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1. Kruskal’s Algorithm for Computing MSTs
• Section 9.2

2. The Minimum Spanning Tree
Give a graph G = (V, E), the minimum spanning tree (MST) is
a weighted graph G’ = (V, E’) such that:
•E’  E
•G’ is connected
•G’ has the minimum cost

3. B
A
C
E
D
Example
12
6
13
20
Which is the
MST?
20
B
A
C
E
D
12
6
13
20 B
A
C
E
D
12
6
13
20
Why MST is
a tree?

4. Naïve Algorithm for MST
(Using Exhaustive Search)
MST  Empty-Graph
cost  
For each subgraph G’ = (V, E’) of G = (V, E) do {
1. Check that G’ is connected
2. Compute cost c of G’
3. If c < cost then {
3.1 MST  G’
3.2 cost  c }}
What is the complexity “in words” of the algorithm?
“the number of subgraphs”  “cost of computing Steps 1-3”
How many subgraphs are there?
2|E|
This is bad!

5. The Prim Algorithm
1. Select a starting node, v
2. T  {v} //the nodes in the MST
3. E  {} //the edges in the MST
4. While not all nodes in G are in the T do
3.1 Choose the edge v’ in G − T such that there is a v in T:
weight(v,v’) is the minimum in
{weight(u,w) : w in G − T and u in T}
3.2 T  T  {v’}
3.3 E  E  {(v,v’)}
5. return E
// input: a graph G
// output: E: a MST for G
Complexity: - O((|E|+|V|)log2|E|)
- In class we show O(|E||V|)

6. T
G-T
If there is an edge e between the 2 groups with minimum cost, then
there is an MST containing e
Why does it works?
Property: Suppose that we divide the nodes of a graph
G = (V, E) in two groups V, V’:
5
7
10
22
66
12
6
13
2
20

7. Visited
Non visited
Example of a non greedy algorithm?
Example of a “Greedy” Algorithm
Dijkstra-Prim is an example of a greedy algorithm since it
looks at a subset of the larger problem
5
7
10
22
66
12
6
13
2
20
Our naïve algorithm for MST
Unfortunately,
not all problems
can be solved
with greedy
algorithms

8. Kruskal Algorithm
B
A
C
E
D
12
6
13
12
20
3
F
2
6
•|V| = 6
•|E| = 8
•A tree with |V| nodes has edges
|V|-1

9. The Kruskal Algorithm
1.
2. E  {} //the edges in the MST
3. i  1 //counter for EG
4. While do
if adding EG[i] to E does not add a cycle then
E  E  {EG[i]}
i  i + 1
5. return E
// input: a graph G with n nodes and m edges
// output: E: a MST for G
EG[1..m]  Sort the m edges in G in increasing weight order
|E| < n - 1
Complexity: O(|E|log2 |E|)
Is this algorithm Greedy?
Yes

10. Why Does the Kruskal Algorithm Works?
Property: If a connected graph G has n nodes, then any
acyclic subgraph G’ of G with n-1 edges is also connected
and includes all n nodes (G’ is a tree)
Corollary: Kruskal’s algorithm returns an MST of G

11. Homework
• 010:
9.1: 3, 6.b, 9 (just explain)
9.2: 1.a, 2.a, 5
• 011:
9.1: 2, 6.a, 9.a
9.2: 1.b, 2.d, 4