens to planarity if the input graph ispartially embedded?

Planarity of Partially Embedded Graphs

� INPUT: A planar graph G and a planar embeddingΓ(H) of a subgraph H of G

� OUTPUT: Can Γ(H) be extended to a planar embedding Γ(G) of G?


embedding Γ(G) of G?

2

143

5 6

7

2

1

4

3

6

7

G Γ(H)

Our Results

Partially Embedded Planaritycan be tested in linear time

� We test in linear time whether, among all the possible


� We test in linear time whether, among all the possibleembeddings of G, there is one that extends Γ(H)

� Instead of trying to directly extend Γ(H)

� P. Angelini, G. Di Battista, F. Frati, V. Jelinek, J. Kratochvil, M. Patrignani, I. Rutter. Testing Planarity of PartiallyEmbedded Graphs. In Symposium On Discrete Algorithms(SODA '10), ACM-SIAM, pages 202-221, 2010.

Topology Descriptors

� If H is disconnected, the rotation schemes are notsufficient to describe the topology of the graph

� Rotation scheme

� Vertex-Cycle containment


Vertex-Cycle containment

2

143

5 6

7

2

1

4

3

6

7

G Γ(H)

5

Key Properties

Lemma 1: ONCAS Property

Obviously Necessary Conditionson scheme rotations and vertex-cycle containments

are Also Sufficient


are Also Sufficient

Lemma 2: Locality Property

A biconnected graph admits an embedding extensionif and only if all its skeletons admit an embedding

extension

Optimal Embeddings

How efficiently can be computed an embedding that is


How efficiently can be computed an embedding that isoptimal with respect to a certain measure?

Objective

Find a planar embedding which minimizes the maximum distance of the internal faces from the external face

Depth


Depth

Two faces are adjacent if they share an edge

� Motivation

� Quality of the drawing

� Pizzonia and Tamassia-2000, Pizzonia-2005

� Asymptotically optimal area

� Dolev, Leighton, and Trickey, 1984

Depth

Depth = 2 Depth = 3


� D. Bienstock and C. Monma

� On the Complexity of Embedding Planar Graphs To Minimize Certain Distance Measures. Algorithmica, 1990

State of the Art

BiconnectedGraph

O(n5 * log n)

Connected GraphO(n5 * log n)

Biconnected graph with a given edge on the external faceO(n4 * log n)


� P.Angelini, G. Di Battista, M, Patrignani

� Finding a Minimum-Depth Embedding of a Planar Graph in O(n^4) Time. Algorithmica, 2010

Our Results

BiconnectedGraph

O(n5 * log n)

Connected GraphO(n5 * log n)

Biconnected graph with a given edge on the external faceO(n4 * log n)

O(n4) O(n4)

O(n3)


Clustering

What happens to planarity if the graph has a cluster structure?


cluster structure?

What is the relationship among planarity, topology, and geometry in this setting?

Clustered Graphs

Underlying GraphUnderlying GraphUnderlying GraphUnderlying Graph

Clustered GraphClustered GraphClustered GraphClustered Graph

Inclusion TreeInclusion TreeInclusion TreeInclusion Tree

Underlying GraphUnderlying GraphUnderlying GraphUnderlying Graph


Clustered Drawing

� A drawing of a clustered graph (G,T) is c-planar if

� G is planar

� Each edge crosses the boundary of each cluster at most once

� No two cluster boundaries intersect


c-Planarity Testing

� INSTANCE: A clustered graph C =(G,T)

� QUESTION: Does C admit a c-planar drawing?

� Unknown complexity

� Many variants and results


� Many variants and results

� High connectivity induced by each cluster

� Feng et al. ’95, Dahlhaus ’98, Cornelsen and Wagner ’06, Cortese et al. ’08, Jelinek et al. ’08

� Flat hierarchy

� Cortese et al. ’05, Di Battista and Frati ’07

� Simple classes of graphs

� Cortese et al. ’05 , Jelinkova et al. ’07

Topology/Geometry in c-Planarity


Convex Clustered Drawing

� Each c-planar clustered graph admits a drawingwhere each cluster is drawn as a convex polygon

� Straight-line drawing algorithms for hierarchical graphsand clustered graphs. Eades, Feng, Lin, Nagamochi, 1996


Our Results

� Each c-planar clustered graph admits a drawing where eachcluster is drawn as a rectangle� P. Angelini, F. Frati, M. Kaufmann. Straight-line Rectangular Drawingsof Clustered Graphs. In 11th Algorithms and Data Structures Symposium (WADS '09), Springer, volume 5664 of LNCS, pages 25-36, 2009.


