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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
Patrizio Angelini - PhD Dissertation
� Geographical and Transportation Maps
� Biological Networks
� Knowledge Representation
� …
Graphs
Patrizio Angelini - PhD Dissertation
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
Patrizio Angelini - PhD Dissertation
� 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
� …
Patrizio Angelini - PhD Dissertation
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
Patrizio Angelini - PhD Dissertation
Planarity
Planarity can be studied in terms of topology
Embedding of a graph: rotation scheme of each vertex
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Topology/Geometry
Some of the asthetic criteria depend on the topological features of the graph (embedding),
Patrizio Angelini - PhD Dissertation
topological features of the graph (embedding), while other criteria also depend on
its geometrical realization (drawing).
Connectivity
� Simply connected graphs
cut-vertex
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Connectivity
� Biconnected graphs
Separation pair
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Connectivity
� Triconnected graphs
Separating triplet
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Connectivity - Embedding
� A triconnected graph admits only 2 planar embeddings, which differ for a flip of the whole graph
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SPQR-Trees
Data structure describing the embeddings of a biconnected graph
� Di Battista and Tamassia, 1996
S P Q R
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skeleton
Planarity Testing
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� 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
Patrizio Angelini - PhD Dissertation
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?
Patrizio Angelini - PhD Dissertation
embedding Γ(G) of G?
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5 6
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G Γ(H)
Our Results
Partially Embedded Planaritycan be tested in linear time
� We test in linear time whether, among all the possible
Patrizio Angelini - PhD Dissertation
� 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
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Vertex-Cycle containment
2
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5 6
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G Γ(H)
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Key Properties
Lemma 1: ONCAS Property
Obviously Necessary Conditionson scheme rotations and vertex-cycle containments
are Also Sufficient
Patrizio Angelini - PhD Dissertation
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
Patrizio Angelini - PhD Dissertation
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
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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
Patrizio Angelini - PhD Dissertation
� 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)
Patrizio Angelini - PhD Dissertation
� 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)
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Clustering
What happens to planarity if the graph has a cluster structure?
Patrizio Angelini - PhD Dissertation
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
Patrizio Angelini - PhD Dissertation
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
Patrizio Angelini - PhD Dissertation
c-Planarity Testing
� INSTANCE: A clustered graph C =(G,T)
� QUESTION: Does C admit a c-planar drawing?
� Unknown complexity
� Many variants and results
Patrizio Angelini - PhD Dissertation
� 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
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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
Patrizio Angelini - PhD Dissertation
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.
Patrizio Angelini - PhD Dissertation
Our Results
c-planarity can be studied in terms of topology
Patrizio Angelini - PhD Dissertation
Analogous of Fary’s Theorem for planarity
Outline (1)
Clustered Outerclustered
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Outline (2)
OuterclusteredThree linearly-ordered outerclustered
Patrizio Angelini - PhD Dissertation
Outline (3)
Triangular-convex-separated drawing
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Three convex-separateddrawings
Outline (4)
Triangular-convex-separated drawing
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Outline (5)
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Outline (6)
Patrizio Angelini - PhD Dissertation
Graph Drawing in Networking
Are Graph Drawing techniques useful in
Patrizio Angelini - PhD Dissertation
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
Patrizio Angelini - PhD Dissertation
� 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
Patrizio Angelini - PhD Dissertation
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]
Patrizio Angelini - PhD Dissertation
� [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
Patrizio Angelini - PhD Dissertation
Our Results
There exists an algorithm to computea greedy drawing of any given triconnected planar
graph in polynomial time
Patrizio Angelini - PhD Dissertation
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.
Patrizio Angelini - PhD Dissertation
• Every binary cactus admits a greedy drawing.
Greedy Drawings
Are greedy drawings really useful in practice?
Patrizio Angelini - PhD Dissertation
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
Patrizio Angelini - PhD Dissertation
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
Patrizio Angelini - PhD Dissertation
� 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
Patrizio Angelini - PhD Dissertation
� 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.
Patrizio Angelini - PhD Dissertation
� 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.
Patrizio Angelini - PhD Dissertation
� 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.
Conference Publications
� 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.
Patrizio Angelini - PhD Dissertation
� 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.
Conference Publications
� 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
Patrizio Angelini - PhD Dissertation
� 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.
Patrizio Angelini - PhD Dissertation
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
Patrizio Angelini - PhD Dissertation
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!
Patrizio Angelini - PhD Dissertation