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Algorithms

Algorithms. Sorting Problem Ch 6: Heapsort Ming-Te Chi. Why sorting ?. 1. Sometimes the need to sort information is inherent in an application. 2. Algorithms often use sorting as a key subroutine. 3. There is a wide variety of sorting algorithms, and they use rich set of techniques.

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Algorithms

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  1. Algorithms Sorting Problem Ch 6: Heapsort Ming-Te Chi Ch6 Heapsort

  2. Why sorting ? 1. Sometimes the need to sort information is inherent in an application. 2. Algorithms often use sorting as a key subroutine. 3. There is a wide variety of sorting algorithms, and they use rich set of techniques. 4. Sorting problem has a nontrivial lower bound 5. Many engineering issues come to fore when implementing sorting algorithms. Ch6 Heapsort

  3. Sorting algorithm • Insertion sort : • Worst case in O(n2) • In place: only a constant number of elements of the input array are even sorted outside the array. • Merge sort : • Worst case in O(n lg n) • not in place. • Heap sort : (Chapter 6) • Sorts n numbers in place in O(n lg n) Ch6 Heapsort

  4. Sorting algorithm • Heap sort : (Chapter 6) • O(n lg n)worst case—like merge sort. • Sorts in place—like insertion sort. • Combines the best of both algorithms. • Introduces another algorithm design technique: the use of data structure. Ch6 Heapsort

  5. Sorting algorithm (preview) • Quick sort : (chapter 7) • worst time complexity O(n2) • Average time complexity O(n logn) • Decision tree model : (chapter 8) • Lower bound O(n logn) • Counting sort • Radix sort • Bucket sort • Order statistics Ch6 Heapsort

  6. Heap data structure • Heap(not garbage-collected storage) is a nearly complete binary tree. • Heightof node = # of edges on a longest simple path from the node down to a leaf • Heightof tree = height of root • Heightof heap = height of root = (log n) Ch6 Heapsort

  7. Heap data structure • A heap can be stored as an array A. • Root of tree is A[1]. • Parent of A[i ] = A[ ]. • Left child of A[i ] = A[2i ]. • Right child of A[i ] = A[2i + 1]. • Computing is fast with binary representation implementation. Ch6 Heapsort

  8. Remarks about a TREE Depth d 0 (root) 1 2 3 (leaves) Height h(of a node) 3 2 1 0 Ch6 Heapsort

  9. 6.1 Heaps (Binary heap) • Thebinary heap data structure is an array object that can be viewed as a complete tree. Parent(i) return LEFT(i) return2i  Right(i) return2i+1 Ch6 Heapsort

  10. Heap property • Two kinds of heap: Max-heap and Min-heap • Max-heap property: • A[parent(i)]  A[i] (Every node i other than the root) • The value of the node is at most the value of its parent. • Min-heap property: • A[parent(i)] ≤A[i] (Every node i other than the root) • The smallest element in a min-heap is the root. Ch6 Heapsort

  11. Heap property • Max-heap is used in heapsort algorithm. • Min-heaps are commonly used in priority queues. • Basic operations on heaps run in time proportional to the height of the tree • O(lg n) time Ch6 Heapsort

  12. Basic procedures on Max-heap • Max-Heapify procedure • O(lg n) , key to maintain the max-heap property • Build-Max-Heap procedure • O(n) , produces a max-heap from an unordered input array. • Heapsort procedure • O(n lg n) , sorts an array in place • Max-Heap-Insert, Heap-Extract-Max, Heap-Increase-Key, and Heap-Maximum procedures • O(lg n) , allow the heap data structure to be used as a priority queue. Ch6 Heapsort

  13. 6.2 Maintaining the heap property • MAX-HEAPIFY is used to maintain the max-heap property. • Before MAX-HEAPIFY, A[i ] may be smaller than its children. • Assume left and right subtrees of i are max-heaps. • After MAX-HEAPIFY, subtree rooted at i is a max-heap. Ch6 Heapsort

  14. MAX-HEAPIFY • Input: • Array • An index i into the array (root of a subtree.) • Find the largest element in the heap subtree. • If it is not the root of the subtree • Exchange it with the root of the subtree • The subtree rooted at largestmay not be a max-heap • Recursively call the MAX-HEAPIFY Ch6 Heapsort

  15. Ch6 Heapsort

  16. Max-Heapify(A,2) heap-size[A] = 10 Ch6 Heapsort

  17. Time Complexity of MAX-HEAPIFY • Running time of Max-heapify on a subtree of size n rooted at given node i • O(1) to fix up relationships among the elements A[i ] , A[LEFT(i )] , and A[RIGHT(i )] • Time to run Max-heapify on a subtree rooted at one of the children of node i . • The children’s subtrees each have size at most 2n/3. • The worst case occurs when the last row of the tree is exactly half full. Ch6 Heapsort

  18. Time Complexity of MAX-HEAPIFY • Running time of Max-heapify (case 2 of the master theorem) • Alternatively O(h) (h: height of the node) Ch6 Heapsort

