[PPT] - Planar Embeddings of Graphs with Specified Edge Lengths Sergio PowerPoint Presentation

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Planar Embeddings of Graphs with Specified Edge Lengths

Sergio Cabello GIVE, University Utrecht Erik D. Demaine MIT G¨ unter Rote FU Berlin

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Problem and motivation

Given a graph G

Sergio Cabello, sergio@cs.uu.nl, 2/11

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Problem and motivation

Given a graph G

1 1 1 2 2 3 3 2 1 2 3 3 2 1 1 2 1 1

and specified edge lengths Question: Can we draw G with these edge lengths?

Sergio Cabello, sergio@cs.uu.nl, 2/11

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Problem and motivation

Applications:

Sensor networks
Structural analysis of molecules
Linear cartograms

Sergio Cabello, sergio@cs.uu.nl, 3/11

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Our results

Restriction to planar embeddings (so planar graphs):

Triangulated graphs

⇒ linear time decision



            3-connected ( → fixed topology) unit edge lengths, bounded face degree, and generically rigid              ⇒ NP-hard. Improves Eades and Wormald ’90    2-connected + unit length, or 3-connected    also in simplicity

Sergio Cabello, sergio@cs.uu.nl, 4/11

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Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed.

Sergio Cabello, sergio@cs.uu.nl, 5/11

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Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O(n) time.

Sergio Cabello, sergio@cs.uu.nl, 5/11

SLIDE 8

Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O(n) time.

1 2 3 4 5 6 7 8

Sergio Cabello, sergio@cs.uu.nl, 5/11

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Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O(n) time.

1 2 3 4 5 6 7 8

Sergio Cabello, sergio@cs.uu.nl, 5/11

SLIDE 10

Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O(n) time.

1 2 3 4 5 6 7 8

Sergio Cabello, sergio@cs.uu.nl, 5/11

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Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O(n) time.

1 2 3 4 5 6 7 8

Sergio Cabello, sergio@cs.uu.nl, 5/11

SLIDE 12

Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O(n) time.

1 2 3 4 5 6 7 8

Sergio Cabello, sergio@cs.uu.nl, 5/11

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Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O(n) time.

1 2 3 4 5 6 7 8

Sergio Cabello, sergio@cs.uu.nl, 5/11

SLIDE 14

Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O(n) time.

1 2 3 4 5 6 7 8

Sergio Cabello, sergio@cs.uu.nl, 5/11

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Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O(n) time.

1 2 3 4 5 6 7 8

Sergio Cabello, sergio@cs.uu.nl, 5/11

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Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O(n) time.

1 2 3 4 5 6 7 8

Longest edge in the outermost face ⇒ Two outer face candidates.

Sergio Cabello, sergio@cs.uu.nl, 5/11

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Triangulated graphs

Planar triangulation ⇒ Planar 3-connected Planar 3-connected ⇒ One topological embedding in S2 [Whitney] ⇒ In R2: fixing outer face, the topology is completely fixed. ⇒ Fixed outer face, incremental treatment of triangles in O(n) time.

1 2 3 4 5 6 7 8

Longest edge in the outermost face ⇒ Two outer face candidates.

L i n e a r t i m e .

Sergio Cabello, sergio@cs.uu.nl, 5/11

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NP-hardness

Th: It is NP-hard for planar 3-connected graphs. Reduction from planar 3-SAT . . . v1 v2 v3 v4 v5 vn

Sergio Cabello, sergio@cs.uu.nl, 6/11

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