on the existenceand optimalityofsome planar …compunet/www/docs/angelini-thesis-slides.pdfon the...
TRANSCRIPT
![Page 1: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/1.jpg)
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
![Page 2: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/2.jpg)
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
![Page 3: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/3.jpg)
Graphs
� Areas of application:
� Computer Networks
� Social Networks
� Interpersonal Relationships
� Geographical and
Patrizio Angelini - PhD Dissertation
� Geographical and Transportation Maps
� Biological Networks
� Knowledge Representation
� …
![Page 4: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/4.jpg)
Graphs
Patrizio Angelini - PhD Dissertation
![Page 5: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/5.jpg)
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
![Page 6: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/6.jpg)
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
![Page 7: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/7.jpg)
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
![Page 8: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/8.jpg)
Planarity
Planarity can be studied in terms of topology
Embedding of a graph: rotation scheme of each vertex
1
Patrizio Angelini - PhD Dissertation
3
1 2
5
6
42
56
3
3
5
6
14
2 5
6
3
![Page 9: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/9.jpg)
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).
![Page 10: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/10.jpg)
Connectivity
� Simply connected graphs
cut-vertex
Patrizio Angelini - PhD Dissertation
![Page 11: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/11.jpg)
Connectivity
� Biconnected graphs
Separation pair
Patrizio Angelini - PhD Dissertation
![Page 12: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/12.jpg)
Connectivity
� Triconnected graphs
Separating triplet
Patrizio Angelini - PhD Dissertation
![Page 13: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/13.jpg)
Connectivity - Embedding
� A triconnected graph admits only 2 planar embeddings, which differ for a flip of the whole graph
Patrizio Angelini - PhD Dissertation
![Page 14: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/14.jpg)
SPQR-Trees
Data structure describing the embeddings of a biconnected graph
� Di Battista and Tamassia, 1996
S P Q R
Patrizio Angelini - PhD Dissertation
skeleton
![Page 15: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/15.jpg)
Planarity Testing
Patrizio Angelini - PhD Dissertation
� 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
![Page 16: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/16.jpg)
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?
![Page 17: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/17.jpg)
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?
2
143
5 6
7
2
1
4
3
6
7
G Γ(H)
![Page 18: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/18.jpg)
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.
![Page 19: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/19.jpg)
Topology Descriptors
� If H is disconnected, the rotation schemes are notsufficient to describe the topology of the graph
� Rotation scheme
� Vertex-Cycle containment
Patrizio Angelini - PhD Dissertation
Vertex-Cycle containment
2
143
5 6
7
2
1
4
3
6
7
G Γ(H)
5
![Page 20: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/20.jpg)
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
![Page 21: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/21.jpg)
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?
![Page 22: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/22.jpg)
Objective
Find a planar embedding which minimizes the maximum distance of the internal faces from the external face
Depth
Patrizio Angelini - PhD Dissertation
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
![Page 23: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/23.jpg)
Depth
Depth = 2 Depth = 3
Patrizio Angelini - PhD Dissertation
![Page 24: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/24.jpg)
� 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
![Page 25: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/25.jpg)
� 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)
Patrizio Angelini - PhD Dissertation
![Page 26: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/26.jpg)
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?
![Page 27: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/27.jpg)
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
![Page 28: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/28.jpg)
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
![Page 29: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/29.jpg)
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
![Page 30: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/30.jpg)
Topology/Geometry in c-Planarity
Patrizio Angelini - PhD Dissertation
![Page 31: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/31.jpg)
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
![Page 32: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/32.jpg)
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
![Page 33: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/33.jpg)
Our Results
c-planarity can be studied in terms of topology
Patrizio Angelini - PhD Dissertation
Analogous of Fary’s Theorem for planarity
![Page 34: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/34.jpg)
Outline (1)
Clustered Outerclustered
Patrizio Angelini - PhD Dissertation
![Page 35: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/35.jpg)
Outline (2)
OuterclusteredThree linearly-ordered outerclustered
Patrizio Angelini - PhD Dissertation
![Page 36: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/36.jpg)
Outline (3)
Triangular-convex-separated drawing
Patrizio Angelini - PhD Dissertation
Three convex-separateddrawings
![Page 37: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/37.jpg)
Outline (4)
Triangular-convex-separated drawing
Patrizio Angelini - PhD Dissertation
![Page 38: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/38.jpg)
Outline (5)
Patrizio Angelini - PhD Dissertation
![Page 39: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/39.jpg)
Outline (6)
Patrizio Angelini - PhD Dissertation
![Page 40: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/40.jpg)
Graph Drawing in Networking
Are Graph Drawing techniques useful in
Patrizio Angelini - PhD Dissertation
Are Graph Drawing techniques useful in networking “just” for visualization?
![Page 41: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/41.jpg)
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
?
![Page 42: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/42.jpg)
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
![Page 43: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/43.jpg)
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
![Page 44: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/44.jpg)
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
![Page 45: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/45.jpg)
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.
![Page 46: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/46.jpg)
Our Results
• Every triconnected planar graph contains a spanningbinary cactus.
Patrizio Angelini - PhD Dissertation
• Every binary cactus admits a greedy drawing.
![Page 47: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/47.jpg)
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?
![Page 48: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/48.jpg)
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
![Page 49: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/49.jpg)
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
![Page 50: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/50.jpg)
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
![Page 51: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/51.jpg)
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.
![Page 52: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/52.jpg)
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.
![Page 53: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/53.jpg)
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.
![Page 54: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/54.jpg)
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.
![Page 55: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/55.jpg)
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.
![Page 56: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/56.jpg)
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.
![Page 57: On the Existenceand Optimalityofsome Planar …compunet/www/docs/angelini-thesis-slides.pdfOn the Existenceand Optimalityofsome Scuola Dottorale di Ingegneria Sezione di Informatica](https://reader033.vdocument.in/reader033/viewer/2022042117/5e956f5a2979e444592b8eaf/html5/thumbnails/57.jpg)
Thanks for your attention!
Patrizio Angelini - PhD Dissertation