Computer Science & IT Applications Mastery Hub: The Industry

In the context of "The Complete Data Structures & Algorithms for PSUs Course 2026: From Zero to Expert!", which data structure is most suitable for implementing a priority queue where elements are frequently inserted and the highest priority element is always retrieved in O(log n) time, considering the need for efficient heapify operations?

Considering the "From Zero to Expert!" progression in the PSU course, when analyzing the time complexity of a recursive algorithm that exhibits overlapping subproblems and optimal substructure, what technique is most crucial for transforming its exponential time complexity into polynomial time?

Within the "Mastery Hub" context for "The Complete Data Structures & Algorithms for PSUs Course 2026", if you are designing a system to detect cycles in a directed graph with a large number of vertices and edges, which algorithmic approach offers the most robust and efficient solution, particularly when dealing with potential disconnected components?