Some Results on Greedy Embeddings in Metric Spaces
Ankur Moitra, Tom Leighton
October 26, 2008
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
1 / 82
Greedy Embedding: Example
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
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Greedy Embedding: Example
t
s
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
3 / 82
Greedy Embedding: Example
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
4 / 82
Greedy Embedding: Example
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
5 / 82
Greedy Embedding: Example
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
6 / 82
Greedy Embedding: Example
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
7 / 82
Greedy Embedding: Example
t
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
8 / 82
Greedy Embedding: Example
t
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
9 / 82
Greedy Embedding: Example
t
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
10 / 82
Greedy Embedding: Example
t
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
11 / 82
Greedy Embedding: Example
t
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
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Greedy Embedding: Example
t
s
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
13 / 82
Greedy Embedding: Example
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
14 / 82
Greedy Embedding: Example
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
15 / 82
Greedy Embedding: Example
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
16 / 82
Greedy Embedding: Example
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
17 / 82
Greedy Embedding: Example
t
s
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
18 / 82
Greedy Embedding: Example
t
s
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
19 / 82
Greedy Embedding: Example
t
s
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
20 / 82
Greedy Embedding: Example
t
s
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
21 / 82
Greedy Embedding: Example
t
s
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
22 / 82
Greedy Routing
Reminder: (X , d) is a metric space, f : V → X
Greedy Routing
Always forward packets to a neighbor that is strictly closer to the
destination
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
23 / 82
Greedy Routing
Reminder: (X , d) is a metric space, f : V → X
Greedy Routing
Always forward packets to a neighbor that is strictly closer to the
destination
(distances are measured using the distance function d applied to the
images of nodes in the metric space)
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
23 / 82
Greedy Embedding: Definition
Fact
For all (s, t) there exists a path connecting s to t in which distances to t
are decreasing ⇐⇒ greedy routing never fails
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
24 / 82
Greedy Embedding: Definition
Fact
For all (s, t) there exists a path connecting s to t in which distances to t
are decreasing ⇐⇒ greedy routing never fails
Definition
A graph G admits a greedy embedding into a metric space (X , d) if there
is a function f : V → X s.t. greedy routing never fails
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
24 / 82
Results on Greedy Embeddings
Lemma (Papadimitriou, Ratajczak, 2005)
K1,7 , K2,13 , ...Kr ,6r +1 admit no greedy embedding into Euclidean plane
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
25 / 82
Results on Greedy Embeddings
Lemma (Papadimitriou, Ratajczak, 2005)
K1,7 , K2,13 , ...Kr ,6r +1 admit no greedy embedding into Euclidean plane
Conjecture (Papadimitriou, Ratajczak, 2005)
All 3-connected planar graph admits a greedy embedding into the
Euclidean plane
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
25 / 82
Results on Greedy Embeddings
Lemma (Papadimitriou, Ratajczak, 2005)
K1,7 , K2,13 , ...Kr ,6r +1 admit no greedy embedding into Euclidean plane
Conjecture (Papadimitriou, Ratajczak, 2005)
All 3-connected planar graph admits a greedy embedding into the
Euclidean plane
Theorem (Kleinberg, 2007)
All connected graphs admit a greedy embedding into the Hyperbolic plane
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
25 / 82
Results on Greedy Embeddings
Lemma (Papadimitriou, Ratajczak, 2005)
K1,7 , K2,13 , ...Kr ,6r +1 admit no greedy embedding into Euclidean plane
Conjecture (Papadimitriou, Ratajczak, 2005)
All 3-connected planar graph admits a greedy embedding into the
Euclidean plane
Theorem (Kleinberg, 2007)
All connected graphs admit a greedy embedding into the Hyperbolic plane
Theorem (Dhandapani, 2008)
All 3-connected, triangulated planar graphs admit a greedy embedding
into the Euclidean plane
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
25 / 82
Our Results
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
26 / 82
Our Results
1
The Papadimitriou-Ratajczak Conjecture is
true!
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
26 / 82
Our Results
1
2
The Papadimitriou-Ratajczak Conjecture is
true!
A combinatorial condition which is sufficient to
guarantee no greedy embedding into Euclidean
plane exists
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
26 / 82
Proof Outline
Theorem
All 3-connected planar graph admits a greedy embedding into the
Euclidean plane
1
All 3-connected planar graphs contain a spanning Christmas Cactus
graph
2
All Christmas Cactus graphs admit a greedy embedding into the
Euclidean plane
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
27 / 82
Proof Outline
Theorem
All 3-connected planar graph admits a greedy embedding into the
Euclidean plane
1
All 3-connected planar graphs contain a spanning Christmas Cactus
graph
2
All Christmas Cactus graphs admit a greedy embedding into the
Euclidean plane
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
28 / 82
Cactus Graphs: Definition
Definition
A cactus graph G = (V , E ) is a connected graph for which each edge is
in at most one simple cycle
not cactus
Ankur Moitra ()
cactus, not Christams cactus
Geometric Graph Theory
Christmas cactus
October 26, 2008
29 / 82
Cactus Graphs: Definition
Definition
A cactus graph G = (V , E ) is a connected graph for which each edge is
in at most one simple cycle
not cactus
Ankur Moitra ()
cactus, not Christams cactus
Geometric Graph Theory
Christmas cactus
October 26, 2008
30 / 82
Christmas Cactus Graphs: Definition
Definition
A Christmas cactus graph G = (V , E ) is a cactus graph for which the
removal of any node v ∈ V disconnects G into at most 2 components.
