Time Complexity of Inserting an Element into a Priority Queue (Binary Heap)

Introduction
A priority queue is an abstract data type that stores a collection of elements, each associated with a priority. The priority of an element determines its position in the queue. A binary heap is a commonly used data structure to implement a priority queue.

Binary Heap
A binary heap is a complete binary tree that satisfies the heap property. In a min-heap, for example, the value of each node is greater than or equal to the values of its children. The root of the heap contains the minimum element. In a max-heap, the values are ordered in the opposite direction.

Insertion Operation in a Binary Heap
When inserting an element into a binary heap, we first add the element to the bottom of the heap, maintaining the complete tree property. Then, we compare the priority of the new element with its parent and swap them if necessary to satisfy the heap property. This process is repeated until the heap property is satisfied or the element reaches the root.

Time Complexity Analysis
The time complexity of inserting an element into a binary heap depends on the height of the tree, which is logarithmic to the number of elements in the heap.

Best Case
In the best case scenario, the element being inserted has the highest priority, and it becomes the new root of the heap. In this case, no swaps are required, and the time complexity of insertion is constant, O(1).

Worst Case
In the worst case scenario, the element being inserted has the lowest priority, and it needs to be swapped with each of its ancestors until it reaches the root. The number of swaps required is equal to the height of the tree, which is logarithmic to the number of elements in the heap. Therefore, the time complexity of insertion in the worst case is O(log n).

Average Case
In the average case scenario, the element being inserted has a priority that falls somewhere in the middle of the priority range. In this case, the number of swaps required is expected to be less than the worst case. Therefore, the average time complexity of insertion is also O(log n).

Conclusion
In conclusion, the time complexity of inserting an element into a priority queue implemented using a binary heap is O(log n) in the worst and average cases. This makes the binary heap an efficient data structure for maintaining a priority queue, as the time complexity of insertion remains relatively low even as the number of elements in the heap increases.