Question

Although it is correct to use a "reverse" comparator with our priority queue ADT so that we retrieve and remove an element with the maximum key each time, it is confusing to have an element with the maximum key returned by a function named "removeMin." Write a short adapter class that can take any priority queue \(P\) and an associated comparator \(C\) and implement a priority queue that concentrates on the element with the maximum key, using functions with names like removeMax. (Hint: Define a new comparator \(C^{\prime}\) in terms of \(C\).)

Short answer

Create a new comparator \( C' \) that reverses \( C \); then design an adapter class using \( C' \), and implement removeMax by calling removeMin from the original priority queue.

Step-by-step solution

Define a New Comparator

Create a new comparator \( C' \) that reverses the logic of the given comparator \( C \). This will allow the priority queue to focus on the maximum key instead of the minimum key.

Create the Adapter Class

Design an adapter class that accepts any priority queue \( P \) and the new comparator \( C' \). This class should implement all the necessary methods of a priority queue, but adapted to work with the maximum key using \( C' \).

Implement the Constructor

In the adapter class, write a constructor that initializes the priority queue \( P \) with the comparator \( C' \).

Implement removeMax Method

Create the removeMax method which internally calls the removeMin method of the original priority queue \( P \). Since \( C' \) reverses the order, removeMin will act as removeMax.

Implement Other Methods

For completeness, implement other priority queue methods such as insert, isEmpty, etc., ensuring they use the new comparator \( C' \).

Testing

Write test cases to ensure that the adapter class correctly handles priority queue operations, especially focusing on methods like removeMax.

Key concepts

Comparator Design

Comparators are powerful tools used in priority queues to control the order in which elements are processed. They determine whether one element should be placed before or after another based on specific criteria. Normally, a comparator returns a negative integer, zero, or a positive integer when comparing two elements. This indicates whether the first element is less than, equal to, or greater than the second element, respectively.
Creating a new comparator, let's call it \( C' \), that reverses this logic is a simple yet effective way to transform a priority queue focusing on minimum keys to one concentrating on maximum keys. By altering the original comparator \( C \) so that it offers the opposite of its default comparison, elements with maximum keys are returned and removed first, despite the original priority queue operations being named removeMin.
Using this reversed comparator is key to our solution, making the confusing and often counter-intuitive removeMin method act as removeMax under the hood.

Adapter Class

An adapter class allows you to change the interface of an existing class without modifying it. This is incredibly useful when you need an interface to mirror a different functionality. In our exercise, the adapter serves as a bridge between the original priority queue operations and the newly defined comparator \( C' \).
To build an effective adapter class:

• Accept an existing priority queue \( P \) and the comparator \( C' \)
• Override methods like removeMin to function as removeMax

Our adapter class essentially translates the operations targeting minimum keys into their maximum key counterparts by utilizing the reversed comparator. The main methods to be adapted include:

• removeMax: Simulates removeMin but focuses on maximum key due to \( C' \)
• insert: Adds elements using the new comparator for consistent ordering
• isEmpty: Checks if the priority queue is empty

Each method in the adapter class ensures that the priority queue operations align with focusing on the maximum key.

Priority Queue Operations

Priority queue operations involve adding, removing, and inspecting elements based on their priority, usually determined by a comparator. The primary operations include:

• insert: Adds a new element while maintaining the queue's order
• removeMin: Removes and returns the element with the minimum key
• findMin: Returns (but does not remove) the element with the minimum key

By adapting our priority queue to focus on the maximum key, we redefine these operations with a new perspective. The critical operation here is removeMax, repurposed from removeMin but oriented to handle maximum keys due to the reversed comparator \( C' \). Similarly, insert ensures that any new element respects the order defined by the maximum key logic.
This approach guarantees that our priority queue remains efficient and consistent, even as we change its fundamental focus from minimum to maximum key handling. Understanding these operations helps demystify the underlying structure and behavior, allowing students to grasp how seemingly simple changes can lead to significant differences in functionality.