1. Chapter 7: Greedy Algorithms
Kruskal’s, Prim’s, and Dijkstra’s Algorithms

2. 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

3. 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 I J

4. Kruskal’s kruskal(e, n) { sort(e); 4 5 A B C 3 6 4 5 6 7 3 4 D E F G 7H I J

5. Kruskal’s kruskal(e, n) { sort(e); A B C D E F G H I J 4 5 7 3 6 A E 5

6. kruskal(e, n) { sort(e); for (i = A to J) makeset(i) A B C D E F G H I4 5 7 3 6 A B C D E F G H I J
kruskal(e, n) {
sort(e);
for (i = A to J)
makeset(i) A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

7. kruskal(e, n) { ... count = 0; i = 1 A B C D E F G H I J 4 5 7 3 6 A BA E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

8. 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 B C D E F G H I J n 10
count i 1 A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

9. 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 B C DH E F G I J n 10
Count 1 i 2 A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

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 B C DH EF G I J n 10
Count 2 i 3 A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

11. 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++;
ADH B C EF G I J n 10
Count 3 i 4 A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

12. 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++;
ADH B C
EFG I J n 10
Count 4 i 5 A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

13. 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++;
ADHB C
EFG I J n 10
Count 5 i 6 A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

14. 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++;
ADHB
CEFG I J n 10
Count 6 i 7 A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

15. 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++;
ADHB
CEFGJ I n 10
Count 7 i 8 A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

16. 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++;
ADHBCEFGJ I n 10
Count 8 i 9 A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

17. 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++;
ADHBCEFGJ I n 10
Count 8 i 10 A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

18. 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++;
ADHBCEFGJI n 10
Count 9 i 11 A E 5 F J 5 F I 6 B E 6 C G 6 F G 4 E F 3 A B 4 D H 3 I J 5 C F 4 G J 4 E H 7 A D 3 B C 5 H I 5 D E 7

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

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

21. 4 5 A B C A 3 6 4 5 6 7 3 4 D E F G D E B 7 6 4 5 3 5 5 H I J H F G C J I

22. Theorem 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

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

24. G C G’ Subset of Minimal Spanning Tree of G S Theorem 7.2.5 pp. 280 43 6 4 5 6 7 3 4 D E F G A 7 6 4 C 5 3 5 5 H I J D E
Theorem 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 4 A E 5 D H 3 D E 7 S

25. G C G’ Subset of Minimal Spanning Tree of G S Theorem 7.2.5 pp. 2804 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 7 6 4 C 5 3 5 5 H I J D E
Theorem 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 4 A E 5 D H 3 D E 7 S

26. 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.

27. 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.