1 / 23

Design and Analysis of Algorithms

Design and Analysis of Algorithms. Lecture # 15 BCS. Build-Max-Heap(A). Running time of Max- Heapify O(h) n-element heap has height floor( lgn ) Max. nodes of height h are Ceiling(n/2 h+1 ). Build-Max-Heap(A). HeapSort(A). Build-Max-Heap(A) for i ← length [A] downto 2

dyre
Download Presentation

Design and Analysis of Algorithms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Design and Analysis of Algorithms Lecture # 15 BCS Muhammad Umair

  2. Build-Max-Heap(A) • Running time of Max-Heapify • O(h) • n-element heap has height • floor(lgn) • Max. nodes of height h are • Ceiling(n/2h+1) Muhammad Umair

  3. Build-Max-Heap(A) Muhammad Umair

  4. HeapSort(A) • Build-Max-Heap(A) • for i ← length [A] downto 2 • do exchange A[1] ↔ A[i] • Heap-size[A] ← heap-size[A] – 1 • Max-HeapIFY(A,1) Muhammad Umair

  5. Heapsort(A)[16,14,10,8,7,9,3,2,4,1] Muhammad Umair

  6. Heapsort(A)[16,14,10,8,7,9,3,2,4,1] Muhammad Umair

  7. Heapsort(A)[16,14,10,8,7,9,3,2,4,1] Muhammad Umair

  8. Heapsort(A)[16,14,10,8,7,9,3,2,4,1] Muhammad Umair

  9. Heapsort(A)[16,14,10,8,7,9,3,2,4,1] Muhammad Umair

  10. Heapsort(A)[16,14,10,8,7,9,3,2,4,1] Muhammad Umair

  11. Heapsort(A)[16,14,10,8,7,9,3,2,4,1] Muhammad Umair

  12. Heapsort(A)[16,14,10,8,7,9,3,2,4,1] Muhammad Umair

  13. Heapsort(A)[16,14,10,8,7,9,3,2,4,1] Muhammad Umair

  14. Heapsort(A)[16,14,10,8,7,9,3,2,4,1] Muhammad Umair

  15. Analysis of Heap Algorithms • Max-HeapIFY(A,i) • O (lg n) = O (h) • Build-Max-Heap(A) • O (n) • HeapSort(A) • O (n lg n) Muhammad Umair

  16. Student Exercise • Priority Queues • Max-priority queue • Min-priority queue Muhammad Umair

  17. Quick Sort (A, p, r) • if p < r then • Q ← Partition (A, p, r) • Quick Sort (A, p, q – 1) • Quick Sort (A, q + 1, r) Muhammad Umair

  18. PARTITION (A, p, r) • x ← A [ r] • i ← p – 1 • for j ← p to r – 1 do • If A [ j] <= x then • i ← i + 1 • exchange A [ i ] & A [ j] • exchange A [ i + 1] & A [ r] • return i + 1 Muhammad Umair

  19. Analysis of Quick Sort • Worst Case Partitioning • Best Case Partitioning • Balanced Partitioning Muhammad Umair

  20. Worst Case Partitioning • The maximum running time • Pivot element is already on its actual location • It means the input data is also sorted form • Its recurrence : T(n) = T(0)+T(n-1)+Θ(n) • The solution : T(n) = Θ(n2) Muhammad Umair

  21. Best Case Partitioning • The best running time • Pivot element is always be placed at the mid location • It means the input data will be split into 2-partitions of approximately same size • Its recurrence : T(n) <= 2T(n /2)+Θ(n) • The solution : T(n) = O( n lg n) Muhammad Umair

  22. Balanced Partitioning • The average running time • It means the input data will be split into 2-partitions of different size • Its recurrence : T (n)<= T(9n/10) + T (n/10) +Θ(n) • The solution : T (n) = O( n lg n) Muhammad Umair

  23. Summary • Quick Sort • Analysis • Worst Case • Best Case • Balanced Partitioning Muhammad Umair

More Related