Kruskal’s and Dijkstra’s Algorithm

INTRODUCTION ABOUT GRAPH

SPANNING TREE

MINIMAL SPANNING TREE

KRUSKAL’S ALGORITHM

DIJKSTRA’S ALGORITHM

Agenda

Graph

A Graph G=(V, E), Where V is a nonempty set of vertices & E is a set of edges. every edge is associated with unordered pair of vertices.

V={V1, V2, V3, V4}

∑={e1, e2}

e1

e2

V1V3

V4 V2

e1=<V4,V1>

e2=<V3,V2>

Tree• A Graph G is called a Tree if G is a connected

acyclic graph

Spanning Tree

Let G be a Graph & H be a Spanning Sub graph of G. If H is a tree then H is called as spanning tree.

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G=(V, E) be a connected graph if there is a function

f : V(G) -> R then it is called weighted connected graph.

A(G),The cost adjacency matrix of G,

whose entries are given by

cij if <i,j> ϵ E(G)

0/∞ otherwise

Weighted Connected Graph and its Cost adjacent Matrix :

aij

=

Result: G is connected iff G has at least one spanning tree.

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Minimal Spanning Tree-MST

Let G be a weighted connected Graph among all

the spanning tree of G, whose weight is Minimal

is called Minimal Spanning Tree(MST).

They are 3 algorithms to find minimal spanning

tree

Prim’s Algorithm (Based on Vertices ) (order is O(E +V lg V)

Kruskal’s Algorithm(Based on Edges)(order is O(E lg E)

Sollin's Algorithm

Kruskal’s Algorithm

Input Connected weighted graph with |v(G)|=n Or Cost adjacency matrix(n*n)

Output Minimal Spanning Tree

Kruskal’s Algorithm contd..

Step-1: Arrange the edges in ascending order according to

their weights choose the minimum weight edge say e1.

Step-2: Having had chosen <e1, e2, …. ,eK>, choose eK+1 as

follows :<e1, e2,…,eK, eK+1> is acyclic & the remaining edges w(eK+1) is minimal.

Step-3: Repeat step-2 until (n-1) edges are included.

Example

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Dijkstra’s Algorithm

[Single Source Shortest Path Problem]Input:

Connected weighted graph G(V|G|=n)Or

Cost adjacency matrix (n*n) with a specified vertex as source say v.

Output: Shortest path from v, to all other vertices in G.

Dijkstra’s Algorithm(Labeling procedure)

Procedure: Dijkstra(G:weighted connected simple graph,with all weights positive)

{G has vertices a=V0,V1,…Vn=Z and weights w(Vi,Vj) where w(Vi,Vj) =∞ if {Vi,Vj} is not an edge in G}

for i:=1 to n L(Vi):= ∞ L(a):= 0 S:=ф{ the labels are now initialized so that the label of a is

0 and all other labels are ∞, and S is the empty set}

While z Є s

Begin

U:=a vertex not in S with L(u) minimal

S:=S { u}

For all vertices V not in S

If L(u)+w(u , v)<L(v) then L(v):=L(u)+w( u,v)

{this adds a vertex to S with minimal label and updates

the labels of vertices not in S}

End{L(z)=length of a shortest path from a

to z}

Dijkstra’s Algorithm contd…

Example( Solved by other method)

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Result L(V2) L(V3) L(V4) L(V5) L(V6) T

<V1,V3>V1-V3

<V1,V2>V1-V2

<V1,V4>V1-V3-V4

<V1,V5>V1-V3-V5

<V1,V6>V1-V3-V4-V6

3 2 ∞ ∞ ∞3 2 7 9 ∞3 2 7 9 ∞3 2 7 9 13

3 2 7 9 13

{V1,V3}

{V1,V3, V2}

{V1,V3, V2, V4}

{V1,V3, V2, V4, V5}

{V1,V3, V2, V4, V5, V6}

<V1,V2><V1,V3><V1,V4><V1,V5><V1,V6>

V1 - V2

V1 – V3V1 – V4

V1 – V5

V1 – V6

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Result Vertices L(V)

1.Avoid Looping in the network

2.In I/p Multicasting

3.Routing

Application of MST & Single Source Shortest Path

1.Discrete Mathematics and its Applications ,6th edition., Tata McGraw-Hill.,2007.,by Kenneth.H.Rosen

2.Graph Theory and its Applications ,2nd edition.,Chapman & Hall/CRC.,2006., By Jonathan L.Gross Jay Yellen

3.Introduction to Algorithms.,Prentice Hall of India.,2004.,by Thomas H.Cormen,Charles E.Leiserson,Ronald L.Rivest

References

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