not cactus
Ankur Moitra ()
cactus, not Christams cactus
Geometric Graph Theory
Christmas cactus
October 26, 2008
31 / 82
Christmas Cactus Graphs: Definition
Definition
A Christmas cactus graph G = (V , E ) is a cactus graph for which the
removal of any node v ∈ V disconnects G into at most 2 components.
not cactus
Ankur Moitra ()
cactus, not Christams cactus
Geometric Graph Theory
Christmas cactus
October 26, 2008
32 / 82
?? What is a Christmas cactus??
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
33 / 82
?? What is a Christmas cactus??
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
33 / 82
Spanning Subgraphs in 3-Connected Planar Graphs
Theorem (this paper)
All 3-connected planar graphs contain a spanning Christmas cactus graph
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
34 / 82
Spanning Subgraphs in 3-Connected Planar Graphs
Theorem (this paper)
All 3-connected planar graphs contain a spanning Christmas cactus graph
⇓
Theorem (Gao, Richter, 1994)
All 3-connected planar graphs contain a spanning 2-walk
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
34 / 82
Spanning Subgraphs in 3-Connected Planar Graphs
Theorem (this paper)
All 3-connected planar graphs contain a spanning Christmas cactus graph
⇓
Theorem (Gao, Richter, 1994)
All 3-connected planar graphs contain a spanning 2-walk
⇓
Theorem (Barnette, 1966)
All 3-connected planar graphs contain a spanning 3-tree
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
34 / 82
Proof Outline
Theorem
All 3-connected planar graph admits a greedy embedding into the
Euclidean plane
1
XAll 3-connected planar graphs contain a spanning Christmas Cactus
graph
2
All Christmas Cactus graphs admit a greedy embedding into the
Euclidean plane
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
35 / 82
Example I
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
36 / 82
Example I
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
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Example I
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
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Example I
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
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Example I
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
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Example I
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
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Example I
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
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Example I
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
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Example I
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Ankur Moitra ()
Geometric Graph Theory
R+e
October 26, 2008
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Example I
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Ankur Moitra ()
Geometric Graph Theory
R+e
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Example II
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
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Example II
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
50 / 82
Example II
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
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Example II
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
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Example II
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
53 / 82
Example II
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Ankur Moitra ()
Geometric Graph Theory
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Example II
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Ankur Moitra ()
Geometric Graph Theory
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Example II
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
56 / 82
Example II
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
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Example II
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
58 / 82
Example II
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Ankur Moitra ()
Geometric Graph Theory
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Example II
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Geometric Graph Theory
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Example II
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Geometric Graph Theory
October 26, 2008
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Example II
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Geometric Graph Theory
October 26, 2008
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Example II
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Geometric Graph Theory
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Example II
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Geometric Graph Theory
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Example II
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Geometric Graph Theory
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Example II
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Ankur Moitra ()
Geometric Graph Theory
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Example II
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Ankur Moitra ()
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Example II
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
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General Christmas Cactus Graphs
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
69 / 82
General Christmas Cactus Graphs
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Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
70 / 82
General Christmas Cactus Graphs
G
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
71 / 82
General Christmas Cactus Graphs
G
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
72 / 82
General Christmas Cactus Graphs
G
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
73 / 82
General Christmas Cactus Graphs
G
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
74 / 82
General Christmas Cactus Graphs
G
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
75 / 82
General Christmas Cactus Graphs
G
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
76 / 82
General Christmas Cactus Graphs
G
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
77 / 82
Proof Outline
Theorem
All 3-connected planar graph admits a greedy embedding into the
Euclidean plane
1
XAll 3-connected planar graphs contain a spanning Christmas Cactus
graph
2
XAll Christmas Cactus graphs admit a greedy embedding into the
Euclidean plane
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
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Open Questions
Our embedding requires exponential sized coordinates:
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
79 / 82
Open Questions
Our embedding requires exponential sized coordinates:
Conjecture
Any greedy embedding scheme for general Christmas cactus graphs
requires exponential sized coordinates
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
79 / 82
Open Questions
Our embedding requires exponential sized coordinates:
Conjecture
Any greedy embedding scheme for general Christmas cactus graphs
requires exponential sized coordinates
OR: Are there greedy embedding schemes for which coordinates are only
polynomial sized?
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
79 / 82
Partial Results
Theorem
There exist graphs that admit a greedy embedding into the Euclidean line,
but all greedy embeddings require exponential sized coordinates
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
80 / 82
Partial Results
Theorem
There exist graphs that admit a greedy embedding into the Euclidean line,
but all greedy embeddings require exponential sized coordinates
Theorem
There are metric spaces s.t. all connected graphs can be greedily
embedded, and average coordinate size is constant
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
80 / 82
Questions?
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
81 / 82
Thanks!
Ankur Moitra ()
Geometric Graph Theory
October 26, 2008
82 / 82