Priority Queues and Heaps

1. Priority Queues and Heaps

2. Outline and Reading • PriorityQueue ADT (§8.1) • Total order relation (§8.1.1) • Comparator ADT (§8.1.2) • Sorting with a Priority Queue (§8.1.5) • Implementing a PQ with a list (§8.2) • Selection-sort and Insertion Sort (§8.2.2) • Heaps (§8.3) • Complete Binary Trees (§8.3.2) • Implementing a PQ with a heap (§8.3.3) • Heapsort (§8.3.5)

3. A priority queue stores a collection of items An item is a pair(key, element) Main methods of the Priority Queue ADT insertItem(k, o)inserts an item with key k and element o removeMin()removes the item with the smallest key Additional methods minKey()returns, but does not remove, the smallest key of an item minElement()returns, but does not remove, the element of an item with smallest key size(), isEmpty() Applications: Standby flyers Auctions Priority Queue ADT

4. A comparatorencapsulates the action of comparing two objects according to a given total order relation A generic priority queue uses a comparator as a template argument, to define the comparison function (<,=,>) The comparator is external to the keys being compared. Thus, the same objects can be sorted in different ways by using different comparators. When the priority queue needs to compare two keys, it uses its comparator Comparator ADT

5. A comparator class overloads the “()” operator with a comparison function. Example: Compare two points in the plane lexicographically. Using Comparators in C++ class LexCompare {public:intoperator()(Point a, Point b) {if (a.x < b.x) return –1;else if (a.x > b.x) return +1;else if (a.y < b.y) return –1;else if (a.y > b.y) return +1;else return 0; }};

6. Keys in a priority queue can be arbitrary objects on which an order is defined Two distinct items in a priority queue can have the same key Mathematical concept of total order relation  Reflexive property:x  x Antisymmetric property:x  yy  x  x = y Transitive property:x  yy  z  x  z Total Order Relation

7. PQ-Sort: Sorting with a Priority Queue • We can use a priority queue to sort a set of comparable elements • Insert the elements one by one with a series of insertItem(e, e) operations • Remove the elements in sorted order with a series of removeMin() operations • Running time depends on the PQ implementation Algorithm PQ-Sort(S, C) • Input sequence S, comparator C for the elements of S • Output sequence S sorted in increasing order according to C P  priority queue with comparator C while !S.isEmpty () e  S.remove (S.first()) P.insertItem(e, e) while !P.isEmpty() e  P.minElement() P.removeMin() S.insertLast(e)

8. Unsorted list implementation Store the items of the priority queue in a list-based sequence, in arbitrary order Performance: insertItem takes O(1) time since we can insert the item at the beginning or end of the sequence removeMin, minKey and minElement take O(n) time since we have to traverse the entire sequence to find the smallest key sorted list implementation Store the items of the priority queue in a sequence, sorted by key Performance: insertItem takes O(n) time since we have to find the place where to insert the item removeMin, minKey and minElement take O(1) time since the smallest key is at the beginning of the sequence 4 5 2 3 1 1 2 3 4 5 List-based Priority Queue

9. 4 5 2 3 1 Selection-Sort • Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted sequence • Running time of Selection-sort: • Inserting the elements into the priority queue with ninsertItem operations takes O(n) time • Removing the elements in sorted order from the priority queue with nremoveMin operations takes time proportional to1 + 2 + …+ n • Selection-sort runs in O(n2) time

10. 4 5 2 3 1 Exercise: Selection-Sort • Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted sequence (first n insertItems, then n removeMins) • Illustrate the performance of selection-sort on the following input sequence: • (22, 15, 36, 44, 10, 3, 9, 13, 29, 25)

11. 1 2 3 4 5 Insertion-Sort • Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted sequence • Running time of Insertion-sort: • Inserting the elements into the priority queue with ninsertItem operations takes time proportional to1 + 2 + …+ n • Removing the elements in sorted order from the priority queue with a series of nremoveMin operations takes O(n) time • Insertion-sort runs in O(n2) time

12. 1 2 3 4 5 Exercise: Insertion-Sort • Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted sequence (first n insertItems, then n removeMins) • Illustrate the performance of insertion-sort on the following input sequence: • (22, 15, 36, 44, 10, 3, 9, 13, 29, 25)

