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Heapsort O(n lg n) worst case Another design paradigm –Use of a data structure (heap) to manage information during execution of algorithm Comparision-based Sorting Algorithm Analysis of Algorithms1

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Heap Data Structure Analysis of Algorithms2

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Heap Property Analysis of Algorithms3

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A Heap Example Analysis of Algorithms4

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Heap Data Structure Analysis of Algorithms5

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Heap Operations Analysis of Algorithms6

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Heap Operations Analysis of Algorithms7

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Maintaining Heap Analysis of Algorithms8

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Runtime Analysis of HEAPIFY Analysis of Algorithms9

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Formal Runtime Analysis of HEAPIFY Analysis of Algorithms10

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Formal Runtime Analysis of HEAPIFY Analysis of Algorithms11

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Building Heap Analysis of Algorithms12

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Proof of Lemma Analysis of Algorithms13

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Building Heap Analysis of Algorithms14

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Build Heap - Example Analysis of Algorithms15

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Build Heap - Example Analysis of Algorithms16

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BuildHeap – Tighter Running Time Analysis Analysis of Algorithms17

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BuildHeap – Tighter Running Time Analysis Analysis of Algorithms18

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BuildHeap – Tighter Running Time Analysis Analysis of Algorithms19

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BuildHeap – Tighter Running Time Analysis Analysis of Algorithms20

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HEAPSORT Analysis of Algorithms21

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HEAPSORT Analysis of Algorithms22

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Heapsort - Example Analysis of Algorithms23

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Heapsort - Example Analysis of Algorithms24

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Heapsort - Example Analysis of Algorithms25

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Heapsort – Running Time Analysis Analysis of Algorithms26

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How fast can we sort? The comparison sorting algorithms such as insertion sort, mergesort, quicksort and heapsort only use comparisons to determine the relative order of elements. The best worst-case running time that we’ve seen for comparison sorting is O(n lg n). Is O(n lg n) the best we can do? Decision trees can help us answer this question. Analysis of Algorithms27

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Decision-tree example Analysis of Algorithms28

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Decision-tree example Analysis of Algorithms29

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Decision-tree model Analysis of Algorithms30

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Lower bound for decision-tree sorting Analysis of Algorithms31 Corollary. Heapsort and merge sort are asymptotically optimal comparison sorting algorithms.

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