Develop A Heap Data Structure Using Linked Nodes

Develop a Heap Data Structure Using Linked Nodes and Implement a Priority Queue

You have been tasked with developing a heap data structure using a linked node-based approach instead of the list-based structure provided in the example code. Using the provided list-based heap implementation as a reference, create a linked structure where each node has pointers to its children and parent to maintain the heap properties. Ensure that the heap supports the essential operations of insertion and removal (specifically, removing the root element), and that it maintains the heap property (for this example, a min-heap). Once the linked heap is implemented, design a priority queue class that uses this linked heap as its underlying data structure. The priority queue should provide methods for enqueue (adding with priority), dequeue (removing the highest priority element), peek (viewing the highest priority element without removing), check if empty, and size.

Finally, demonstrate the functionality of your priority queue by performing the following sequence of operations:

• Enqueue the items: 4, 7, 5, 11, 8, 6, 9
• Dequeue three times; the removed items should be in the order of the smallest three, specifically 4, 5, and 6

Paper For Above instruction

The implementation of a heap data structure using a linked node-based approach offers a more dynamic and pointer-based alternative to the conventional array-based heap. This approach facilitates more flexible memory management, especially for applications requiring frequent insertions and deletions, and avoids the limitations tied to contiguous memory allocation inherent in array-based structures. The goal of this paper is to develop such a data structure – a linked heap – and embed it within a priority queue that supports common operations, demonstrated through specific use cases.

Design and Implementation of a Linked Heap

The primary step involves defining the structure of a node in the linked heap. Each node will contain a value, a reference to its parent node, and references to its two children: left and right. This structure allows traversal from any node to its parent or children and supports maintaining the complete binary tree necessary for a heap. The heap itself will maintain a reference to the root node and track the number of elements to facilitate index-based operations conceptually.

Inserting a new element into the linked heap involves locating the appropriate position to maintain the complete binary tree property, typically the next available position at the deepest level. The insertion process includes inserting at the next available spot and then "floating" the new node upward if necessary to maintain the heap property—comparing with its parent and swapping values if the heap order is violated. This process resembles sift-up operation in array-based heaps but adapted to pointer manipulation. Similarly, removing the root involves replacing it with the last node in level order, removing that node, and then sinking down the new root to restore the heap property.

Implementing the Priority Queue

The priority queue exploits the linked heap as its backbone, providing methods to enqueue elements with specified priorities, dequeue the highest priority element, peek at the highest priority element without removal, and check if the queue is empty or determine its size. When enqueuing, the element is inserted into the linked heap; for dequeuing, the root is removed, which always guarantees the highest-priority element in a min-heap context. The approach ensures all operations preserve the heap property.

Operational Demonstration

Using the designed priority queue, enqueue the items 4, 7, 5, 11, 8, 6, and 9. Following the insertion phase, perform three dequeue operations. Because the structure is a min-heap, the first three items removed should be the smallest three: 4, 5, and 6, respectively. This sequence exemplifies the property of the heap-based priority queue, where the highest priority (minimum value in this case) is always dequeued first.

Conclusion

Implementing a heap using linked nodes enhances the robustness and flexibility of the data structure, especially suitable for applications requiring efficient dynamic memory management. The integration of this linked heap into a priority queue offers an intuitive and effective way to manage prioritized data, with operations that uphold the core properties of heaps. The demonstrated sequence confirms that the design functions correctly, fulfilling the requirements of adding, removing, and peeking at elements based on priority.

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