What is spanning tree and minimum spanning tree , methods to create minimum spanning tree, prim's algorithm , kruskal's algorithm.

1. GRAPH
APPLICATION - MST
o Prim’s
o Kruskal’s
1

2. WHAT IS SPANNING TREE
 A spanning tree of a graph is a tree that has all the
vertices of the graph connected by some edge
 A graph can have one or more number of spanning
trees.
 If the graph has N vertices then the spanning tree
will have n-1 edges
2

3. SPANNING TREE EXAMPLE
3

4. SPANNING TREE - CREATION
 Either DFS or BFS can be used to create a spanning
tree
 When DFS is used, the resulting spanning tree is
known as a depth first spanning tree
 When BFS is used, the resulting spanning tree is
known as a breadth first spanning tree
 While adding a non-tree edge into any spanning tree,
this will create a cycle
4

5. MINIMUM SPANNING TREE
 A minimum spanning tree is a spanning tree that has
the minimum weight than all other spanning trees of a
graph
 Three different algorithms can be used
 Prim’s algorithm
 Kruskal’s algorithm 5

6. PRIM’S ALGORITHM - INTRODUCTION
 Prim's algorithm is a greedy algorithm.
 It finds a minimum spanning tree for a weighted
undirected graph.
 This means it finds a subset of the edges that forms a
tree that includes every vertex, where the total weight of
all the edges in the tree is minimized.
6

7. PRIM’S STEPS
1.Select any vertex
2.Select the shortest edge connected to that vertex
3.Select the shortest edge connected to any vertex
already connected
4.Repeat step 3 until all vertices have been
connected
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8. PRIM’S ALGORITHM
8
Initialization:
 Pick a vertex r to be the root
 Set D(r) = 0, parent(r) = null
 For all vertices v  V, v  r, set D(v) = 
 Insert all vertices into priority queue P, using
distances as the keys

9. PRIM’S ALGORITHM (CONT..)
9
a
c
e
d
b
2
45
9
6
4
5
5
e a b c d
0    
Vertex Parent
e -
Vertex
Distance

10. PRIM’S ALGORITHM (CONT..)
While P is not empty:
 Select the next vertex u to add to the tree
u = P.deleteMin()
 Update the weight of each vertex w adjacent to u
which is not in the tree (i.e., w  P)
If weight(u,w) < D(w),
a. parent(w) = u
b. D(w) = weight(u,w)
c. Update the priority queue to
reflect new distance for w
10

11. PRIM’S ALGORITHM (CONT..)
11
d b c a
4 5 5 
Vertex Parent
e -
b e
c e
d e
Vertex Parent
e -
b -
c -
d -
d b c a
   
e
0

12. PRIM’S ALGORITHM (CONT..)
12
a c b
2 4 5
Vertex Parent
e -
b e
c d
d e
a d
d b c a
4 5 5 
Vertex Parent
e -
b e
c e
d e

13. PRIM’S ALGORITHM (CONT..)
13
c b
4 5
Vertex Parent
e -
b e
c d
d e
a d
a c b
2 4 5
Vertex Parent
e -
b e
c d
d e
a d

14. PRIM’S ALGORITHM (CONT..)
14
b
5
Vertex Parent
e -
b e
c d
d e
a d
c b
4 5
Vertex Parent
e -
b e
c d
d e
a d

15. PRIM’S ALGORITHM (CONT..)
15
Vertex Parent
e -
b e
c d
d e
a d
b
5
Vertex Parent
e -
b e
c d
d e
a d

16. MST- USING PRIM’S ALGORITHM
16
a
c
e
d
b
2
45
9
6
4
5
5

17. RUNNING TIME OF PRIM’S ALGORITHM
 Initialization of priority queue (array): O(|V|)
 Update loop:
• Choosing vertex with minimum cost edge: O(|V|)
• Updating distance values of unconnected vertices:
each edge is considered only once during entire
execution, for a total of O(|E|) updates
 Overall cost :
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O(|E| + |V| 2)

18. KRUSKAL’S ALGORITHM - INTRODUCTION
 Create a forest of trees from the vertices
 Repeatedly merge trees by adding “safe edges”
until only one tree remains
 A “safe edge” is an edge of minimum weight which
does not create a cycle
18

19. 19
a
c
e
d
b
2
45
9
6
4
5
5
forest: {a}, {b}, {c}, {d}, {e}
DEFINING FOREST

20. 1. Select the shortest edge in a network
2. Select the next shortest edge which does not
create a cycle
3. Repeat step 2 until all vertices have been
connected
20
KRUSKAL’S STEPS

21. 21
Initialization
 a. Create a set for each vertex v  V
 b. Initialize the set of “safe edges” A comprising
the MST to the empty set
 c. Sort edges by increasing weight
KRUSKAL’S ALGORITHM

22. 22
a
c
e
d
b
2
45
9
6
4
5
5
F = {a}, {b}, {c}, {d}, {e}
A = 
E = {(a,d), (c,d), (d,e), (a,c),
(b,e), (c,e), (b,d), (a,b)}
KRUSKAL’S ALGORITHM (CONT.)

23. 23
 For each edge (u,v)  E in increasing order
while more than one set remains:
 If u and v, belong to different sets U and V
a. add edge (u,v) to the safe edge set
A = A  {(u,v)}
b. merge the sets U and V
F = F - U - V + (U  V)
 Return A
KRUSKAL’S ALGORITHM (CONT.)

24. 24
E = {(a,d), (c,d), (d,e), (a,c),
(b,e), (c,e), (b,d), (a,b)}
Forest
{a}, {b}, {c}, {d}, {e}
{a,d}, {b}, {c}, {e}
{a,d,c}, {b}, {e}
{a,d,c,e}, {b}
{a,d,c,e,b}
A

{(a,d)}
{(a,d), (c,d)}
{(a,d), (c,d), (d,e)}
{(a,d), (c,d), (d,e), (b,e)}
KRUSKAL’S ALGORITHM (CONT.)

25. 25
 Running time bounded by sorting (or findMin)
O(|E|log|E|) , or equivalently, O(|E|log|V|)
RUNNING TIME OF KRUSKAL’S ALGORITHM

26. 26

27. APPLICATION OF MST
 Any time you want to visit all vertices in a graph at
minimum cost (e.g., wire routing on printed circuit boards,
sewer pipe layout, road planning…)
 Internet content distribution
 Idea: publisher produces web pages, content
distribution network replicates web pages to many
locations so consumers can access at higher speed
 MST may not be good enough!
content distribution on minimum cost tree may take a
long time!
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