The tightest lower bound on the number of comparisons, in the worst case, for comparison-based sorting is of the order of

Suppose we run Dijkstra’s single source shortest-path algorithm on the following edge-weighted directed graph with vertex P as the source.
In what order do the nodes get included into the set of vertices for which the shortest path distances are finalized?

The elements 32, 15, 20, 30, 12, 25, 16, are inserted one by one in the given order into a max Heap. The resultant max Heap is

The recurrence equation
T(1) = 1
T(n) = 2T(n - 1)+n, $$n \ge 2$$
Evaluates to