Embed Size (px)
Citation preview
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
57
8
4
3
1
6
9
2
7
Min EdgePick a root
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
= in heap
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
7
Min Edge = 2
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Min Edge = 3
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Min Edge = 4
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Min Edge = 3
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Min Edge = 4
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Min Edge = 6
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
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
7
Min Edge = 2
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Min Edge = 3
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Now have 2 disjoint components:
ABFG and CH
Min Edge = 3
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Min Edge = 4
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Two components now merged into one.
Min Edge = 4
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
Min Edge = 5
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7Rejected due to a cycle BFGB
Min Edge = 6
A
B
D
G
C
F
IE
H
2 3
4
57
8
4
3
1
6
9
2
7
