2. What is A Spanning Tree?
• A spanning tree for an
undirected graph G=(V,E)
is a subgraph of G that is
a tree and contains all the
vertices of G
• Can a graph have more
than one spanning tree?
• Can an unconnected graph
have a spanning tree?
a
b
u
e
c
v
f
d
3. Minimal Spanning Tree.
• The weight of a subgraph
is the sum of the weights
of it edges.
a
4
9
3
b
• A minimum spanning tree
for a weighted graph is a
spanning tree with
minimum weight.
4
u
14
2
10
c
v
3
Mst T: w( T )=
15
f
8
d
• Can a graph have more
then one minimum
spanning tree?
e
(u,v)
T
w(u,v ) is minimized
4. Example of a Problem that
Translates into a MST
The Problem
• Several pins of an electronic circuit must be
connected using the least amount of wire.
Modeling the Problem
• The graph is a complete, undirected graph
G = ( V, E ,W ), where V is the set of pins, E
is the set of all possible interconnections
between the pairs of pins and w(e) is the
length of the wire needed to connect the
pair of vertices.
• Find a minimum spanning tree.
5. Greedy Choice
We will show two ways to build a minimum
spanning tree.
• A MST can be grown from the current
spanning tree by adding the nearest vertex
and the edge connecting the nearest
vertex to the MST. (Prim's algorithm)
• A MST can be grown from a forest of
spanning trees by adding the smallest edge
connecting two spanning trees. (Kruskal's
algorithm)
6. Notation
• Tree-vertices: in the tree constructed so far
• Non-tree vertices: rest of vertices
Prim’s Selection rule
• Select the minimum weight edge between a treenode and a non-tree node and add to the tree
7. The Prim algorithm Main Idea
Select a vertex to be a tree-node
while (there are non-tree vertices) {
if there is no edge connecting a tree node
with a non-tree node
return “no spanning tree”
select an edge of minimum weight between a
tree node and a non-tree node
add the selected edge and its new vertex to
the tree
}
return tree
8. 5
A
4
6
2
C
2
E
3
D
1
3
B
2
4
Prim's Algorithm
F
9. A
B
D
C
E
Prim's Algorithm
F
10. A
C
2
B
D
E
F
Prim's Algorithm
11. 5
A
4
2
6
2
C
B
D
1
3
E
2
4
Prim's Algorithm
F
12. A
B
2
2
C
1
3
3
D
E
2
F
Prim's Algorithm
13. A
B
2
2
C
1
3
3
D
E
2
F
Prim's Algorithm
14. A
B
2
2
C
D
1
3
E
2
F
Prim's Algorithm
15. A
B
2
2
C
D
1
3
E
2
F
Prim's Algorithm
16. minimum- spanning tree
A
B
2
2
C
D
1
3
E
2
F
Prim's Algorithm
17. Kruskal„s Algorithm
1. Each vertex is in its own cluster
2. Take the edge e with the smallest weight
- if e connects two vertices in different clusters,
then e is added to the MST and the two clusters,
which are connected by e, are merged into a
single cluster
- if e connects two vertices, which are already
in the same cluster, ignore it
3.
Continue until n-1 edges were selected
Kruskal's Algorithm
18. 5
A
4
6
2
C
2
E
3
D
1
3
B
2
4
Kruskal's Algorithm
F
19. 5
A
4
6
2
C
2
E
3
D
1
3
B
2
4
Kruskal's Algorithm
F
20. 5
A
4
6
2
C
2
E
3
D
1
3
B
2
4
Kruskal's Algorithm
F
21. 5
A
4
6
2
C
2
E
3
D
1
3
B
2
4
Kruskal's Algorithm
F
22. 5
A
4
6
2
C
2
E
3
D
1
3
B
2
4
Kruskal's Algorithm
F
23. 5
A
4
6
2
C
2
E
3
D
1
3
B
2
4
Kruskal's Algorithm
F
cycle!!
24. 5
A
4
6
2
C
2
E
3
D
1
3
B
2
4
Kruskal's Algorithm
F
25. 5
A
4
6
2
C
2
E
3
D
1
3
B
2
4
Kruskal's Algorithm
F
26. minimum- spanning tree
A
B
2
2
C
D
1
3
E
2
F
Kruskal's Algorithm
27. Graph Traversal
Traversing a graph means visiting all the
vertices in the graph exactly once.
Breadth First Search (BFS)
Depth First Search (DFS)
28. DFS
Similar to in-order traversal of a binary
search tree
Starting from a given node, this traversal
visits all the nodes up to the deepest
level and so on.
31. DFS Traversal
Visit the vertex v
Visit all the vertices along the path which
begins at v
Visit the vertex v, then the vertex
immediate adjacent to v, let it be vx . If vx
has an immediate adjacent vy then visit it
and so on till there is a dead end.
Dead end: A vertex which does not have an
immediate adjacent or its immediate
adjacent has been visited.
32. After coming to an dead end we
backtrack to v to see if it has an
another adjacent vertex other than vx
and then continue the same from it else
from the adjacent of the adjacent
(which is not visited earlier) and so on.
