Dijkstra's algorithm, Data Structure & Algorithms

Q. Explain Dijkstra's algorithm for finding the shortest path in the graph given to us.

 Ans:

The Dijkstra's algorithm: This is a problem which is concerned with finding the least cost path from an originating node in a weighted graph to a destination node in that particular graph.

The Dijkstra's algorithm is stated below:

Let G=(V,E) be the simple graph. Let a and z be any two vertices on the graph. Let L(x ) denotes the label of the vertex x which represent the length of shortest path from the vertex a to the vertex x and wij which denotes the weight of the edge eij=

1.  Let P=φ where  P is the set of those vertices which

Are having permanent labels and T={all vertices in graph}

 

Set L(a)=0,

L(x)=∞ for all x≠a

 

2.

Select the

label. This

vertex v in T which has

label is called permanent

the smallest

label of v.

Also set P=P {v} and T=T-{v}

If v=z then L(z) is the length of the shortest path from the vertex a to z and stop.

3.  If  v is  not equal to z    then   revise      the   labels    of vertices of T that is the vertices which do not consist of permanent labels.

The new label of vertex x in T can be given by

L(x)=min(old L(x), L(v)+w(v, x))

Here w(v, x) is the weight of the edge joining the vertex v and x. If there exists no direct edge joining v and x then take w(v, x)= ∞

4.  Repeat steps 2 and 3 until get the permanent z label.

Posted Date: 7/10/2012 6:38:52 AM | Location : United States







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