Our Results

c-planarity can be studied in terms of topology


Analogous of Fary’s Theorem for planarity

Outline (1)

Clustered Outerclustered


Outline (2)

OuterclusteredThree linearly-ordered outerclustered


Outline (3)

Triangular-convex-separated drawing


Three convex-separateddrawings

Outline (4)

Triangular-convex-separated drawing


Outline (5)


Outline (6)


Graph Drawing in Networking

Are Graph Drawing techniques useful in


Are Graph Drawing techniques useful in networking “just” for visualization?

Greedy Routing

� A sensor knows its own location, the location of its neighbors, and the location of the destination

� A sensor sends packets to any neighbor that is closer than itself to the destination, in terms of Euclidean distance

� If there is no neighbor that is closer to the destination the


� If there is no neighbor that is closer to the destination the routing fails

s t

?

Greedy Drawing - Motivation

� A graph represents the network

� Greedy routing is performed using virtual coordinates instead of real coordinatescoordinates instead of real coordinates� Coordinates of the vertices representing the sensors� A drawing is greedy if the coordinates of the vertices are

such that greedy routing always succeeds

� Rao, Papadimitriou, Shenker, Stoica. Geographic routing

without location information. MOBICOM’03


State of the Art

� Graphs not admitting any Greedy drawing

� Complete bipartite graph K1,6

� [Papadimitriou, Ratacjzak-05]

� Complete binary tree with 31 vertices

� [Leigthon,Moitra-08]


� [Leigthon,Moitra-08]

� Graphs always admitting a Greedy drawing

� Paths

� Hamiltonian graphs

� Complete graphs

� Delaunay triangulations

� Triangulations [Dhandapani-08]

� Existential proof

The Conjecture

� Conjecture: Any 3-connected planar graph admits a Greedy Drawing

� Papadimitriou and Ratajczak. On a Conjecture Related to Geometric Routing. TCS, 2005 Geometric Routing. TCS, 2005

� If a graph G admits a greedy drawing, then any graph containing G as a spanning subgraph admits a greedy drawing


Our Results

There exists an algorithm to computea greedy drawing of any given triconnected planar

graph in polynomial time


Independently proved by Leigthon and Moitra at FOCS’08 with very similar techniques

� P. Angelini, F. Frati, L. Grilli. An Algorithm to Construct GreedyDrawings of Triangulations. Journal of Graph Algorithms and Applications, 14(1):19-51, 2010. Special Issue on Selected Papers fromGD '08.

Our Results

• Every triconnected planar graph contains a spanningbinary cactus.


• Every binary cactus admits a greedy drawing.

Greedy Drawings

Are greedy drawings really useful in practice?


Are greedy drawings really useful in practice?

How many bits are needed to represent the location of the vertices?

Graph Drawing Perspective

� The problem can be stated as an area-minimization problem.

� If the drawing has exponential area, then a polynomialnumber of bits are needed to represent the Cartesiancoordinates of the vertices


coordinates of the vertices

�We would like a logarithmic number of bits

� The constructions known so far of [Angelini et al.] and [Moitra et al.] produce exponential-areadrawings

Our Results

� There exist greedy-drawable graphs that require exponential area in any greedy drawing in the Euclidean plane


� There exist greedy-drawable graphs such that the Cartesian coordinates of the vertices require a polynomial number of bits in any greedy drawing

Other Research Activities

� Topological Morphing of Planar Graphs

� Simultaneous Embedding of a tree and a path


� Generalization of c-planarity problem� Clusters can be split in order to get c-planarity

� Right-Angle Crossing drawings

� Acyclic vertex-coloring of graphs

Journal Publications

� P. Angelini, Giuseppe Di Battista, Maurizio Patrignani. Finding a Minimum-Depth Embedding of a Planar Graph in O(n4) Time. Algorithmica, 2010.


� P. Angelini, F. Frati, L. Grilli. An Algorithm to Construct GreedyDrawings of Triangulations. Journal of Graph Algorithms and Applications, 14(1):19-51, 2010. Special Issue on Selected Papers from GD '08.

