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Data Structures and Algorithms

Submitted By nats14
Words 269
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Q1. The above represents a random looking undirected graph where the circles represent nodes and the boundary circumscribes all the nodes of the graph. The length of the segment joining two nodes represents the length of the shortest path from one node to another. Let’s assume nodes A and B are the farthest pair of nodes according to the definition in the question. Thus A and B have to be on the circumscribing boundary. Thus AB represents the diameter of the graph. Let C be any random node in the graph. Let D be a point which is farthest away from C. So if a BFS is launched on node C, the depth(in terms of the summation of the edges) of the BFS tree rooted at C will be equal to CD. As a result CD >= CA and CD>=CB.
Now let AB = d.
So from triangle inequality CA+CB>=d. Thus d/2 <= max(CA,CB)<=d. Now since CD is greater than or equals to both CA and CB, it has to be greater than or equals to d/2 as well. CD has to be less than d as well because if not then CD would have been the diameter. But that is not the case.

Hence d/2<=CD<=d thus if we can find the length of CD we are done with approximating the value of the diameter within a factor of 2 and that we can do by creating BFS tree from C and calculating depth of the tree which is nothing but CD.

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