Implementing a Priority Queue with Heaps

"Learn about implementing a priority queue using heaps. Priority queues are essential data structures that maintain a special ordering property. The use of binary trees and heaps is explained in detail, focusing on maintaining shape and heap properties during insertions."

• Data Structure
• Priority Queue
• Binary Tree
• Heaps

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1. Implementing a Priority Queue Heaps

2. Priority Queues Recall: 3 Container Adapters Stack Queue Priority Queue implementation? Priority Queue Uses std::vector Maintains complete tree w/special ordering property Vector w/property is Heap

3. Binary Trees V > V <= V

4. Heaps (Max Heap) V <= V <= V

5. Binary Tree Heap vs. 6 13 9 3 3 9 1 3 1 6 7 7 1

6. Heap Shape Property A heap must be a complete binary tree This means the levels of the tree will always be filled from left-to-right Level 0 13 9 3 Level 1 6 7 1 Level 2

7. Insertion 13 9 3 6 7 1

8. Insertion 13 9 3 To maintain the Shape property, we must insert at the end of level 2 6 7 1 8

9. Insertion 13 9 3 But if we insert the 8 there the heap ordering property isn t maintained! 8 <= 3 6 7 1 8

10. Insertion 13 8 <= 13 9 8 So we do the only sensible thing and swap the values! 3 <= 8 6 7 1 3

11. Insertion 13 9 8 This strategy applies for more than one step , too 6 7 1 3 1 8

12. Insertion 13 18 <= 7 9 8 6 7 1 3 1 8

13. Insertion 13 18 <= 9 9 8 1 8 6 1 3 7

14. Insertion 13 1 8 18 <= 13 8 6 9 1 3 7

15. Insertion 18 13 <= 18 8 13 6 9 1 3 7

16. Insertion Done 18 The 13 node bubbled up from the bottom to its final position 8 13 This operation is known as upheap() 6 9 1 3 while n.data > parent(n).data: swap(n.data, parent(n).data) n = parent(n) if n == null: break 7

17. Removal 18 We can only remove from the top We need to maintain the shape property We need to maintain ordering property 8 13 6 9 1 3 7

18. Removal 7 Swap the root with the last We compare the node with its two children Choose the larger one and swap Repeat until in place 8 13 6 9 1 3 18 removed

19. Removal 13 Swap the root with the last We compare the node with its two children Choose the larger one and swap Repeat until in place 7 8 6 9 1 3 18 removed

20. Removal 13 Swap the root with the last We compare the node with its two children Choose the larger one and swap Repeat until in place 9 8 6 7 1 3 1 8 removed

21. Removal 13 The 7 node bubbled down from the top to its final position 9 8 This operation is known as downheap() 6 7 1 3 18 removed

22. Heaps Are Arrays 0 13 parent(i) = (i 1) / 2 9 8 1 2 left(i) = 2 * i + 1 right(i) = 2 * i + 2 6 4 7 1 3 3 5 6

23. Complete Binary Tree Why store tree in vector? 5 v[0] 1 3 v[2] v[1] 6 9 2 4 v[4] v[3] v[5] v[6] 8 7 0 parent (i) = p (i) = ? leftChild (i) = lc (i) = ? rightChild (i) = rc (i) = ? v[7] v[8] v[9]

24. Max and Min Heaps Max Heap property: ( nodes X) [ Value (X) >= Value (Child (X)) ] 55 40 52 30 50 15 11 10 10 25 5 20 22 (A) Maximum Heap (9 nodes) (B) Maximum Heap (4 nodes) 10 5 30 15 50 10 40 52 11 20 55 25 22 (C) Minimum Heap (9 nodes) (D) Minimum Heap (4 nodes)

25. priority_queue::push 63 63 v[0] v[0] 30 40 30 40 v[2] v[1] v[2] v[1] 25 8 38 10 25 8 38 10 v[4] v[3] v[5] v[6] v[4] v[3] v[5] v[6] 18 5 3 18 5 3 50 v[7] v[8] v[9] v[7] v[8] v[9] v[10] (a) (b) v.push_back (50);

26. Upheap (restore heap property) upHeap (v.size () 1); 63 63 63 v[0] v[0] . . . . . . v[0] 30 50 . . . 50 v[1] v[1] . . . . . . v[1] 50 . . . 30 30 v[4] v[4] v[4] 18 25 18 25 18 25 v[9] v[9] v[10] v[10] v[9] v[10] Step 1 Compare 50 and 25 (Exchange v[10] and v[4]) Step 2 Compare 50 and 30 (Exchange v[4] and v[1]) Step 3 Compare 50 and 63 (50 in correct location)

27. Push void PQ::push (const T& item) { v.push_back (item); upHeap (v.size () 1); }

28. upHeap (Helper Method) void PQ::upHeap (size_t pos) { T item = v[pos]; size_t i; // Move parent down for (i = pos; i != 0 && item > v[p (i)]; i = p (i)) v[i] = v[p (i)]; // swap unnecessary v[i] = item; }