Conference Publications

� P. Angelini, F. Frati. Acyclically 3-Colorable Planar Graphs. In Workshop on Algorithms and Computation (WALCOM '10), volume 5942 of LNCS, pages113-124, 2010.

� P. Angelini, G. Di Battista, F. Frati, V. Jelinek, J. Kratochvil, M. Patrignani, I.


� P. Angelini, G. Di Battista, F. Frati, V. Jelinek, J. Kratochvil, M. Patrignani, I. Rutter. Testing Planarity of Partially Embedded Graphs. In Symposium On Discrete Algorithms (SODA '10), ACM-SIAM, pages 202-221, 2010.

� P. Angelini, G. Di Battista, F. Frati. Succinct Greedy Drawings Do NotAlways Exist. In 17th International Symposium on Graph Drawing (GD '09), LNCS, 2009. To appear.


� P. Angelini, L. Cittadini, G. Di Battista, W. Didimo, F. Frati, M. Kaufmann, A. Symvonis. On the Perspectives Opened by Right Angle CrossingDrawings. In 17th International Symposium on Graph Drawing (GD '09), LNCS, 2009. To appear.


� P. Angelini, F. Frati, M. Patrignani. Splitting Clusters To Get C-Planarity. In 17th International Symposium on Graph Drawing (GD '09), LNCS, 2009. Toappear.

� P. Angelini, F. Frati, M. Kaufmann. Straight-line Rectangular Drawingsof Clustered Graphs. In 11th Algorithms and Data Structures Symposium (WADS '09), volume 5664 of LNCS, pages 25-36, 2009.


� P. Angelini, P. F. Cortese, G. Di Battista, M. Patrignani. TopologicalMorphing of Planar Graphs. In 16th International Symposium on GraphDrawing (GD '08), volume 5417 of LNCS, pages 145-156, 2008.

� P. Angelini, F. Frati, L. Grilli. An Algorithm to Construct Greedy


� P. Angelini, F. Frati, L. Grilli. An Algorithm to Construct GreedyDrawings of Triangulations. In 16th International Symposium on GraphDrawing (GD '08), volume 5417 of LNCS, pages 26-37, 2008.

� P. Angelini, G. Di Battista, M. Patrignani. Computing a Minimum-DepthPlanar Graph Embedding in O(n^4) Time. In 10th Workshop on Algorithms and Data Structures (WADS '07), volume 4619 of LNCS, pages 287-299, 2007.

Technical Reports

� P. Angelini, M. Geyer, M. Kaufmann, D. Neuwirth. On a Tree and a Path withno Geometric Simultaneous Embedding CoRR, arXiv:1001.0555v1, 2010.

� P. Angelini, G. Di Battista, F. Frati. Succinct Greedy Drawings May Be Unfeasible. Technical Report RT-DIA-148-2009, Dept. of Computer Science and Automation, University of Roma Tre, 2009.


and Automation, University of Roma Tre, 2009.

� P. Angelini, F. Frati. Acyclically 3-Colorable Planar Graphs. TechnicalReport RT-DIA-147-2009, Dept. of Computer Science and Automation, University of Roma Tre, 2009.

� P. Angelini, F. Frati, M. Kaufmann. Straight-Line Rectangular Drawings ofClustered Graphs. Technical Report RT-DIA-144-2009, Dept. of Computer Science and Automation, Roma Tre University, 2009.

Technical Reports

� P. Angelini, F. Frati, L. Grilli. An Algorithm to Construct GreedyDrawings of Triangulations. Technical Report RT-DIA-140-2009, Dept. ofComputer Science and Automation, Roma Tre University, 2009.

� P. Angelini, P. F. Cortese, G. Di Battista, M. Patrignani. TopologicalMorphing of Planar Graphs. Technical Report RT-DIA-134-2008, Dept. of


Morphing of Planar Graphs. Technical Report RT-DIA-134-2008, Dept. ofComputer Science and Automation, Roma Tre University, 2008.

� P. Angelini, G. Di Battista, M. Patrignani. Computing a Minimum-DepthPlanar Graph Embedding in O(n^4) Time. Technical Report RT-DIA-116-2007, Dept. of Computer Science and Automation, University of Roma Tre, 2007.

Thanks for your attention!


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