minimum spanning tree dr. bernard chen ph.d. university of central arkansas fall 2008
TRANSCRIPT
Minimum Spanning Tree
Dr. Bernard Chen Ph.D.University of Central Arkansas
Fall 2008
Minimum Spanning Tree
Example of MST
Problem: Laying Telephone Wire
Central office
Wiring: Naïve Approach
Central office
Expensive!
Wiring: Better Approach
Central office
Minimize the total length of wire connecting the customers
Growing an MST: general idea
GENERIC-MST(G,w)
1. A{}2. while A does not form a spanning tree3. do find an edge (u,v) that is safe
for A4. A A U {(u,v)}5. return A
Tricky part
How do you find a safe edge? This safe edge is part of the minimum
spanning tree
Algorithms for MST
Prim’s Grow a MST by adding a single edge at a
time
Kruskal’s Choose a smallest edge and add it to the
forest If an edge is formed a cycle, it is
rejected
Prim’s greedy algorithm Start from some (any) vertex.
Build up spanning tree T, one vertex at a time.
At each step, add to T the lowest-weight edge in G that does not create a cycle.
Stop when all vertices in G are touched
Prim’s MST algorithm
Example
A
B
D
G
C
F
IE
H
2 3
4
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Min EdgePick a root
A
B
D
G
C
F
IE
H
2 3
4
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= in heap
Min Edge = 1
A
B
D
G
C
F
IE
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2 3
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Min Edge = 2
A
B
D
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C
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IE
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2 3
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Min Edge = 2
A
B
D
G
C
F
IE
H
2 3
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Min Edge = 3
A
B
D
G
C
F
IE
H
2 3
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Min Edge = 4
A
B
D
G
C
F
IE
H
2 3
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Min Edge = 3
A
B
D
G
C
F
IE
H
2 3
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3
1
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2
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Min Edge = 4
A
B
D
G
C
F
IE
H
2 3
4
57
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3
1
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Min Edge = 6
A
B
D
G
C
F
IE
H
2 3
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57
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Example II
Kruskal’s Algorithm
Choose the smallest edge and add it to a forest
Keep connecting components until all vertices connected
If an edge would form a cycle, it is rejected.
Example
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Min Edge = 1
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Min Edge = 2
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
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Min Edge = 2
A
B
D
G
C
F
IE
H
2 3
4
57
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3
1
6
9
2
7
Min Edge = 3
A
B
D
G
C
F
IE
H
2 3
4
57
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3
1
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2
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Now have 2 disjoint components:
ABFG and CH
Min Edge = 3
A
B
D
G
C
F
IE
H
2 3
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Min Edge = 4
A
B
D
G
C
F
IE
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2 3
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Two components now merged into one.
Min Edge = 4
A
B
D
G
C
F
IE
H
2 3
4
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Min Edge = 5
A
B
D
G
C
F
IE
H
2 3
4
57
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2
7Rejected due to a cycle BFGB
Min Edge = 6
A
B
D
G
C
F
IE
H
2 3
4
57
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9
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