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KUVEMPU UNIVERSITY KUVEMPU UNIVERSITY Department of Computer ScienceDepartment of Computer Science

Jnana SahyadriJnana SahyadriShankarghattaShankarghatta

Seminar onSeminar on

““Kruskal’s Algorithm ”Kruskal’s Algorithm ”

Presented by, Presented by, Chaitra.M.S Chaitra.M.S 33rdrd sem , M.Sc, sem , M.Sc, Dept. Of Computer Science,Dept. Of Computer Science, Shankarghatta.Shankarghatta.

Under the guidance of,Under the guidance of, Suresh.M,Suresh.M,

Dept. Of Computer Science,Dept. Of Computer Science, Shankarghatta.Shankarghatta.

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Contents

• Spanning trees• Finding a spanning tree• Minimum-cost spanning trees• Kruskal’s algorithm And example

Spanning Trees

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Spanning trees

• Suppose you have a connected undirected graph– Connected: every node is reachable from every other node

– Undirected: edges do not have an associated direction

• ...then a spanning tree of the graph is a connected subgraph in which there are no cycles

A connected,undirected graph

Four of the spanning trees of the graph

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Finding a spanning tree

• To find a spanning tree of a graph, pick an initial node and call it part of the spanning tree do a search from the initial node:

each time you find a node that is not in the spanning tree, add to the spanning tree both the new node and the edge you followed to get to it

An undirected graph

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Minimizing costs

• Suppose you want to supply a set of houses (say, in a new subdivision) with:– electric power

– water

– sewage lines

– telephone lines

• To keep costs down, you could connect these houses with a spanning tree (of, for example, power lines)– However, the houses are not all equal distances apart

• To reduce costs even further, you could connect the houses with a minimum-cost spanning tree

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Minimum-cost spanning trees

• Suppose you have a connected undirected graph with a weight (or cost) associated with each edge

• The cost of a spanning tree would be the sum of the costs of its edges

• A minimum-cost spanning tree is a spanning tree that has the lowest cost

A B

E D

F C

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A connected, undirected graph

A B

E D

F C

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A minimum-cost spanning tree

A B

E D

F C

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A connected, undirected graph

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Kruskal algorithm(E,cost,n,t)//E is set of edges in G .G has n vertices.Cost[u,v] is //cost of edge(u,v).t is the set of edges in minimum-cost //spanning tree.the final cost is returned.

{ //construct a heap out of the edge costs using heapify For i=:1 to n do parent[i]:=-1;//each vertex is in a different set.i:=0;mincost:=0.0;

While((i<n-1) and (heap not empty)) do{ //delete a minimum cost edje(u,v) from the heap and reheapify using

Adjust; j:=Find(u); k:=Find(v);

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If (j!=k) then { i:=i+1; t[i,1]:=u; t[i,2]:=v; mincost:=mincost+cost[u,v]; union(j,k); } } if(i!=n-1) then write(“no spanning tree”); else return mincost;

}

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Example graph

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Minimum cost spanning tree we obtained finally

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Any queries???????????

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Thank U………………