Welcome to our exploration of the intriguing worlds of heaps and priority queues. These are powerful data structures used extensively across a range of applications, from job scheduling systems to modeling the stock market. Heaps can efficiently solve problems involving intervals, n-th largest elements, and even sorting. C#'s PriorityQueue class provides the functionality to interact with a collection as if it were a heap.

Heaps are a category of binary trees where every node has a value less than or equal (in the case of min-heap) or greater than or equal (in the case of max-heap) to its children. This property allows us to repeatedly access the smallest or largest element, respectively, enabling us to solve numerous problems effortlessly. For example, if you want to find the n-th largest number in a list, using sorting can be costly. By leveraging C#'s PriorityQueue implementation, the heap data structure lets us do this efficiently.

C#'s PriorityQueue class, by default, implements a min-heap, meaning the smallest element is given the highest priority.

Here is how you can do it in C#:

Priority queues are an abstraction over heaps that store elements according to their priorities. They are used when objects need to be processed based on priority. For instance, scheduling CPU tasks based on priority is a real-life scenario for priority queues.

Let's plunge into the exercises, trusting that the best way to learn is by doing. As we solve various problems, you'll build a solid understanding of how these powerful tools can be employed to simplify complex tasks. This will not only prepare you for your interviews but also cultivate a mindset for problem-solving. Welcome aboard!