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What is Kruskal Algorithm?


1) Kruskal's algorithm is defines as it is an graph theory that finds a minimum spanning tree(MST) for a connected weighted graph.

2) If the graph is not connected, then it finds a minimum spanning forest

3) Kruskal algorithm is an example of a greedy algorithm


Kruskal (K, E, cost):
sort edges in E by increasing cost
while |T|<|K|-1:
let (a, b) be the next edge in E
if aandb are on different components:
join the components ofaandb
T = T U {(a, b)}
return T


1.   Given a network
2.   Choose the shortest edge (if there is more than one, choose any of the   shortest)
3.   Choose the next shortest edge and add it.





4.   Choose the next shortest edge which would not create a cycle and add it.
5.  Choose the next shortest edge which would   not create a cycle and add it.
6.   Repeat until you have a minimal spanning tree.







Example: Find the total weight. All vertices are connected with the cost. i.e AB( 3 ), BC( 5 ), CD( 4 ), DE( 2 ), EA( 4 ), AF( 7 ), EF( 5 ), DF( 8 ), CF( 6 ), BF( 8 )

Edge list, according to the order of size i.e,

E, D, 2

A, E, 4

B, C, 5

C, F, 6

B, F, 8

A, B, 3

C, D, 4

E, F, 5

A, F, 7

C, F, 8

SOLUTION:  

Step 1: select the shorted edge in the network. Here we found the shortest edge is ED. i.e ED is 2

Step 2: Select the next shortest edge which does not create a cycle. Here we got AB i.e, 3

Step 3: Select the next shortest edge which does not create a cycle. Here we got CD and AE i.e 4

 

Step 4: Select the next shortest edge which does not create a cycle. Here we got BC or EF i.e 5

All vertices have been connected now, we found the total weight of the tree is

The solution is,

E D : 2

A B : 3

C D : 4

A E : 4

E F : 5

The total weight of the tree is 18.


Example: Find the total weight.


Edge
Cost
Spanning   Forest






1,   2
10


3, 6
15


4,   6
20


2,   6
25


1,   4
30
reject
3,   5
35





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