33. Push the starting vertex into the STACK
While STACK not empty do
POP a vertex V
If V is not visited
Visit the vertex V
Store V in VISIT
PUSH all adjacent vertex of V
onto STACK
End of IF
End of While
STOP
34. A
Adjacency List
C
D
J
B
E
F
G
K
A: F,C,B
B: G,C
C: F
D: C
E: D,C,J
F: D
G: C,E
J: D,K
K: E,G
35. DFS of G starting at J
[1] Initially push J onto STACK
STACK : J
VISIT: Ø
[2] POP J from the STACK, add it in
VISIT and PUSH onto the STACK all
neighbor of J
STACK: D, K
VISIT: J
36. [3] POP the top element K, add it in
VISIT and PUSH all neighbor of K onto
STACK
STACK: D,E,G
VISIT: J, K
[4] POP the top element G, add it in
VISIT and PUSH all neighbor of G onto
STACK
STACK: D,E, E, C,
VISIT: J, K, G
37. [5] POP the top element C, add it in
VISIT and PUSH all neighbor of C onto
STACK
STACK: D,E,E, F
VISIT: J, K, G, C
[6] POP the top element F, add it in
VISIT and PUSH all neighbor of F onto
STACK
STACK: D,E, E, D
VISIT: J, K, G, C, F
38. [5] POP the top element D, add it in
VISIT and PUSH all neighbor of D onto
STACK
STACK: D,E,E, C
VISIT: J, K, G, C, F,D
[6] POP the top element C, which is
already in VISIT
STACK: D,E, E
VISIT: J, K, G, C, F,D
39. [5] POP the top element E, add it in
VISIT which is already in VISIT and
its neighbor onto STACK
STACK: D,E, D, C, J
VISIT: J, K, G, C, F,D,E
[6] POP the top element J, C, D,E, D
which is already in VISIT
STACK:
VISIT: J, K, G, C, F, D, E
40. A
C
D
J
B
E
F
J, K, G, C, F, D, E
Adjacency List
G
K
A: F,C,B
B: G,C
C: F
D: C
E: D,C,J
F: D
G: C,E
J: D,K
K: E,G
41. BFS Traversal
Any vertex in label i will be visited only
after the visiting of all the vertices in
its preceding level that is at level i – 1
42. BFS Traversal
[1] Enter the starting vertex v in a queue
Q
[2] While Q is not empty do
Delete an item from Q, say u
If u is not in VISIT store u in
VISIT
Enter all adjacent vertices of u
into Q
[3] Stop
43. v1
v2
v8
v3
v4
v6
v5
v7
44. [1] Insert the starting vertex V1 in Q
Q = V1
VISIT = Ø
[2] Delete an item from Q, let it be u = V1
u is not in VISIT. Store u in VISIT
and its adjacent element in Q
Q = V2 , V 3
VISIT = V1
45. [3] Delete an item from Q, let it be u = V2
u is not in VISIT. Store u in VISIT
and its adjacent element in Q
Q = V2 , V3 , V4 , V5
VISIT = V1 , V2
[4] Delete an item from Q, let it be u = V3
u is not in VISIT. Store u in VISIT
and its adjacent element in Q
Q = V3 , V4 , V5 , V4 , V6
VISIT = V1 , V2 , V3
46. [5] Delete an item from Q, let it be u = V4
u is not in VISIT. Store u in VISIT
and its adjacent element in Q
Q = V4 , V5 , V4 , V6 , V8
VISIT = V1 , V2 , V3 , V4
[6] Delete an item from Q, let it be u =V5
u is not in VISIT. Store u in VISIT
and its adjacent element in Q
Q = V5 , V4 , V6 , V8 , V7
VISIT = V1 , V2 , V3 , V4 , V5
47. [7] Delete an item from Q, let it be u =V4
u is in VISIT.
Q = V4 , V6 , V8 , V7
VISIT = V1 , V2 , V3 , V4 , V5
[8] Delete an item from Q, let it be u =V6
u is not in VISIT. Store u in VISIT
and its adjacent element in Q
Q = V6 , V8 , V7
VISIT = V1 , V2 , V3 , V4 , V5 , V6
48. [9] Delete an item from Q, let it be u =V8
u is not in VISIT. Store u in VISIT and
its adjacent element in Q
Q = V8 , V7 , V1
VISIT = V1 , V2 , V3 , V4 , V5 , V6 ,
V8
[10] Delete an item from Q, let it be u =V7
u is not in VISIT. Store u in VISIT and
its adjacent element in Q
Q = V7 , V1
VISIT = V1 , V2 , V3 , V4 , V5 , V6 ,
V8 , V7
49. [11] Delete an item from Q, let it be u =V1
u is in VISIT.
Q = V1
VISIT = V1 , V2 , V3 , V4 , V5 , V6 ,
V8 , V 7
[12] Q is empty, Stop
Q=
VISIT = V1 , V2 , V3 , V4 , V5 , V6 ,
V8 , V 7
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