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) Sell100IBM$122 Sell300IBM$120 Buy500IBM$119 Buy400IBM$118

Priority Queues7/2/2015 9:52 AM Trees2 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 Queues7/2/2015 9:52 AM Trees3 Priority Queue ADT 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 Stock market

Priority Queues7/2/2015 9:52 AM Trees4 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 Queues7/2/2015 9:52 AM Trees5 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.y b.y) return +1 else return 0; } };

Priority Queues7/2/2015 9:52 AM Trees6 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

Priority Queues7/2/2015 9:52 AM Trees7 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 Queues7/2/2015 9:52 AM Trees8 List-based Priority Queue 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

Priority Queues7/2/2015 9:52 AM Trees9 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 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(n 2 ) time 45231

Priority Queues7/2/2015 9:52 AM Trees10 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) 45231

Priority Queues7/2/2015 9:52 AM Trees11 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 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(n 2 ) time 12345

Priority Queues7/2/2015 9:52 AM Trees12 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) 45231

Priority Queues7/2/2015 9:52 AM Trees13 In-place Insertion-sort 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

Priority Queues7/2/2015 9:52 AM Trees14 Heaps and Priority Queues

Priority Queues7/2/2015 9:52 AM Trees15 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 2 i 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 last node

Priority Queues7/2/2015 9:52 AM Trees16 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 2 i keys at depth i  0, …, h  2 and at least one key at depth h  1, we have n  1  2  4  …  2 h  2  1 Thus, n  2 h  1, i.e., h  log n  h22h2 1 keys 0 1 h2h2 h1h1 depth

Priority Queues7/2/2015 9:52 AM Trees17 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 Queues7/2/2015 9:52 AM Trees18 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) (6, Mark)(5, Pat) (9, Jeff)(7, Anna)

Priority Queues7/2/2015 9:52 AM Trees19 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) insertion node z z

Priority Queues7/2/2015 9:52 AM Trees20 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 z z

Priority Queues7/2/2015 9:52 AM Trees21 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) last node w w

Priority Queues7/2/2015 9:52 AM Trees22 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 w w

Priority Queues7/2/2015 9:52 AM Trees23 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 Queues7/2/2015 9:52 AM Trees24 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 Queues7/2/2015 9:52 AM Trees25 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) 45231

Priority Queues7/2/2015 9:52 AM Trees26 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 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

Priority Queue Sort Summary PQ-Sort consists of n insertions followed by n removeMin ops 7/2/2015 9:52 AM Vectors 27 InsertRemove Min PQ-Sort total Insertion Sort (ordered sequence) O(n)O(1)O(n 2 ) Selection Sort (unordered sequence) O(1)O(n)O(n 2 ) Heap Sort (binary heap, vector-based implementation) O(logn) O(nlogn)

Priority Queues7/2/2015 9:52 AM Trees28 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

Priority Queues7/2/2015 9:52 AM Trees29 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 2 i  1 keys are merged into heaps with 2 i  1  1 keys Bottom-up Heap Construction 2 i  1 2i112i11

Priority Queues7/2/2015 9:52 AM Trees30 Example

Priority Queues7/2/2015 9:52 AM Trees31 Example (cont’d)

Priority Queues7/2/2015 9:52 AM Trees32 Example (cont’d)

Priority Queues7/2/2015 9:52 AM Trees33 Example (end)

Priority Queues7/2/2015 9:52 AM Trees34 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 Queues7/2/2015 9:52 AM Trees35 Locators 3 a 1 g 4 e

Priority Queues7/2/2015 9:52 AM Trees36 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: 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 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 When an order is placed, a locator to it is returned Given a locator, an order can be canceled or modified

Priority Queues7/2/2015 9:52 AM Trees37 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 Queues7/2/2015 9:52 AM Trees38 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 3 a 6 d 9 b 1 g 4 e 8 c

Priority Queues7/2/2015 9:52 AM Trees39 Positions vs. Locators 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