Chapter 7: Greedy Algorithms

Kruskal’s, Prim’s, and Dijkstra’s Algorithms

Kruskal’s 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

A B C

D E F G

H I J

4 5

7 3 4

5 5

3 6

7

4

6 43

5 6

5

Kruskal’skruskal(e, n) {

sort(e);

A B C

D E F G

H I J

4 5

7 3 4

5 5

3 6

7

4

6 43

5 6

5

Kruskal’skruskal(e, n) {

sort(e);

A B C

D E F G

H I J

4 5

7 3 4

5 5

3 6

7

4

6 43

5 6

5

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

kruskal(e, n) {

sort(e);

for (i = A to J)

makeset(i)

A B C

D E F G

H I J

4 5

7 3 4

5 5

3 6

7

4

6 43

5 6

5

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

A B C D E F G H I J

kruskal(e, n) {

...

count = 0;

i = 1

A B C

D E F G

H I J

4 5

7 3 4

5 5

3 6

7

4

6 43

5 6

5

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

A B C D E F G H I J

Count

0

i

1

kruskal(e, n) {

while (count < n-1) {

if (findset(e[i].v) != findset(e[i].w)) {

print(e[i].v + “ ”+ e[i].w);

count++;

union(e[i].v, e[i].w);

}

i++;

}

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

A B C D E F G H I J

count

0

i

1

n

10

kruskal(e, n) {

while (count < n-1) {

if (findset(e[i].v) != findset(e[i].w)) {

print(e[i].v + “ ”+ e[i].w);

count++;

union(e[i].v, e[i].w);

}

i++;

}

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

A B C DH E F G I J

Count

1

i

2

n

10

kruskal(e, n) {

while (count < n-1) {

if (findset(e[i].v) != findset(e[i].w)) {

print(e[i].v + “ ”+ e[i].w);

count++;

union(e[i].v, e[i].w);

}

i++;

}

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

A B C DH EF G I J

Count

2

i

3

n

10

kruskal(e, n) {

while (count < n-1) {

if (findset(e[i].v) != findset(e[i].w)) {

print(e[i].v + “ ”+ e[i].w);

count++;

union(e[i].v, e[i].w);

}

i++;

}

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

ADH B C EF G I J

Count

3

i

4

n

10

kruskal(e, n) {

while (count < n-1) {

if (findset(e[i].v) != findset(e[i].w)) {

print(e[i].v + “ ”+ e[i].w);

count++;

union(e[i].v, e[i].w);

}

i++;

}

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

ADH B C EFG I J

Count

4

i

5

n

10

kruskal(e, n) {

while (count < n-1) {

if (findset(e[i].v) != findset(e[i].w)) {

print(e[i].v + “ ”+ e[i].w);

count++;

union(e[i].v, e[i].w);

}

i++;

}

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

ADHB C EFG I J

Count

5

i

6

n

10

kruskal(e, n) {

while (count < n-1) {

if (findset(e[i].v) != findset(e[i].w)) {

print(e[i].v + “ ”+ e[i].w);

count++;

union(e[i].v, e[i].w);

}

i++;

}

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

ADHB CEFG I J

Count

6

i

7

n

10

kruskal(e, n) {

while (count < n-1) {

if (findset(e[i].v) != findset(e[i].w)) {

print(e[i].v + “ ”+ e[i].w);

count++;

union(e[i].v, e[i].w);

}

i++;

}

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

ADHB CEFGJ I

Count

7

i

8

n

10

kruskal(e, n) {

while (count < n-1) {

if (findset(e[i].v) != findset(e[i].w)) {

print(e[i].v + “ ”+ e[i].w);

count++;

union(e[i].v, e[i].w);

}

i++;

}

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

ADHBCEFGJ I

Count

8

i

9

n

10

kruskal(e, n) {

while (count < n-1) {

if (findset(e[i].v) != findset(e[i].w)) {

print(e[i].v + “ ”+ e[i].w);

count++;

union(e[i].v, e[i].w);

}

i++;

}

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

ADHBCEFGJ I

Count

8

i

10

n

10

kruskal(e, n) {

while (count < n-1) {

if (findset(e[i].v) != findset(e[i].w)) {

print(e[i].v + “ ”+ e[i].w);

count++;

union(e[i].v, e[i].w);

}

i++;

}

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

ADHBCEFGJI

Count

9

i

11

n

10

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

A B C

D E F G

H I J

4 5

7 3 4

5 5

3 6

7

4

6 43

5 6

5

A

D

3

D

H

3

E F3 A B4

C

F

4

F G4

G

J

4

A

E

5

H I5

I J5

F

J

5

B C5

B

E

6

C

G

6

F

I

6

D E7

E

H

7

A B C

D E F G

H I J

4 5

7 3 4

5 5

3 6

7

4

6 43

5 6

5

A B C

D E F G

H I J

4 5

7 3 4

5 5

3 6

7

4

6 43

5 6

5

A

B

C

D E

F

G

H

I

J

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

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

A B C

D E F G

H I J

4 5

7 3 4

5 5

3 6

7

4

6 43

5 6

5

G A

B

C

D E

F

G

H

I

J

Minimal Spanning Tree of G

Theorem 7.2.5 pp. 280• 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.

A B C

D E F G

H I J

4 5

7 3 4

5 5

3 6

7

4

6 43

5 6

5

G G’ Subset of Minimal

Spanning Tree of G

S

C A

D E

A B4

A

E

5 D E7D

H

3

Theorem 7.2.5 pp. 280• If we add a minimum

weight edge in S to G’, the resulting graph is also contained in a minimal spanning tree of G

A B C

D E F G

H I J

4 5

7 3 4

5 5

3 6

7

4

6 43

5 6

5

G G’ Subset of Minimal

Spanning Tree of G

S

C A

D E

A B4

A

E

5 D E7D

H

3

Theorem 7.2.6: Kruskal’s Algorithm finds minimum spanning tree

Proof by induction

• G’ is a sub-graph constructed by Kruskal’s Algorithm

• G’ is initially empty but each step of the Algorithm increases the size of G’

• Inductive Assumption: G’ is contained in the MST.

Theorem 7.2.6: Kruskal’s Algorithm finds minimum spanning tree

Proof by induction• Let (v,w) be the next edge selected by Kruskal’s

Algorithm• Kruskal’s 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.