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Heaps and Priority Queues

2. 5. 6. 9. 7. Heaps and Priority Queues. 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.

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Heaps and Priority Queues

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  1. 2 5 6 9 7 Heaps and Priority Queues Heaps and Priority Queues

  2. 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(k, o)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 Stock market Priority Queue ADT (§7.1) Heaps and Priority Queues

  3. 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 Heaps and Priority Queues

  4. A comparator encapsulates 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 (§7.1.4) Heaps and Priority Queues

  5. A comparator class overloads the “()” operator with a comparison function. Example: Compare two points in the plane lexicographically.class LexCompare {public:int operator()(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; }}; Using Comparators in C++ To use the comparator, define an object of this type, and invoke it using its “()” operator: Example of usage:Point p(2.3, 4.5);Point q(1.7, 7.3);LexCompare lexCompare;if (lexCompare(p, q) < 0) cout << “p less than q”;else if (lexCompare(p, q) == 0) cout << “p equals q”;else if (lexCompare(p, q) > 0) cout << “p greater than q”; Heaps and Priority Queues

  6. Sorting with a Priority Queue (§7.1.2) • 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 • The running time of this sorting method depends on the priority queue implementation AlgorithmPQ-Sort(S, C) • Inputsequence S, comparator C for the elements of S • Outputsequence 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) Heaps and Priority Queues

  7. Implementation with an unsorted list 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 Implementation with a sorted list 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 Sequence-based Priority Queue Heaps and Priority Queues

  8. 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 Heaps and Priority Queues

  9. 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 Heaps and Priority Queues

  10. 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 external nodes The last node of a heap is the rightmost internal node of depth h- 1 What is a heap? (§7.3.1) 2 5 6 9 7 last node Heaps and Priority Queues

  11. 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 Heaps and Priority Queues

  12. 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) Heaps and Priority Queues

  13. 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 (§7.3.2) z insertion node 2 5 6 z 9 7 1 Heaps and Priority Queues

  14. 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 Heaps and Priority Queues

  15. 2 5 6 9 7 Removal from a Heap (§7.3.2) • 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 Heaps and Priority Queues

  16. 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-order property by swapping key k along a downward path from the root • Upheap 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 Heaps and Priority Queues

  17. 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 Heaps and Priority Queues

  18. Implementing a Priority Queue with a Heap comp last < + > heap (4,C) (5,A) (6,Z) (15,K) (9,F) (7,Q) (20,B) (16,X) (25,,j) (14,E) (12,H) (11,S) (18,W) Heaps and Priority Queues

  19. Insertion (4,C) (5,A) (6,Z) (15,K) (9,F) (7,Q) (20,B) (16,X) (25,,j) (14,E) (12,H) (11,S) (18,W) Heaps and Priority Queues

  20. Insertion (4,C) (5,A) (6,Z) (15,K) (9,F) (7,Q) (20,B) (16,X) (25,,j) (14,E) (12,H) (11,S) (18,W) (2,T) Heaps and Priority Queues

  21. Insertion (4,C) (5,A) (6,Z) Swap (2,T) and (20,B) (15,K) (9,F) (7,Q) (20,B) (16,X) (25,,j) (14,E) (12,H) (11,S) (18,W) (2,T) Heaps and Priority Queues

  22. Insertion (4,C) Swap (2,T) and (6,Z) (5,A) (6,Z) (15,K) (9,F) (7,Q) (2,T) (16,X) (25,,j) (14,E) (12,H) (11,S) (18,W) (20,B) Heaps and Priority Queues

  23. Insertion Swap (2,T) and (4,C) (4,C) (5,A) (2,T) (15,K) (9,F) (7,Q) (6,Z) (16,X) (25,,j) (14,E) (12,H) (11,S) (18,W) (20,B) Heaps and Priority Queues

  24. Insertion In O (log n)  Tree is complete (2,T) (5,A) (2,T) (15,K) (9,F) (7,Q) (6,Z) (16,X) (25,,j) (14,E) (12,H) (11,S) (18,W) (20,B) Heaps and Priority Queues

  25. Removal (removeMin( )) (4,C) (5,A) (6,Z) (18,W) (15,K) (9,F) (7,Q) (20,B) (16,X) (25,,j) (14,E) (12,H) (11,S) Heaps and Priority Queues

  26. Removal (removeMin( )) Swap (18,W) and (5,A) (18,W) (5,A) (6,Z) (15,K) (9,F) (7,Q) (20,B) (16,X) (25,,j) (14,E) (12,H) (11,S) Heaps and Priority Queues

  27. Removal (removeMin( )) (5,A) Swap (18,W) and (9,F) (18,W) (6,Z) (15,K) (9,F) (7,Q) (20,B) (16,X) (25,,j) (14,E) (12,H) (11,S) Heaps and Priority Queues

  28. Removal (removeMin( )) (5,A) (9,F) (6,Z) (15,K) Swap (18,W) and (12,H) (18,W) (7,Q) (20,B) (16,X) (25,,j) (14,E) (12,H) (11,S) Heaps and Priority Queues

  29. Removal (removeMin( )) Last node (5,A) (9,F) (6,Z) (15,K) (12,H) (7,Q) (20,B) (16,X) (25,,j) (14,E) (18,W) (11,S) Heaps and Priority Queues

  30. Complexity Space Complexity If linked structure then O (n) If the vector structure is used then it is proportional to N ( the size of the array) Time complexity: Heaps and Priority Queues

  31. 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 (§7.3.4) Heaps and Priority Queues

  32. 2 5 6 9 7 2 5 6 9 7 0 1 2 3 4 5 Vector-based Heap Implementation (§7.3.3) • We can represent a heap with n keys by means of a vector of length n +1 • For the node at rank i • the left child is at rank 2i • the right child is at rank 2i +1 • Links between nodes are not explicitly stored • The leaves are not represented • The cell of at rank 0 is not used • Operation insertItem corresponds to inserting at rank n +1 • Operation removeMin corresponds to removing at rank n • Yields in-place heap-sort Heaps and Priority Queues

  33. 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 Heaps and Priority Queues

  34. 2i -1 2i -1 Bottom-up Heap Construction (§7.3.5) • 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 Heaps and Priority Queues

  35. Example 16 15 4 12 6 7 23 20 25 5 11 27 20 16 15 4 12 6 7 23 Heaps and Priority Queues

  36. Example (contd.) 25 5 11 27 16 15 4 12 6 9 23 20 15 4 6 20 16 25 5 12 11 9 23 27 Heaps and Priority Queues

  37. Example (contd.) 7 8 15 4 6 23 16 25 5 12 11 9 27 20 4 6 15 5 8 20 16 25 7 12 11 9 23 27 Heaps and Priority Queues

  38. Example (end) 10 4 6 15 5 8 23 16 25 7 12 11 9 27 20 4 5 6 15 7 8 20 16 25 10 12 11 9 23 27 Heaps and Priority Queues

  39. Analysis • We visualize the worst-case time of a downheap with a proxy path that goes first right and then repeatedly goes left until the bottom of the heap (this path may differ from the actual downheap path) • Since each node is traversed by at most two proxy paths, the total number of nodes of the proxy paths is O(n) • 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 Heaps and Priority Queues

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