Weighted Graphs and Dijkstra's Algorithm

8/8/2019 Weighted Graphs and Dijkstra's Algorithm

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Andrew Pereira

1st Year M.Sc Computer Science

Roll-1025001


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We may need to find the fastest way to route a data packet between 2

computers. We note that some computers in a computer network are

faster than others.

We may want to find the fastest way to travel cross country. Someintercity distances are larger than others.

Thus it is natural to consider graphs whose edges are not weighted

equally.


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Aweighted graph is a graph that has a numeric label w(e) associated with

each edge e called the weight of the edge e.

The weight of a path or the weight of a tree in a weighted graph is the

sum of the weights of the selected edges.

Sometimes the wordc

ost is used instead of weight

UNWEIGHTEDGRAPH WEIGHTEDGRAPH


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Dijkstas Algorithm is used to find the shortest path between any pair of

vertices in a connected weighted graph.

Works on both directed and undirected graphs. However, all edges must

have nonnegative weights.

Input: Weighted graph G={E,V} and source vertex vV, such that all edge

weights are nonnegative

Output: Lengths of shortest paths (or the shortest paths themselves) froma given source vertex vV to all other vertices


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The algorithms begins by assigning a permanent label 0 to the starting

vertex and a temporary label to all other n-1 vertices. At every

iteration another vertex gets a permanent label according to :

Each vertex j that is not permanently labelled gets a new temporary

label whose value is given by label j={min of j, old label of i+dij

} where

i is the latest vertex permanently labelled in the previous iteration and

dij is the distance between i and j. If i and j are not joined by an edge

then dij = .

The smallest value between all the temporary labels is found and the

this becomes the permanent label of the corresponding vertex.

Repeat the above two steps until we reach the terminal vertex with ashortest distance.


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` Using Dijkstras algorithm find the shortest path

length between the vertices a and z in this

weighted graph.(Qn 37)b

a

z

c d

8

4

6

2

2


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` Using Dijkstras algorithm find the shortest path

length between the vertices a and z in this

weighted graph.(Qn 39)

a

b

c e

d

z

3

7

6

6

8

5

2 21


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As mentioned, Dijkstras algorithm calculates the shortest path to every

vertex.

However, it is about as computationally expensive to calculate the

shortest path from vertexu

to every vertex usingD

ijkstras as it is tocalculate the shortest path to some particular vertex v.

Therefore, anytime we want to know the optimal path to some other

vertex from a determined origin, we can use Dijkstras algorithm.

Weighted Graphs and Dijkstra's Algorithm

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