13. Instead of using an external data structure, we can implement selection-sort and insertion-sort in-place (only O(1) extra storage) A portion of the input sequence itself serves as the priority queue For in-place insertion-sort We keep sorted the initial portion of the sequence We can use swapElements instead of modifying the sequence 5 4 2 3 1 5 4 2 3 1 4 5 2 3 1 2 4 5 3 1 2 3 4 5 1 1 2 3 4 5 1 2 3 4 5 In-place Insertion-sort

14. 2 5 6 9 7 Heaps and Priority Queues

15. A heap is a binary tree storing keys at its internal nodes and satisfying the following properties: Heap-Order: for every internal node v other than the root,key(v)key(parent(v)) Complete Binary Tree: let h be the height of the heap for i = 0, … , h - 1, there are 2i nodes of depth i at depth h- 1, the internal nodes are to the left of the leaf nodes The last node of a heap is the rightmost internal node of depth h- 1 What is a heap? 2 5 6 9 7 last node

16. Height of a Heap • Theorem: A heap storing nkeys has height O(log n) • Proof: (we apply the complete binary tree property) • Let h be the height of a heap storing n keys • Since there are 2i keys at depth i=0, … , h - 2 and at least one key at depth h - 1, we have n1 + 2 + 4 + … + 2h-2 + 1 • Thus, n2h-1 , i.e., hlog n + 1 depth keys 0 1 1 2 h-2 2h-2 h-1 1

17. Exercise: Heaps • Where may an item with the largest key be stored in a heap? • True or False: • A pre-order traversal of a heap will list out its keys in sorted order. Prove it is true or provide a counter example. • Let H be a heap with 7 distinct elements (1,2,3,4,5,6, and 7). Is it possible that a preorder traversal visits the elements in sorted order? What about an inorder traversal or a postorder traversal? In each case, either show such a heap or prove that none exists.

18. Heaps and Priority Queues • We can use a heap to implement a priority queue • We store a (key, element) item at each internal node • We keep track of the position of the last node • For simplicity, we show only the keys in the pictures (2, Sue) (5, Pat) (6, Mark) (9, Jeff) (7, Anna)

19. Method insertItem of the priority queue ADT corresponds to the insertion of a key k to the heap The insertion algorithm consists of three steps Find the insertion node z (the new last node) Store k at z and expand z into an internal node Restore the heap-order property (discussed next) 2 5 6 9 7 Insertion into a Heap z insertion node 2 5 6 z 9 7 1

20. Upheap • After the insertion of a new key k, the heap-order property may be violated • Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node • Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k • Since a heap has height O(log n), upheap runs in O(log n) time 2 1 5 1 5 2 z z 9 7 6 9 7 6

21. 2 5 6 9 7 Removal from a Heap • Method removeMin of the priority queue ADT corresponds to the removal of the root key from the heap • The removal algorithm consists of three steps • Replace the root key with the key of the last node w • Compress w and its children into a leaf • Restore the heap-order property (discussed next) w last node 7 5 6 w 9

22. 5 7 6 w 9 Downheap • After replacing the root key with the key k of the last node, the heap-order property may be violated • Algorithm downheap restores the heap property by swapping key kwith the child with the smallest key along a downward path from the root • Downheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k • Since a heap has height O(log n), downheap runs in O(log n) time 7 5 6 w 9

23. Updating the Last Node • The insertion node can be found by traversing a path of O(log n) nodes • Go up until a left child or the root is reached • If a left child is reached, go to the right child • Go down left until a leaf is reached • Similar algorithm for updating the last node after a removal

24. Consider a priority queue with n items implemented by means of a heap the space used is O(n) methods insertItem and removeMin take O(log n) time methods size, isEmpty, minKey, and minElement take time O(1) time Using a heap-based priority queue, we can sort a sequence of n elements in O(n log n) time The resulting algorithm is called heap-sort Heap-sort is much faster than quadratic sorting algorithms, such as insertion-sort and selection-sort Heap-Sort

25. 4 5 2 3 1 Exercise: Heap-Sort • Heap-sort is the variation of PQ-sort where the priority queue is implemented with a heap (first n insertItems, then n removeMins) • Illustrate the performance of heap-sort on the following input sequence: • (22, 15, 36, 44, 10, 3, 9, 13, 29, 25)

