Chapter 7: Greedy Algorithms

1
Chapter 7 Greedy Algorithms

• Kruskals, Prims, and Dijkstras Algorithms

2
Kruskals Algorithm

• Solves the Minimum Spanning Tree Problem
• Input
• List of edges in a graph
• n the number of vertices
• Output
• Prints the list of edges in the Minimum Spanning
  Tree

3
4
5
A
B
C
3
6
4
5
6
7
3
4
G
D
E
F
7
6
4
5
3
5
5
J
H
I
4
Kruskals
4
5
A
B
C
3
6
4
5
6

• kruskal(e, n)
• sort(e)

7
3
4
D
E
F
G
7
6
4
5
3
5
5
H
J
I
5
Kruskals

• kruskal(e, n)
• sort(e)

6
A
B
C
D
E
F
G
H
I
J

• kruskal(e, n)
• sort(e)
• for (i A to J)
• makeset(i)

7
A
B
C
D
E
F
G
H
I
J

• kruskal(e, n)
• ...
• count 0
• i 1

Count 0
i 1
8

• kruskal(e, n)
• while (count lt n-1)
• if (findset(ei.v) ! findset(ei.w))
• print(ei.v ei.w)
• count
• union(ei.v, ei.w)
• i

A
B
C
D
E
F
G
H
I
J
count 0
n 10
i 1
9

• kruskal(e, n)
• while (count lt n-1)
• if (findset(ei.v) ! findset(ei.w))
• print(ei.v ei.w)
• count
• union(ei.v, ei.w)
• i

A
B
C
DH
E
F
G
I
J
Count 1
n 10
i 2
10

• kruskal(e, n)
• while (count lt n-1)
• if (findset(ei.v) ! findset(ei.w))
• print(ei.v ei.w)
• count
• union(ei.v, ei.w)
• i

A
B
C
DH
EF
G
I
J
Count 2
n 10
i 3
11

• kruskal(e, n)
• while (count lt n-1)
• if (findset(ei.v) ! findset(ei.w))
• print(ei.v ei.w)
• count
• union(ei.v, ei.w)
• i

ADH
B
C
EF
G
I
J
Count 3
n 10
i 4
12

• kruskal(e, n)
• while (count lt n-1)
• if (findset(ei.v) ! findset(ei.w))
• print(ei.v ei.w)
• count
• union(ei.v, ei.w)
• i

ADH
B
C
EFG
I
J
Count 4
n 10
i 5
13

• kruskal(e, n)
• while (count lt n-1)
• if (findset(ei.v) ! findset(ei.w))
• print(ei.v ei.w)
• count
• union(ei.v, ei.w)
• i

ADHB
C
EFG
I
J
Count 5
n 10
i 6
14

• kruskal(e, n)
• while (count lt n-1)
• if (findset(ei.v) ! findset(ei.w))
• print(ei.v ei.w)
• count
• union(ei.v, ei.w)
• i

ADHB
CEFG
I
J
Count 6
n 10
i 7
15

• kruskal(e, n)
• while (count lt n-1)
• if (findset(ei.v) ! findset(ei.w))
• print(ei.v ei.w)
• count
• union(ei.v, ei.w)
• i

ADHB
CEFGJ
I
Count 7
n 10
i 8
16

• kruskal(e, n)
• while (count lt n-1)
• if (findset(ei.v) ! findset(ei.w))
• print(ei.v ei.w)
• count
• union(ei.v, ei.w)
• i

ADHBCEFGJ
I
Count 8
n 10
i 9
17

• kruskal(e, n)
• while (count lt n-1)
• if (findset(ei.v) ! findset(ei.w))
• print(ei.v ei.w)
• count
• union(ei.v, ei.w)
• i

ADHBCEFGJ
I
Count 8
n 10
i 10
18

• kruskal(e, n)
• while (count lt n-1)
• if (findset(ei.v) ! findset(ei.w))
• print(ei.v ei.w)
• count
• union(ei.v, ei.w)
• i

ADHBCEFGJI
Count 9
n 10
i 11
19
4
5
A
B
C

3
6
4
5
6
7
3
4
D
E
F
G
7
6
4
5
3
5
5
H
J
I
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4
5
A
B
C

3
6
4
5
6
7
3
4
D
E
F
G
7
6
4
5
3
5
5
H
J
I
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4
5
A
B
C

A
3
6
4
5
6
7
3
4
D
E
F
G
B
D
E
7
6
4
5
3
5
5
H
J
I
H
F
C
G
J
I
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Theorem 7.2.5 pp. 280

• Let G be a connected, weighted graph, and let G
  be a sub-graph of a minimal spanning tree of G.
  Let C be a component of G, and let S be the set
  of all Edges with one vertex in C and the other
  not in C. If we add a minimum weight edge in S to
  G, the resulting graph is also contained in a
  minimal spanning tree of G

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Theorem 7.2.5 pp. 280
4
5
A
B
C

G
Minimal Spanning Tree of G
3
6
4
5
6
A
7
3
4
D
E
F
G
7
6
4
5
B
D
E
3
5
5
H
J
I
H
F

• Let G be a connected, weighted graph, and let G
  be a sub-graph of a minimal spanning tree of G.
  Let C be a component of G, and let S be the set
  of all Edges with one vertex in C and the other
  not in C. If we add a minimum weight edge in S to
  G, the resulting graph is also contained in a
  minimal spanning tree of G

C
G
J
I
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Theorem 7.2.5 pp. 280
4
5
G Subset of Minimal Spanning Tree of G
A
B
C

G
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7
3
4
D
E
F
G
A
C
7
6
4
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3
5
5
H
J
I
D
E

• G be a sub-graph of a minimal spanning tree of
  G. Let C be a component of G, and let S be the
  set of all Edges with one vertex in C and the
  other not in C.

S
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Theorem 7.2.5 pp. 280
4
5
G Subset of Minimal Spanning Tree of G
A
B
C

G
3
6
4
5
6
7
3
4
D
E
F
G
A
C
7
6
4
5
3
5
5
H
J
I
D
E

• If we add a minimum weight edge in S to G, the
  resulting graph is also contained in a minimal
  spanning tree of G

S
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Theorem 7.2.6 Kruskals Algorithm finds minimum
spanning tree

• Proof by induction
• G is a sub-graph constructed by Kruskals
  Algorithm
• G is initially empty but each step of the
  Algorithm increases the size of G
• Inductive Assumption G is contained in the MST.

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Theorem 7.2.6 Kruskals Algorithm finds minimum
spanning tree

• Proof by induction
• Let (v,w) be the next edge selected by Kruskals
  Algorithm
• Kruskals algorithm finds the minimum weight edge
  (v,w) such that v and w are not already in G
• C can be any subset of the MST, so you can always
  construct a C such that v is in C and w is not.
• Therefore, by Theorem 7.2.5, when (v,w) is added
  to G, the resulting graph is also contained in
  the MST.