Question

Suppose that queue is a queueType object and the size of the array implementing queue is \(100 .\) Also, suppose that the value of queueFront is 50 and the value of queueRear is 99 a. What are the values of queueFront and queueRear after adding an element to queue? b. What are the values of queueFront and queueRear after removing an element from queue?

Short answer

After adding an element: `queueFront = 50`, `queueRear = 0`. After removing an element: `queueFront = 51`, `queueRear = 99`.

Step-by-step solution

Understanding the Queue Structure

The queue is implemented using an array of size 100. This means it has indices ranging from 0 to 99. The front of the queue is indicated by `queueFront`, and the rear of the queue is indicated by `queueRear`. Currently, `queueFront` is 50 and `queueRear` is 99.

Adding an Element to the Queue

When an element is added to the queue, `queueRear` is incremented by 1. Since `queueRear` is currently at 99, which is the last index, it will wrap around to 0 (beginning of the array) due to the circular nature of the queue. Therefore, after adding an element: - `queueRear = (queueRear + 1) % 100 = 0` - `queueFront` remains unchanged at 50.

Removing an Element from the Queue

When an element is removed from the queue, `queueFront` is incremented by 1. Therefore, after removing an element: - `queueFront = queueFront + 1 = 51` - `queueRear` remains unchanged at 99.

Key concepts

Circular Queue

A circular queue is an advanced form of the regular queue data structure. It's designed as a circular arrangement using an array, where the last position is connected back to the first. This special arrangement helps efficiently utilize storage by reusing vacant spaces once occupied by dequeued elements. Think of it like a merry-go-round where you can keep going round without stopping at the end.

In a circular queue, you never run into an issue of hitting the end of an array because the positions loop back to the beginning. Its main advantage is that it mitigates the "waste" of unused space that occurs when elements are removed from a linear queue, and those positions could not be reused. This wrapping around is what allows the queue to overcome the fixed size limitation of a normal queue based on a linear data structure.

Array Implementation

The implementation of a circular queue using an array is a common choice due to the simplicity of using arrays. The key to its implementation is managing the wrap-around, or "circular" aspect, using modular arithmetic. This operation helps connect the end of the array back to the beginning, creating what's known as a "circular buffer."

In practical terms, when you add an element to the rear, if the rear index is at the last position in the array, it immediately wraps to the first index (position 0) to add the new element. This behavior uses the formula: \[ \text{new\_rear} = (\text{current\_rear} + 1) \% \text{array\_size} \]This simple mathematical operation ensures that the array is utilized fully and no space is indefinitely lost.

This efficiency can be particularly beneficial when operating under constraints of memory or when precision regarding constant time complexity for enqueue and dequeue operations is required.

Queue Operations

The primary operations of a queue are enqueuing (adding an element) and dequeuing (removing an element). In the context of a circular queue, these operations gain efficiency due to its design.

• Enqueue: When adding an element, you increase the rear position. In a circular queue implemented with an array, you check if the rear is at the last index of the array, and if so, set it to zero. Otherwise, increment rear by one. This ensures new elements are placed correctly.
• Dequeue: Upon removing an element, you simply move the front pointer forward. If it reaches the end of the array, it also wraps around to the starting position, demonstrating the circular feature again.

These operations maintain the queue’s order and ensure constant time complexity, allowing operations to be completed in linear time, no matter the queue's size.

Front and Rear Pointers

The front and rear pointers are integral to managing a circular queue. They signify the current beginning and end of the queue respectively, and play a vital role in understanding the state of the queue.

• Front Pointer: This pointer indicates the position of the first or front element in the queue. When dequeueing, this pointer is incremented to point to the new "head" of the queue. In a circular queue, once it reaches the array's end, it wraps back to the start.
• Rear Pointer: Similarly, the rear pointer marks where new elements will be added. When you enqueue an element, this pointer acts to identify the "end" of the queue. Just as with the front, if the rear pointer reaches the last index of the array, it wraps around to the first index to accommodate continuous additions.

Managing these pointers correctly is pivotal for maintaining the integrity of data within the circular buffer and avoiding errors such as overwriting existing data or misreading the queue's current state.