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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 BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)1

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Heap Data Structure BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)2

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Heap Property BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)3

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A Heap Example BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)4

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Heap Data Structure BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)5

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Heap Operations BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)6

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Heap Operations BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)7

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Maintaining Heap BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)8

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Runtime Analysis of HEAPIFY BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)9

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Formal Runtime Analysis of HEAPIFY BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)10

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Formal Runtime Analysis of HEAPIFY BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)11

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Building Heap BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)12

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Proof of Lemma BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)13

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Building Heap BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)14

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Build Heap - Example BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)15

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Build Heap - Example BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)16

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BuildHeap – Tighter Running Time Analysis BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)17

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BuildHeap – Tighter Running Time Analysis BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)18

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BuildHeap – Tighter Running Time Analysis BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)19

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BuildHeap – Tighter Running Time Analysis BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)20

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HEAPSORT BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)21

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HEAPSORT BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)22

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Heapsort - Example BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)23

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Heapsort - Example BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)24

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Heapsort - Example BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)25

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Heapsort – Running Time Analysis BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)26

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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. BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)27

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Decision-tree example BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)28

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Decision-tree example BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)29

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Decision-tree model BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)30

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Lower bound for decision-tree sorting BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)31 Corollary. Heapsort and merge sort are asymptotically optimal comparison sorting algorithms.

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