Introduction

Priority Queues are an abstract data structure that allows for efficient access to the smallest (or largest) element in a set. They are utilized in a variety of algorithms, such as Dijkstra's and Prim's algorithms, where it's essential to repeatedly remove elements of highest or lowest priority quickly. In C++, the Standard Template Library (STL) provides the `std::priority_queue` class template which implements a max-heap by default. Understanding the time complexity of operations on a priority queue is crucial for leveraging its power in solving computational problems efficiently.

Basic Operations

A priority queue supports a set of standard operations, each with a specific time complexity:

1. Insertion (`push`): Inserts a new element into the priority queue.
2. Deletion (`pop`): Removes the element with the highest priority.
3. Peek (`top`): Retrieves (but does not remove) the element with the highest priority.
4. Size: Returns the number of elements in the priority queue.
5. Empty: Checks if the priority queue is empty.

Let's explore the time complexities of these operations when implemented as a binary heap, the standard underlying structure of a `std::priority_queue` in C++.

Time Complexity Analysis

The time complexity of operations in a priority queue implemented using a binary heap is as follows:

Example in C++

Below is a simple example demonstrating how to use the `std::priority_queue` in C++: