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Design and Analysis of Algorithms Heapsort

Design and Analysis of Algorithms Heapsort. Haidong Xue Fall 2013, at GSU. Max-Heap. A complete binary tree, and …. every level is completely filled , except possibly the last , which is filled from left to right . Yes. No. Yes. Max-Heap.

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Design and Analysis of Algorithms Heapsort

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  1. Design and Analysis of AlgorithmsHeapsort HaidongXue Fall 2013, at GSU

  2. Max-Heap • A complete binary tree, and … every level is completely filled,except possibly the last, which is filled from left to right Yes No Yes

  3. Max-Heap • Satisfy max-heap property: parent >= children 16 14 10 8 7 9 3 2 Since it is a complete tree, it can be put into an array without lose its structure information.

  4. Max-Heap 16 14 10 8 7 9 3 2 1 2 3 5 7 8 4 6

  5. Max-Heap • Use an array as a heap 1 For element at i: Parent index =parent(i)= floor(i/2); Left child index = left(i)=2*i; Right child index =right(i)=2*i +1 Last non-leaf node = floor(length/2) 16 2 3 14 10 4 5 6 7 8 7 9 3 8 i=3 2 floor(i/2)=floor(1.5)=1 2*i = 6 2*i+1=7 floor(length/2)=4 7 9 3 2 14 16 10 8 1 2 3 5 7 8 4 6

  6. Max-Heapify • Input: A compete binary tree A, rooted at i, ended at t, whose left and right sub trees are max-heaps; last node index • Output: A max-heap rooted at i. • Algorithm: MAX-HEAPIFY (A, i, t) 1. if(right(i)>t and left(i)>t) return; 2. Choose the largest node among node i, left(i), right(i) . 3. if(the largest node is not i){ • m = the index of the larger node • Exchange i with the largest node • MAX-HEAPIFY (A, m, t) }

  7. Max-Heapify Example 2 16 10 14 7 9 3 8

  8. Heapsort for a heap • Input: a max-heap in array A • Output: a sorted array A HEAP-SORT Algorithm: 1. BUILD-MAX-HEAP(A) 2. Last node index i = A’s last node index 3. From the last element to the second in A{ exchange (i, root); i--; MAX-HEAPIFY(A, root, i); }

  9. Heapsort example 16 14 10 3 9 8 7 2

  10. Heapsort example 14 8 10 3 9 2 7 16

  11. Heapsort example 10 8 9 14 3 2 7 16

  12. Heapsort example 9 8 3 14 10 2 7 16

  13. Heapsort example 8 7 3 14 10 2 9 16

  14. Heapsort example 7 2 3 14 10 8 9 16

  15. Heapsort example 3 2 7 14 10 8 9 16

  16. Array -> Max-Heap • Input: a array A • Output: a Max-Heap A BUILD-MAX-HEAP(A): Considering A as a complete binary tree, from the last non-leaf node to the first one i{ MAX-HEAPIFY(A, i, A.lastIndex); }

  17. Build Heap Example 7 10 8 16 14 14 8 16 10 7

  18. Heapsort • Input: array A • Output: sorted array A Algorithm: 1. BUILD-MAX-HEAP(A) 2. Last node index i = A’s last node index 3. From the last element to the second in A{ exchange (i, root); i--; MAX-HEAPIFY(A, root, i); }

  19. Analysis of Heapsort • Input: array A • Output: sorted array A Algorithm: 1. BUILD-MAX-HEAP(A) 2. Last node index i = A’s last node index 3. From the last element to the second in A{ exchange (i, root); i--; MAX-HEAPIFY(A, root, i); } O(nlgn) (or O(n) see page 157-159 for why) O(n) O(nlgn)

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