1. Chapter 15 Heaps

2. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-2 Chapter Objectives Define a heap abstract data structure Demonstrate how a heap can be used to solve problems Examine various heap impmentations Compare heap implementations

3. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-3 Heaps A heap is a binary tree with two added properties: – It is a complete tree – For each node, the node is less than or equal to both the left child and the right child This definition described a minheap

4. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-4 Heaps In addition to the operations inherited from a binary tree, a heap has the following additional operations: – addElement – removeMin – findMin

5. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-5 FIGURE 15.1 The operations on a heap

6. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-6 Listing 15.1

7. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-7 FIGURE 15.2 UML description of the HeapADT

8. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-8 The addElement Operation The addElement method adds a given element to the appropriate location in the heap A binary tree is considered complete if all of the leaves are level h or h-1 where h = log 2 n and n is the number of elements in the tree

9. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-9 The addElement Operation Since a heap is a complete tree, there is only one correct location for the insertion of a new node – Either the next open position from the left at level h – Or the first position in level h+1 if level h is full

10. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-10 The addElement Operation Once we have located the new node in the proper position, then we must account for the ordering property We simply compare the new node to its parent value and swap the values if necessary We continue this process up the tree until either the new value is greater than its parent or the new value becomes the root of the heap

11. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-11 FIGURE 15.3 Two minheaps containing the same data

12. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-12 FIGURE 15.4 Insertion points for a heap

13. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-13 FIGURE 15.5 Insertion and reordering in a heap

14. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-14 The removeMin Operation The removeMin method removes the minimum element from the heap The minimum element is always stored at the root Thus we have to return the root element and replace it with another element

15. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-15 The removeMin Operation The replacement element is always the last leaf The last leaf is always the last element at level h

16. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-16 FIGURE 15.6 Examples of the last leaf in a heap

17. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-17 The removeMin Operation Once the element stored in the last leaf has been moved to the root, the heap will have to reordered This is accomplished by comparing the new root element to the smaller of its children and the swapping them if necessary This process is repeated down the tree until the element is either in a leaf or is less than both of its children

18. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-18 FIGURE 15.7 Removal and reordering in a heap

19. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-19 Using Heaps: Heap Sort Given the ordering property of a heap, it is natural to think of using a heap to sort a list of objects One approach would be to simply add all of the objects to a heap and then remove them one at a time in ascending order

20. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-20 Using Heaps: Heap Sort Insertion into a heap is O(log n) for any given node and thus would O(n log n) for n nodes to build the heap However, it is also possible to build a heap in place using an array Since we know the relative position of each parent and its children in the array, we simply start with the first non-leaf node in the array, compare it to its children and swap if necessary

21. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-21 Using Heaps: Heap Sort However, it is also possible to build a heap in place using an array Since we know the relative position of each parent and its children in the array, we simply start with the first non-leaf node in the array, compare it to its children and swap if necessary

22. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-22 Using Heaps: Heap Sort We then work backward in the array until we reach the root Since at most, this will require us to make two comparisons for each non-leaf node, this approach is O(n) to build the heap We will revisit the analysis of HeapSort at the end of the chapter

23. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-23 Using Heaps: Priority Queue A priority queue is a collection that follows two ordering rules: – Items which have higher priority go first – Items with the same priority use a first in, first out method to determine their ordering A priority queue could be implemented using a list of queues where each queue represents items of a given priority

24. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-24 Using Heaps: Priority Queue Another solution is to use a minheap Sorting the heap by priority accomplishes the first ordering However, the first in, first out ordering for items with the same priority has to be manipulated

25. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-25 Using Heaps: Priority Queue The solution is to create a PriorityQueueNode object that stores the element to be placed on the queue, the priority of the element and the arrival order of the element Then we simply define a compareTo method for the PriorityQueueNode class that first compares priority then arrival time The PriorityQueue class then extends the Heap class and stores PriorityQueueNodes

26. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-26 Listing 15.2

27. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-27 Listing 15.2 (cont.)

28. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-28 Listing 15.2 (cont.)

29. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-29 Listing 15.2 (cont.)

30. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-30 Listing 15.3

31. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-31 Listing 15.3 (cont.)

32. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-32 Listing 15.3 (cont.)

33. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-33 Implementing Heaps with Links A linked implementation of a minheap would simply be an extension of our LinkedBinaryTree class However, since we need each node to have a parent pointer, we will create a HeapNode class to extend our BinaryTreeNode class we used earlier

34. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-34 Listing 15.4

35. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-35 Implementing Heaps with Links The addElement method must accomplish three tasks: – Add the new node at the appropriate location – Reorder the heap – Reset the lastNode pointer to point to the new last node

36. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-36 LinkedHeap - the addElement Operation

37. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-37 LinkedHeap - the addElement Operation

38. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-38 LinkedHeap - the addElement Operation The addElement operation makes use of two private methods: – getNextParentAdd that returns a reference to the node that will be the parent of the new node – heapifyAdd that reorders the heap after the insertion

39. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-39 LinkedHeap - the addElement Operation

40. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-40 LinkedHeap - the addElement Operation

41. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-41 LinkedHeap - the removeMin Operation The removeMin operation must accomplish three tasks: – Replace the element stored in the root with the element stored in the last leaf – Reorder the heap if necessary – Return the original root element

42. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-42 LinkedHeap - the removeMin Operation

43. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-43 LinkedHeap - the removeMin Operation

44. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-44 LinkedHeap - the removeMin Operation Like the addElement operation, the removeMin operation makes use of two private methods – getNewLastNode that returns a reference to the new last node in the heap – heapifyRemove that reorders the heap after the removal

45. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-45 LinkedHeap - the removeMin Operation

46. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-46 LinkedHeap - the removeMin Operation

47. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-47 LinkedHeap - the removeMin Operation

48. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-48 Implementing Heaps with Arrays An array implementation of a heap may provide a simpler alternative In an array implementation, the location of parent and child can always be calculated Given that the root is in postion 0, then for any given node stored in position n of the array, its left child is in position 2n + 1 and its right child is in position 2(n+1) This means that its parent is in position (n- 1)/2

49. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-49 Implementing Heaps with Arrays Like the linked version, the addElement operation for an array implementation of a heap must accomplish three tasks: – Add the new node, – Reorder the heap, – Increment the count by one The ArrayHeap version of this method only requires one private method, heapifyAdd which reorders the heap after the insertion

50. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-50 ArrayHeap - the addElement Operation

51. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-51 ArrayHeap - the addElement Operation

52. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-52 ArrayHeap - the removeMin Operation The removeMin operation must accomplish three tasks: – Replace the element stored at the root with the element stored in the last leaf – Reorder the heap as necessary – Return the original root element Like the addElement operation, the removeMin operation makes use of a private method, heapifyRemove to reorder the heap

53. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-53 ArrayHeap - the removeMin Operation

54. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-54 ArrayHeap - the removeMin Operation

55. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-55 ArrayHeap - the removeMin Operation

56. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-56 Analysis of Heap Implementations The addElement operation is O(log n) for both implementations The removeMin operation is O(log n) for both implementations The findMin operation is O(1) for both implementations

57. Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 15-57 Analysis of Heap Implementations One might conclude then that HeapSort is O(log n) since both adding and removing elements is O(log n) Keep in mind that for HeapSort, we must perform both operations for each of n elements Thus the time complexity of HeapSort is 2*n*log n or O(n log n)