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CPSC 319 (12)
Lecture 11

CPSC 319 Lecture 11: 11 Heaps and Heapsort

4 Pages

Computer Science
Course Code
CPSC 319
Leonard Manzara

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Heaps and Heapsort Heaps - A max heap is a binary tree where: o The value of each node is ≥ the values of its children o The tree is complete 268 ■ Chapter 6 Binary Trees § The tree is perfectly balanced o The leaf nodes in the last level are all pushed to the left o Height is < lg# o The largest element is always the root node FIGURE 6.51 Examples of (a) heaps and (b–c) nonheaps. o E.g. Drozdek p. 268 15 10 21 6 8 7 10 12 2 3 6 (a) - A min heap is similar except the value of each node is ≤ the values of its children o The root contains the smallest element 6 10 12 6 10 Section 6.9 Heaps ■ 269 - Heaps are not perfectly ordered o Order is maintained only through linear lines of descent 15 2 7 10 21 15 8 7 o Lateral lines may be out of order 8 3 FIGUo E.g. Drozdek p. 269 nt heaps construct2d wit3 the same el6ments. (b) Different heaps constructed with the same(c)ements: 10 10 10 2 9 9 8 7 9 FIGURE 6.52 The array [2 8 6 1 10 15 3 12 11] seen as a tree. 1 0 8 7 7 2 0 1 0 1 2 8 (a) (b) (c) - Heaps are normally implemented using2arrays (or vectors) - Elements are stored sequentially in the array: 8 6 o Level by levenode. But the relation between sibling nodes or, to continue the kinship terminology, o From left to right at each level betwee1 uncl10and n15hew n3des is not determined. The order of the elements obeys - The root node is alwa linear line of descent, disregarding lateral.or this reason, lal the trees in Figure - The position of the left child of a node at ! is: 2! + 1, where the position is < # best. - The position of the right child of a node is 2! + 2, where the position is < # - The position of the parent of a node at ! is: (! − 1)/2, where 1 ≤ ! < # o Note: assumesA heap is an excellent way to implement a priority queue.Section 4.3 used linked lists placed at sequential locato implement priority queues, structures for which the complexity was expressed in level from left to right.terms ofO(n) or O(▯n ▯).For large n, this may be too ineffective.On the other hand,a with no gaps.Now,a heap can be defined as an arrayheap of length n in which heap is a perfectly balanced tree; hence,reaching a leaf requiresO(lg n) searches.This heap[i] ≥ heap[2 · i + 1]y prom,for0.≤ i
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