3. Write a modified version of the Dijkstra's algorithm so that if there is more than one
minimum path from the source vertex s to a vertex v, a path with the fewest number of
edges is chosen.
4. Given a directed graph G with positive edge weights and a landmark vertex x, your goal
is to find the length of the shortest path from one vertex v to another vertex w that passes
through the landmark x. (For example, assume that Yurtiçi Kargo packages in central
Turkey are routed through x = Ankara.)
(a) Describe an O(E log V) algorithm for the problem. Justify briefly why your proposed
algorithm is correct.
(b) Now suppose that you will perform many such shortest path queries for the same
landmark x, but different values of v and w. Describe how to build a data structure in
O(E log V) time so that, given the data structure, you can process each query in con-
stant time.
