Question

Let G be a graph and T be its MST, suppose we increase the weight of an edge e € T by a positive quantity 8. Give an algorithm that gets G,T,e and 8 as input and returns a MST of the new graph. Prove the correctness of your algorithm and state the run-time

Answer 1

solution:

given data:

Algorithm

Steps

1. Remove the edge e =(u,v)

2, After removing we get two connected sub tree lets say T1 and T2

3. Now use BFS Traversal in both sub-trees starting from u and v respectively (ignore edge weights) to know which vertices are there in T1 and Which are there in T2

4. Now examine each Edge of Graph G and keep the minimum weight edge e' with one endpoint in T1 and the other in T2

5.Once we find the min edges connecting any vertex in T1 with any vertex in T2 Then out MST T is again completed  

Proof

As we know that An edge e in MST in increased so to check that MST is still minimum or not

Proof

Assume that

MST is still Minimum only with same edge e connected

Proof

We remove the edge e =(u,v)

Now we get two different connected components and acc to MST we need one more edge to connect them

Let the graph be G

1 .Traverse each edge see if it connects the components of not

2 . If Yes then Mark it minimum

else

Check other edge

3. Repeat above steps until all edges have been visited and

If any other edge that connects the Components

So Compare Minimum with new edge weight

and updated minimum if new edge weight < minimum

Now we have edge connecting two connecting and weight is stored in Minimum

4. Check that removed edge e weight and minimum is same or not

If not then our assumption is wrong

And by contradiction we say Graph is not still a MST that means that edge e has to replaced .

So By using above algorithm we will get correct MST

Hence Proved

Time Complexity of algorithm

1 .We are applying BST to check the vertexes So it will take O(V+E)

2. IN step 4 we are traversing each edge in graph to get minimum edge weight connection between sub trees So O(E) time is needed

hence Total Time Complexity = O( V+ 2 E) = O(V+E)

where V is the number of Vertices and E is number of Edges in G

This is how using above we can get the New MST if any edge weight in MST is increased  

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