Spanning G is a subgraph that has all the vertices of G. Trees - - PowerPoint PPT Presentation

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SpanningSubgraphs MathematicsforComputerScience MIT6.042J/18.062J A spanning subgraph of graph Spanning G is a subgraph that has all the vertices of G. Trees Albert R Meyer, April 8, 2013 Albert R Meyer, April 8, 2013 spanning.1

SLIDE 1

Albert R Meyer, April 8, 2013

Spanning Trees

spanning.1

Mathematics for Computer Science MIT 6.042J/18.062J

Albert R Meyer, April 8, 2013

Spanning Subgraphs

A spanning subgraph of graph G is a subgraph that has all the vertices of G.

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Albert R Meyer, April 8, 2013

Spanning Subgraphs

A spanning subgraph of graph G is a subgraph that has all the vertices of G. A spanning tree is a spanning subgraph that is a tree.

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Albert R Meyer, April 8, 2013

Spanning Trees

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SLIDE 2

Albert R Meyer, April 8, 2013

Spanning Trees

a spanning tree

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Albert R Meyer, April 8, 2013

Spanning Trees

another spanning tree (can have many)

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Albert R Meyer, April 8, 2013

Spanning Trees

Lemma: G connected implies

G has a spanning tree Pf: Namely, any minimum edge connected spanning graph.

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Albert R Meyer, April 8, 2013

Minimum Weight Spanning Trees

Suppose edges have weights:

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Find min weight spanning tree?

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SLIDE 3

Albert R Meyer, April 8, 2013

Build MST using gray edges

Start with vertices
Color components black & white
gray edge ::=
add min weight gray edge

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Albert R Meyer, April 8, 2013

Minimum Spanning Trees

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color components

4 4 1 3 6 9 2 1 7

Albert R Meyer, April 8, 2013

Minimum Spanning Trees

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color components

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Albert R Meyer, April 8, 2013

Minimum Spanning Trees

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gray edges

4 4 1 3 6 9 2 1 7

SLIDE 4

Albert R Meyer, April 8, 2013

Minimum Spanning Trees

min-tree.14

gray edges: min weight

4 4 1 3 6 9 2 1 7

Albert R Meyer, April 8, 2013

4 4 1 3 6 9 2 1 7

Minimum Spanning Trees

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re-color components

Albert R Meyer, April 8, 2013

Minimum Spanning Trees

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re-color components

4 4 1 3 6 9 2 1 7

Albert R Meyer, April 8, 2013

Minimum Spanning Trees

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gray edges

4 4 1 3 6 9 2 1 7

SLIDE 5

Albert R Meyer, April 8, 2013

Minimum Spanning Trees

min-tree.18

gray edges: min weight

4 4 1 3 6 9 2 1 7

Albert R Meyer, April 8, 2013

Minimum Spanning Trees

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4 4 1 3 6 9 2 1 7

re-color components

Albert R Meyer, April 8, 2013

Minimum Spanning Trees

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4 4 1 3 6 9 2 1 7

re-color components

Albert R Meyer, April 8, 2013

Minimum Spanning Trees

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etc

4 4 1 3 6 9 2 1 7

SLIDE 6

Ways to grow an MST

start at any vertex, keep

building one tree. (Prim)

keep choosing min weight edge

between diff components (Kruskal)

grow trees in parallel (Meyer)

Albert R Meyer, April 8, 2013

min-tree.22

SLIDE 7

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