GATE CS 2020 Set 1 — Question 50
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Algorithms → Graph Algorithms
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Question
Let be a directed, weighted graph with weight function . For some function , for each edge , define as .Which one of the options completes the following sentence so that it is TRUE?"The shortest paths in under are shortest paths under too, __________".
Correct answer
(A) for every f: V → R
Solution
The reweighting technique described is used in Johnson's algorithm. The new weight of a path is:This sum telescopes:For any two nodes and , the term is constant for all paths from to . Therefore, if a path minimizes , it also minimizes .This property holds for any function , regardless of whether is positive, negative, or related to distances. Thus, the statement is true for every .
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