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Answer to directed weighted graph should be for every vertex the paths will be same as because  w'(u,v) will be w(u,v)+f(u)-f(v) means depednent on only source and destination and not on intermediate step as they will cancel out so paths will reamain same and values can  be different.

eg.

w(1,3)=w(1,2)+w(2,3) and let other candidate be w(1,4)+w(4,3) then

          w'(1,3)=w(1,2)+f(1)-f(2)+w(2,3)+f(2)-f(3) ==w(1,2)+w(2,3)+f(1)-f(3)

          w(1,3)=w(1,4)+f(1)-f(4)+w(3,4)+f(4)-f(3)==w(1,4)+w(4,3)+f(1)-f(3)

      so exteding similarly for any path ,the shortest path will not change  but its value can be changed.
ago in Algorithms by (6 points) | 17 views

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