26. 2 5 6 9 7 Vector-based Heap Implementation • We can represent a heap with n keys by means of a vector of length n • For the node at rank i • the left child is at rank 2i+1 • the right child is at rank 2i +2 • parent is • Links between nodes are not explicitly stored • The leaves are not represented • Operation insertItem corresponds to inserting at rank n • Operation removeMin corresponds to removing at rank n-1 • Yields in-place heap-sort 2 5 6 9 7 0 1 2 3 4

27. Priority Queue Sort Summary • PQ-Sort consists of n insertions followed by n removeMin ops

28. 3 2 8 5 4 6 Merging Two Heaps • We are given two two heaps and a key k • We create a new heap with the root node storing k and with the two heaps as subtrees • We perform downheap to restore the heap-order property 7 3 2 8 5 4 6 2 3 4 8 5 7 6

29. 2i -1 2i -1 Bottom-up Heap Construction • We can construct a heap storing n given keys in using a bottom-up construction with log n phases • In phase i, pairs of heaps with 2i -1 keys are merged into heaps with 2i+1-1 keys 2i+1-1

30. Example 16 15 4 12 6 9 23 20

31. Example 25 5 11 27 16 15 4 12 6 9 23 20

32. Example 15 4 6 20 16 25 5 12 11 9 23 27

33. Example 7 8 15 4 6 20 16 25 5 12 11 9 23 27

34. Example 4 6 15 5 8 20 16 25 7 12 11 9 23 27

35. Example 10 4 6 15 5 8 20 16 25 7 12 11 9 23 27

36. Example 4 5 6 15 7 8 20 16 25 10 12 11 9 23 27

37. Analysis • At each level we insert nodes • Each node can generate h-i swaps

38. Analysis • At each level we insert nodes • Each node can generate h-i swaps

39. Analysis • At each level we insert nodes • Each node can generate h-i swaps

40. Analysis • At each level we insert nodes • Each node can generate h-i swaps

41. Analysis • At each level we insert nodes • Each node can generate h-i swaps

42. Analysis • At each level we insert nodes • Each node can generate h-i swaps

43. Analysis • At each level we insert nodes • Each node can generate h-i swaps • Thus, bottom-up heap construction runs in O(n) time • Bottom-up heap construction is faster than n successive insertions and speeds up the first phase of heap-sort

44. Exercise: Bottom-up Heap • Build a heap from the bottom up on the following input sequence: • (22, 15, 36, 44, 10, 3, 9, 13, 29, 25, 2, 11, 7, 1, 17)

45. a 3 Locators g e 1 4

46. A locator identifies and tracks an item (key, element) within a data structure A locator sticks with a specific item, even if that element changes its position in the data structure Intuitive notion: claim check reservation number Methods of the locator ADT: key(): returns the key of the item associated with the locator element(): returns the element of the item associated with the locator Locators

47. Locator-based priority queue methods: insert(k, o): inserts the item (k, o) and returns a locator for it min(): returns the locator of an item with smallest key remove(l): remove the item with locator l replaceKey(l, k): replaces the key of the item with locator l replaceElement(l, o): replaces with o the element of the item with locator l locators(): returns an iterator over the locators of the items in the priority queue Locator-based dictionary methods: insert(k, o): inserts the item (k, o) and returns its locator find(k): if the dictionary contains an item with key k, returns its locator, else return a special null locator remove(l): removes the item with locator l and returns its element locators(), replaceKey(l, k), replaceElement(l, o) Locator-based Methods

48. Possible Implementation • The locator is an object storing • key • element • position (or rank) of the item in the underlying structure • In turn, the position (or array cell) stores the locator • Example: • binary search tree with locators d 6 a b 3 9 g e c 1 4 8

49. Position represents a “place” in a data structure related to other positions in the data structure (e.g., previous/next or parent/child) often implemented as a pointer to a node or the index of an array cell Position-based ADTs (e.g., sequence and tree) are fundamental data storage schemes Locator identifies and tracks a (key, element) item unrelated to other locators in the data structure often implemented as an object storing the item and its position in the underlying structure Key-based ADTs (e.g., priority queue and dictionary) can be augmented with locator-based methods Positions vs. Locators