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and Heaps Priority Queues 2 5 9 7 6 Heaps and Priority Queues 1 Priority Queue ADT ( 7.1) 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 Heaps and Priority Queues 2 Total Order Relation 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 Heaps and Priority Queues 3 Comparator ADT ( 7.1.4) 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 Heaps and Priority Queues 4 Using Comparators in C++ A comparator class overloads the () operator with a comparison function. Example: Compare two points in the plane lexicographically. class LexCompare { public: 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 ; }; 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; } Heaps and Priority Queues 5 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 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) 6 Heaps and Priority Queues Sequence-based Priority Queue Implementation with an unsorted list 4 5 2 3 1 Implementation with a sorted list 1 2 3 4 5 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 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 Heaps and Priority Queues 7 Selection-Sort Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted sequence 4 5 2 3 1 Running time of Selection-sort: Inserting the elements into the priority queue with n insertItem operations takes O(n) time Removing the elements in sorted order from the priority queue with n removeMin operations takes time proportional to 1 + 2 + + n Selection-sort runs in O(n2) time Heaps and Priority Queues 8 Insertion-Sort Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted sequence 1 2 3 4 5 Running time of Insertion-sort: Inserting the elements into the priority queue with n insertItem operations takes time proportional to 1 + 2 + + n Removing the elements in sorted order from the priority queue with a series of n removeMin operations takes O(n) time Insertion-sort runs in O(n2) time Heaps and Priority Queues 9 What is a heap? ( 7.3.1) 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 Heaps and Priority Queues The last node of a heap is the rightmost internal node of depth h 1 2 5 9 7 6 last node 10 Height of a Heap Theorem: A heap storing n keys 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 h 2 h 1 2 2h 2 1 Heaps and Priority Queues 11 Heaps and Priority Queues We We We For can use a heap to implement a priority queue store a (key, element) item at each internal node keep track of the position of the last node simplicity, we show only the keys in the pictures (2, Sue) (5, Pat) (9, Jeff) (7, Anna) (6, Mark) Heaps and Priority Queues 12 Insertion into a Heap ( 7.3.2) 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 9 7 z 6 insertion node 2 5 9 7 6 z 1 Heaps and Priority Queues 13 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 5 9 7 1 5 9 7 1 2 z 6 z 6 Heaps and Priority Queues 14 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) 2 5 9 7 6 w last node 7 5 9 w 6 Heaps and Priority Queues 15 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 The swapping is done with the sibling with the smallest key 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 9 5 6 9 7 w w 6 Heaps and Priority Queues 16 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 17 Heap-Sort ( 7.3.4) 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 18 Heaps and Priority Queues Merging Two Heaps We are given 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 heaporder property 3 8 5 4 2 6 7 3 8 5 4 2 6 2 3 8 5 7 4 6 19 Heaps and Priority Queues 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 2i 1 2i+1 1 Heaps and Priority Queues 20 Example 16 15 4 12 6 7 23 20 25 16 15 4 5 12 6 11 7 23 27 20 Heaps and Priority Queues 21 Example (contd.) 25 16 15 4 5 12 6 11 9 23 27 20 15 16 25 5 4 12 11 6 9 27 23 20 Heaps and Priority Queues 22 Example (contd.) 7 15 16 25 5 4 12 11 6 9 27 8 23 20 4 15 16 25 7 5 12 11 8 9 6 23 27 20 Heaps and Priority Queues 23 Example (end) 10 4 15 16 25 7 5 12 11 8 9 27 6 23 20 4 5 15 16 25 10 7 12 11 8 9 27 6 23 20 Heaps and Priority Queues 24 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 25 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 9 2 5 7 6 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 2 0 1 5 2 6 3 9 4 7 5 Heaps and Priority Queues 26

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