29. priority_queue::pop v.front () = v.back (); v.pop_back (); 18 63 v[0] v[0] 30 40 30 40 v[2] v[2] v[1] v[1] 25 8 38 25 10 8 38 10 v[4] v[3] v[5] v[6] v[4] v[3] v[5] v[6] 63 18 5 3 18 5 3 v[7] v[8] v[9] v[7] v[8] v[9] After exchanging the root and last element in the heap Before a deletion

30. downHeap 40 40 v[0] v[0] . . . . . . 38 18 v[2] v[2] 8 18 8 38 v[5] v[6] v[5] v[6] Step 2: Exchange 18 and 38 Step 1: Exchange 18 and 40

31. Pop void PQ::pop () { v[0] = v.back (); v.pop_back (); // O (1) // Move elem down to proper place downHeap (0); }

32. downHeap (Helper Method) void PQ::downHeap (size_t pos) { // Move v[pos] down, max child up size_t i, mc; T val = v[pos]; for (i = pos; (mc = lc (i)) < v.size (); i = mc) { if (mc + 1 < v.size () && v[mc] < v[mc + 1]) ++mc; if (val ??) // Move child up if necessary else ? } // Place val in correct spot }

33. Heap Sort Overview Heaps O(N*lg (N)) sort in worst case Can use heaps to sort in two ways 1) pqSort Push all elements Pop and place in vector back to front Complexity? 2*Sum (i =1:N) [ lg(i) ]

34. Heap Sort Overview 2) True heap sort (better, why?) Heapify vector (O(N)) With STL assistance (make_heap) Roll our own buildHeap elemsToPlace = v.size () - 1 while (elemsToPlace > 0) Swap front and back of v --elemsToPlace Restore heap property at root (taking into consideration heap is one elem. smaller)

35. make_heap (STL) int arr[] { 50, 20, 75, 35, 25 }; // Header <algorithm> make_heap (arr, arr + 5); 75 35 50 25 20 Heapified Tree

36. buildHeap Potential algorithms (several don t work!) Call downHeap (0) N times Call downHeap (i) varying i from 0 to N-1 Call downHeap (i) varying i from N-1 to 0 Call upHeap (i) varying i from 0 to N-1 Call upHeap (i) varying i from N-1 to 0

37. buildHeap 9 17 12 20 30 50 60 65 4 19 Initial Vector Start at last internal node: position = ? 9 17 12 Then do downHeap (position) 20 30 50 60 65 4 19 adjustHeap() at 4 causes no changes (A)

38. buildHeap Contd 9 9 60 12 17 12 20 65 50 17 20 65 50 60 30 4 19 30 4 19 adjustHeap() at 2 moves 17 down (C) adjustHeap() at 3 moves 30 down (B) 65 9 60 50 60 65 20 30 19 17 20 30 50 17 12 4 9 12 4 19 adjustHeap() at 0 moves 9 down three levels (E) adjustHeap() at 1 moves 12 down two levels (D)

39. Heap Sorted Vector Now convert heap to sorted vector Swap front and back Decrease size of heap by 1 downheap (0)

40. Heap Sorted Vector 0 1 2 3 4 5 6 7 a[ ]: 10 9 6 7 2 4 5 3 Heap 10 Heap is in a[0..7] and the sorted region is empty 9 6 Swap front and back 7 2 4 5 3

41. Heap Sorted Vector 0 1 2 3 4 5 6 7 a[ ]: 3 9 6 7 2 4 5 10 Sorted Semiheap downHeap a[0..6] now represents a semiheap 3 a[7] is the sorted region 9 6 7 2 4 5

42. Heap Sorted Vector 0 1 2 3 4 5 6 7 a[ ]: 9 3 6 7 2 4 5 10 Sorted Becoming a Heap downHeap 9 3 6 7 2 4 5

43. Heap Sorted Vector 0 1 2 3 4 5 6 7 a[ ]: 9 7 6 3 2 4 5 10 Sorted Heap 9 7 6 3 2 4 5

44. Heap Sorted Vector 0 1 2 3 4 5 6 7 a[ ]: 5 7 6 3 2 4 9 Sorted 10 Semiheap downHeap 5 7 6 3 2 4

45. Heap Sorted Vector 0 1 2 3 4 5 6 7 a[ ]: 7 5 6 3 2 4 9 Sorted 10 Heap downHeap 7 5 6 3 2 4

46. Heap Sorted Vector 0 1 2 3 4 5 6 7 a[ ]: 4 5 6 3 2 7 9 10 Semiheap Sorted downHeap 4 5 6 3 2

47. Heap Sorted Vector 0 1 2 3 4 5 6 7 a[ ]: 6 5 4 3 2 7 9 10 Heap Sorted 6 5 4 3 2

48. Heap Sorted Vector 0 1 2 3 4 5 6 7 a[ ]: 2 5 4 3 6 7 Sorted 9 10 Semiheap downHeap 2 5 4 3

49. Transform a Heap Into a Sorted Array: Example 0 1 2 3 4 5 6 7 a[ ]: 5 2 4 3 6 7 Sorted 9 10 Becoming a Heap downHeap 5 2 4 3

50. Heap Sorted Vector 0 1 2 3 4 5 6 7 a[ ]: 5 3 4 2 6 7 Sorted 9 10 Heap 5 3 4 2

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