Independent and Dominating Sets

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Independent and Dominating Sets

mercredi 11 février 2015

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Why dominating or independent sets ?

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Independent Sets

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Independent Set

• Given a graph G=(V,E), an independent set is a subset of vertices, U,

such that no two nodes in U are neighbors in G.

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Independent Set

• Given a graph G=(V,E), an independent set is a subset of vertices, U,

such that no two nodes in U are neighbors in G.

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Maximal Independent Set (MIS)

• Given a graph G=(V,E), a MIS is a subset of

vertices, U, such that :

• Independence: no two nodes in U are neighbors in G

• Maximality: no vertex can be added to U without violating the independence

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Maximal Independent Set (MIS)

• Given a graph G=(V,E), a MIS is a subset of

vertices, U, such that :

• Independence: no two nodes in U are neighbors in G

• Maximality: no vertex can be added to U without violating the independence

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Maximal Independent Set (MIS)

• Given a graph G=(V,E), a MIS is a subset of

vertices, U, such that :

• Independence: no two nodes in U are neighbors in G

• Maximality: no vertex can be added to U without violating the independence

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Maximal Independent Sets ?

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Maximal Independent Sets ?

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Maximal Independent Sets ?

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Maximal Independent Sets ?

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Maximal Independent Sets ?

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MIS Implementation

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Centralized MIS

1 2 12

5 7

6

9

8 3 10

11 13

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Centralized MIS

1 2 12

5 7

6

9

8 3 10

11 13

Pick a node

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Centralized MIS

1 2 12

5 7

6

9

8 3 10

11 13

Pick a node

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Centralized MIS

1 2 12

5 7

6

9

8 3 10

11 13

Pick a node

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Centralized MIS

1 2 12

5 7

6

9

8 3 10

11 13

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Centralized MIS

1 12

5

8 3 10

11 13

Remove its

neighborhood from the network

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Centralized MIS

1 12

5

8 3 10

11 13

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Centralized MIS

1 12

5

8 3 10

11 13

Pick another node

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Centralized MIS

1 12

5

8 3 10

11 13

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Centralized MIS

1 12

5

8 3 10

11 13

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Centralized MIS

1 12

5

8 3 10

11 13

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Centralized MIS

1

5

3

11 Remove its

neighborhood from the network

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Centralized MIS

1

5

3

11

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Centralized MIS

1

5

3

11 Pick another node

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Centralized MIS

1

5

3

11

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Centralized MIS

1

5

3

11

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Centralized MIS

1

5

3

11

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1 2 12

5 7

6

9

8 3 10

11 13

Centralized MIS

1

5

3

11

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1 2

5 7

6

9

Distributed MIS

1

5

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1 2

5 7

6

9

Distributed MIS

1

5

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Distributed MIS

1 2

5 7

6 9 1

5

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Distributed MIS

• Each node waits until neighbors with

greater «id» decide

1 2

5 7

6 9 1

5

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Distributed MIS

• Each node waits until neighbors with

greater «id» decide

1 2

5 7

6 9 1

5 Complexity ?

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Distributed MIS

• Each node waits until neighbors with

greater «id» decide

1 2

5 7

6 9 1

5

(39)

Distributed MIS

• Each node waits until neighbors with

greater «id» decide

1 2

5 7

6 9 1

5 Other ideas ?

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Dominating Sets

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Dominating Set

• Given a graph G=(V,E), a dominating set (DS) is a subset of vertices,

U, such that each node in V is either in U or

neighbor of a node in U.

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Dominating Set

• Given a graph G=(V,E), a dominating set (DS) is a subset of vertices,

U, such that each node in V is either in U or

neighbor of a node in U.

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Dominating Set

• Given a graph G=(V,E), a dominating set (DS) is a subset of vertices,

U, such that each node in V is either in U or

neighbor of a node in U.

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Dominating Sets ?

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Dominating Sets ?

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DS Implementation

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DS on Trees

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DS on Trees

Mark leafs as L0

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DS on Trees

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DS on Trees

Mark first level L1

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DS on Trees

Mark first level L1

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DS on Trees

Mark first level L1

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DS on Trees

Mark first level L1

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DS on Trees

Mark first level L1

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DS on Trees

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DS on Trees

Mark second level L2

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DS on Trees

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DS on Trees

Choose nodes in L1 as DS members

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DS on Trees

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DS on Trees

Compute a MIS over the rest of the tree

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DS on Trees

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DS on Trees

Compute a MIS over the rest of the tree

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1 2

5 7

6

9

Connected DS

1

5

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1 2

5 7

6

9

Connected DS

1

5

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1 2

5 7

6

9

Connected DS

1

5

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1 2

5 7

6

9

Connected DS

1

5

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1 2

5 7

6

9

Connected DS

1

5 RULE ?

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Matching

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Matching

• Given a graph G=(V,E), a matching is a subset of edges, M, such that no two edges in M are

adjacent in G.

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Maximal Matching

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Reductions

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From Coloring to MIS

1 2

5 7

6 9 1

5

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From Coloring to MIS

• Start with a valid coloring

1 2

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6 9 1

5

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From Coloring to MIS

1 2

5 7

6 9 1

5

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From Coloring to MIS

• Start with a valid coloring

• Assume a total order on colors

1 2

5 7

6 9 1

5

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From Coloring to MIS

1 2

5 7

6 9 1

5

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From Coloring to MIS

1 2

5 7

6 9 1

5 • Choose in MIS all nodes of first color

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From Coloring to MIS

1 2

5 7

6 9 1

5

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From Coloring to MIS

• Choose in MIS all

nodes of second color if they have no

neighbors in MIS

1 2

5 7

6 9 1

5

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From Coloring to MIS

1 2

5 7

6 9 1

5

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From Coloring to MIS

• Repeat until no color left

1 2

5 7

6 9 1

5

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From Coloring to MIS

• Repeat until no color left

1 2

5 7

6 9 1

5

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From Coloring to MIS

1 2

5 7

6 9 1

5

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From Coloring to MIS

• MIS nodes in red

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6 9 1

5

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Theorem

Given a graph G and a coloring of G with f(G) colors in time T(G) it is possible to construct a MIS for G in time T(G)+f(G).

From Coloring to MIS

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Corollary

There exists a deterministic MIS

algorithms for trees and bounded degree graphs with time complexity O(log ^* (n)).

From Coloring to MIS

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Corollary

There exists a deterministic DS algorithms for trees with time complexity O(log ^* (n)).