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.

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 Yurti$$i Kargo packages in central Turkey are routed through $x=$ Ankara.$)$

$($a$)$ Describe an $O(ElogV)$ 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 $O(ElogV)$ time so that, given the data structure, you can process each query in constant time.