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Sell 100 IBM $122 Sell 300 IBM $120 Buy 500 IBM $119 Buy 400 IBM $118 Chapter 8: Priority Queues and Heaps Nancy Amato Parasol Lab, Dept. CSE, Texas A&M University Acknowledgement: These slides are adapted from slides provided with Data Structures and Algorithms in C++, Goodrich, Tamassia and Mount (Wiley 2004) http://parasol.tamu.edu 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) Priority Queues Trees 2 4/8/2015 5:25 AM Priority Queue ADT • Additional methods • A priority queue stores a collection of items • An item is a pair (key, element) • Main methods of the Priority Queue ADT • 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() • insertItem(k, o) inserts an item with key k and • Applications: element o • Standby flyers • removeMin() • Auctions removes the item with the • Stock market smallest key Priority Queues Trees 3 4/8/2015 5:25 AM Comparator ADT • 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 Priority Queues Trees 4 4/8/2015 5:25 AM 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: 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; } }; Priority Queues Trees 5 4/8/2015 5:25 AM Total Order Relation • Keys in a priority • Mathematical concept of queue can be total order relation arbitrary objects on • Reflexive property: which an order is xx defined • Antisymmetric property: • Two distinct items in xyyxx=y a priority queue can • Transitive property: have the same key xyyzxz Priority Queues Trees 6 4/8/2015 5:25 AM 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) Priority Queues Trees 7 4/8/2015 5:25 AM List-based Priority Queue • Unsorted list implementation • sorted list implementation • Store the items of the priority queue in a list-based sequence, in arbitrary order 4 5 2 3 1 • Store the items of the priority queue in a sequence, sorted by key 1 2 3 4 5 • Performance: • 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 • 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 Priority Queues Trees 8 4/8/2015 5:25 AM 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 Priority Queues Trees 9 4/8/2015 5:25 AM 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) 4 5 2 3 1 • Illustrate the performance of selection-sort on the following input sequence: • (22, 15, 36, 44, 10, 3, 9, 13, 29, 25) Priority Queues Trees 10 4/8/2015 5:25 AM Insertion-Sort • Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted sequence 1 2 3 4 Running time of Insertion-sort: • • 5 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 Priority Queues Trees 11 4/8/2015 5:25 AM 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) 4 5 2 3 1 • Illustrate the performance of insertion-sort on the following input sequence: • (22, 15, 36, 44, 10, 3, 9, 13, 29, 25) Priority Queues Trees 12 4/8/2015 5:25 AM In-place Insertion-sort • Instead of using an external data structure, we can implement selectionsort and insertion-sort inplace (only O(1) extra storage) • A portion of the input sequence itself serves as the priority queue • For in-place insertion-sort 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 2 3 4 5 2 3 • We keep sorted the initial portion of the sequence 1 • We can use swapElements instead of modifying the sequence 1 Priority Queues Trees 4 5 13 4/8/2015 5:25 AM Heaps and Priority Queues 2 5 9 6 7 Priority Queues Trees 14 4/8/2015 5:25 AM What is a heap? • 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 2 5 9 Priority Queues Trees 6 7 last node 15 4/8/2015 5:25 AM 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 2 h-2 2h-2 h-1 1 Priority Queues Trees 16 4/8/2015 5:25 AM 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. Priority Queues Trees 17 4/8/2015 5:25 AM 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) (9, Jeff) (6, Mark) (7, Anna) Priority Queues Trees 18 4/8/2015 5:25 AM Insertion into a Heap • 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 6 z 7 insertion node 2 5 9 Priority Queues Trees 6 7 z 1 19 4/8/2015 5:25 AM 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 9 1 7 z 6 5 9 Priority Queues Trees 2 7 z 6 20 4/8/2015 5:25 AM 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) 2 5 6 w 9 7 last node 7 5 w 6 9 Priority Queues Trees 21 4/8/2015 5:25 AM 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 w 5 6 9 7 w 6 9 Priority Queues Trees 22 4/8/2015 5:25 AM 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 Priority Queues Trees 23 4/8/2015 5:25 AM Heap-Sort • 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 Priority Queues Trees 24 4/8/2015 5:25 AM 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) 4 5 2 3 1 • Illustrate the performance of heap-sort on the following input sequence: • (22, 15, 36, 44, 10, 3, 9, 13, 29, 25) Priority Queues Trees 25 4/8/2015 5:25 AM Vector-based Heap Implementation • 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 2 5 6 9 • Links between nodes are not explicitly stored • The leaves are not represented • The cell of at rank 0 is not used • Operation insertItem corresponds 0 to inserting at rank n + 1 • Operation removeMin corresponds to removing at rank n • Yields in-place heap-sort Priority Queues Trees 7 2 5 6 9 7 1 2 3 4 5 26 4/8/2015 5:25 AM Priority Queue Sort Summary • PQ-Sort consists of n insertions followed by n removeMin ops Vectors Insert Remove PQ-Sort Min total Insertion Sort (ordered sequence) O(n) O(1) O(n2) Selection Sort (unordered sequence) O(1) O(n) O(n2) Heap Sort (binary heap, vector-based implementation) O(logn) O(logn) O(nlogn) 4/8/2015 5:25 AM 27 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 heaporder property 3 8 2 5 4 6 7 3 8 2 5 4 6 2 3 8 Priority Queues Trees 4 5 7 6 28 4/8/2015 5:25 AM 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 2i -1 2i+1-1 Priority Queues Trees 29 4/8/2015 5:25 AM Example 16 15 4 25 16 12 6 5 15 4 7 23 11 12 6 Priority Queues Trees 20 27 7 23 20 30 4/8/2015 5:25 AM Example (cont’d) 25 16 5 15 4 15 16 11 12 6 4 25 5 27 9 23 6 12 11 Priority Queues Trees 20 23 9 27 20 31 4/8/2015 5:25 AM Example (cont’d) 7 8 15 16 4 25 5 6 12 11 23 9 4 5 25 20 6 15 16 27 7 8 12 11 Priority Queues Trees 23 9 27 20 32 4/8/2015 5:25 AM Example (end) 10 4 6 15 16 5 25 7 8 12 11 23 9 27 20 4 5 6 15 16 7 25 10 8 12 11 Priority Queues Trees 23 9 27 20 33 4/8/2015 5:25 AM 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 Priority Queues Trees 34 4/8/2015 5:25 AM 3 a Locators 1 g Priority Queues Trees 4 e 35 4/8/2015 5:25 AM Locators • A locator identifies and tracks a (key, element) item within a data structure • A locator sticks with a specific item, even if that element changes its position in the data structure • Intuitive notion: • Application example: • Orders to purchase and sell a given stock are stored in two priority queues (sell orders and buy orders) • the key of an order is the price • the element is the number of shares • 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 • When an order is placed, a locator to it is returned • Given a locator, an order can be canceled or modified Priority Queues Trees 36 4/8/2015 5:25 AM Locator-based Methods • 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) Priority Queues Trees 37 4/8/2015 5:25 AM Possible Implementation • The locator is an object storing • key • element • position (or rank) of the item in the underlying structure 6 d 3 a 9 b • In turn, the position (or array cell) stores the locator • Example: • binary search tree with locators 1 g Priority Queues Trees 4 e 8 c 38 4/8/2015 5:25 AM Positions vs. Locators • Position • Locator • 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 • 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 Priority Queues Trees 39 4/8/2015 5:25 AM