  19. 6.3 Building a heap • The elements in the subarray A[(length[A]/2) … n ] are all leaves of the tree. • Each one is a 1-element heap • No need to exam these elements • The Build-max-heap procedure exams the remaining nodes of the tree and applies MAX-HEAPIFY on each one. Ch6 Heapsort

  20. (given an unordered array, will produce a max-heap.) Ch6 Heapsort

  21. Ch6 Heapsort

  22. Ch6 Heapsort

  23. Ch6 Heapsort

  24. Analysis • Simple bound: (A good approach to analysis in general is to start by proving easy bound, then try to tighten it.) • O(n) calls to MAX-HEAPIFY • Each of which takes O( lg n) time ⇒ O(n lg n). Ch6 Heapsort

  25. Analysis • Tighter analysis: • Time to run MAX-HEAPIFY is linear in the height of the node it’s run on, and most nodes have small heights. • n-element heap: • height • has ≤ ceiling(n/2h+1) nodes of height h • Time required by MAX-HEAPIFY when called on a node of height h is O(h). Ch6 Heapsort

  26. n/2h+1 analysis n = total number of nodes = 20 + 21 + 22 + 23 + 24 + 25 = 26 -1 = 63 Ch6 Heapsort

  27. Analysis─ Tighter analysis(continue) • The running time of BUILD-MAX-HEAP is O(n) Ch6 Heapsort

  28. 6.4 The Heapsort algorithm • Given an input array, the heapsort algorithm acts as follows: • Builds a max-heap from the array. • Starting with the root (the maximum element), the algorithm places the maximum element into the correct place in the array by swapping it with the element in the last position in the array. • “Discard” this last node (knowing that it is in its correct place) by decreasing the heap size, and calling MAX-HEAPIFY on the new (possibly incorrectly-placed) root. • Repeat this “discarding” process until only one node (the smallest element) remains, and therefore is in the correct place in the array. Ch6 Heapsort

  29. 6.4 The Heapsort algorithm Ch6 Heapsort

  30. The operation of Heapsort 16, 14, 10, 8, 7, 9, 3, 2, 4, 1 Ch6 Heapsort

  31. Analysis: O(n logn) Ch6 Heapsort

  32. Analysis • BUILD-MAX-HEAP: O(n) • for loop: n - 1 times, O(n) • exchange elements: O(1) • MAX-HEAPIFY: O(lg n) • Total time (for loop): O(n lg n). • Total time (heapsort): O(n lg n). Ch6 Heapsort

  33. Analysis • Though heapsort is a great algorithm, a well-implemented quicksort usually beats it in practice. • Heap data structure is useful. • Priority queue is one of the most popular applications of a heap. Ch6 Heapsort

  34. 6.5 Priority queues • A queue is a data structure with the FIFO property. • A priority queue is a queue and each element of it has an associated value call a key (the priority). • Insertion: inserts an element to the queue according to its priority. • Removal: removes the element from the queue also according to its priority. • Max-priority queue vs. min-priority queue. Ch6 Heapsort

  35. Priority queues • Max-priority queue: • Based on max-heap • Supports the following operations: insert,maximum, extract-max, andincrease-key. • Example max-priority queue application: • Schedule jobs on shared computer • Greedy search • Example min-priority queue application: • Event-driven simulator. Ch6 Heapsort

  36. Operations • Insert(S, x) O(log n) Inserts the element x into a set S • Maximum (S) O(1) Returns the element of S with the largest key • Extract-Max (S) O(log n) Removes and returns the elements of S with the largest key • Increase-Key (S, x, k) O(log n) Increases the value of element x’s key to the new value k. (assume that k is no less than x’s current key value) Ch6 Heapsort

  37. Heap-Maximum(A ) Time O(1) Ch6 Heapsort

  38. Heap_Extract-Max(A ) Why do we need max-heapify at step 6? Ch6 Heapsort

  39. Analysis: Heap_Extract-Max(A ) • Constant time assignments plus time for MAX-HEAPIFY. • Time required:O(lg n). Ch6 Heapsort

  40. Heap-Increase-Key (A, i, key ) Ch6 Heapsort

  41. Analysis: Heap-Increase-Key (A, i, key ) • Upward path from node i has length O(lg n)in an n-element heap. • Time required:O(lg n). Ch6 Heapsort

  42. Heap-Increase-Key (key = 15) Ch6 Heapsort

  43. Max_Heap_Insert(A, key ) Ch6 Heapsort

  44. Analysis: Max_Heap_Insert(A, key) • Constant time assignments + time for HEAP-INCREASE-KEY. • Time required:O(lg n). Ch6 Heapsort

  45. Summary • A heap gives a good compromise between fast insertion but slow extraction. Both operations take O(lg n) time. • A heap can support any priority-queue operation on a set of size n in O(lg n ) time. • Min-priority queue operations are implemented similarly with min-heaps. Ch6 Heapsort

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