shortest path

Question

S1: In a weighted graph, assume that the shortest path from a source 's' to a destination 't' is correctly calculated using a shortest path algorithm. Is the following statement true? If we increase weight of every edge by 1 , the shortest path always remains same.

S2: Given a graph, suppose we have calculated shortest path from a source to all other vertices. If we modify the graph such that weights of all edges become double of the original weight, then the shortest path remains same, only the total weight of path changes.

• A. True, true
  
• B. True, false
  
• C. False, true
  
• D. False, false

Comments

C, 1st statement false because if distance b/w  s-a is 1, a-b=1, b-t=1 & s-t=4; So shortest distance is s-a-b-t= 3; & if we add 1 in every edge so s-a-b-t=6 & s-t=5 .In this case shortest path is change.          2nd statement is true; take same data now double each edge so new distance s-a-b-t=6 & s-t=8; So shortest distance remain same.

Answer 1

By using counter Example:

Here shortest path from s TO t is A to B and B to D. Let's increase edge weight of each edge by 3. Now the shortes path from s To t is A to D that is 7. So S1 is false.

if we multiply edge weights with same number the edge weights are increased in same ratio so No change in path but total weight has been changed.

So ans is option C.

Comments

Nice explanation @Gabbar..

Answer 2

Consider above graph,

S1)

Now suppose shortest path from S to A was S-B and then B-A. shortest path length = SB + BA = 5+5 =10

Now if we increase weight of each edge by 1, then new shortest path will be

shortest path length = min (SB + BA, SA)= min(6+6,11) =11

that means shortest path has changed. So S1 is wrong

S2)

Now even if we double the edge weight shortest path remains the same

shortest path length = min(SB + BA, SA) = min(10+10,20) = 20

that means shortest path is still S-B and then B-A

So S2 is correct

Hence answer should be C)