Trees:Heaps

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Heaps

Priority Queues

Heaps are used to implement priority queues. In a priority queue, the element with highest (or lowest) priority is deleted from the queue, while elements with arbitrary priority are inserted. A data structure that supports these operations is called a max(min) priority queue. Henceforth, in this chapter, we restrict our discussion to a max priority queue. A priority queue can be implemented by a simple, unordered linked list. Insertions can be performed in O(1) time. However, a deletion requires a search for the element with the largest priority followed by its removal. The search requires time linear in the length of the linked list. When a max heap is used, both of these operations can be performed in O(log n) time.

Definition of a Max-Heap

A max heap is a complete binary tree such that for each node, the key value in the node is greater than or equal to the value in its children. Observe that this implies that the root contains the largest value in the tree. Figure 3.12 shows some examples of max heaps.

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We define a class Heap with the following data members.

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The heap is represented using an array (a consequence of the complete binary tree property) which is dynamically allocated.

Insertion

Suppose that the max heap initially has n elements. After insertion, it will have n + 1 elements. Thus, we need to add a node so that the resulting tree is a complete binary tree with n + 1 nodes. The key to be inserted is initially placed in this new node. However, the key may be larger than its parent resulting in a violation of the max property with its parent. In this case, we swap keys between the two nodes and then repeat the process at the next level. Figure 3.13 demonstrates two cases of an insertion into a max heap.

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The algorithm is described below. In the worst case, the insertion algorithm moves up the heap from leaf to root spending O(1) time at each level. For a heap with n elements, this takes O(log n) time.

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Deletion

The element to be deleted (i.e., the maximum element in the heap) is removed from the root node. Since the binary tree must be restructured to become a complete binary tree on n − 1 elements, the node in position n is deleted. The element in the deleted node is placed in the root. If this element is less than either of the root’s (at most) two children, there is a violation of the max property. This is fixed by swapping the value in the root with its larger child. The process is repeated at the other levels until there is no violation. Figure 3.14 illustrates deletion from a max heap.

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The deletion algorithm is described below. In the worst case, the deletion algorithm moves down the heap from root to leaf spending O(1) time at each level. For a heap with n elements, this takes O(log n) time.

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