(a) Construct a directed graphs G with positive link weights so that (i) there is a unique shortest path between two selected nodes on G (the nodes are subject to your choice) and (ii) if the weight of every link is increased by 10 units, then the shortest path between the same pair of nodes will change to a different one. [10pts]

(b) Given any directed graph G with positive link weights and a shortest path between two given nodes. Prove that the shortest path between the two nodes remain the same if weights of all links are multiplied by a positive constant. [10pts]

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