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(A) (15%) Assume that By the assumptions and above definition and theorem, prove that D is NP-complete.

(B) (5%) By the above definitions and theorem, show the following statement is true or false. You should also show your reasons. If the reasons are incorrect, then you get no point. Statement: An NPC problem can polynomially reduce to any NPC problem.

(C) (5%) By the above definitions and theorem, show the following staten ement is true or false. You should also show your reasons. If the reasons are incorrect, then you get no point. Statement: If an NPC problem can be solved in polynomial time by a deterministic algorithm, then all NP problems can also be solved in polynomial time by a deterministic algorithm.

(A)(12%) For a weighted graph G-(V, E) and two vertices x, y in Y, give an efficient algorithn to find a shortest path from x to y. Analyses the time complexity of the algorithm.

(B) (13%) Now you a are given a a weighted graph G-(V, E), a specified vertex s in V, an array d ] of size V, and it is claimed that for each v in Y, dty is the length of the shortest path from s to v. The claim may be correct or may be not. Design a linear time algorithm to check whether the claim is correct. Analyses the time complexity of your algorithm and make sure it is linear, that is, the running time of your algorithm must be O(|V|+|E|).

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