Priority Queues and Heaps Outline and Reading Priority

Priority Queues and Heaps

Outline and Reading • Priority. Queue 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 Queue ADT • A priority queue stores a collection of items • An item is a pair (key, element) • Main methods of the Priority Queue ADT • insert. Item(k, o) inserts an item with key k and element o • remove. Min() removes the item with the smallest key • Additional methods • min. Key() returns, but does not remove, the smallest key of an item • min. Element() returns, but does not remove, the element of an item with smallest key • size(), is. Empty() • Applications: • Standby flyers • Auctions

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

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

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

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 insert. Item(e, e) operations Remove the elements in sorted order with a series of remove. Min() 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. is. Empty () e S. remove (S. first()) P. insert. Item(e, e) while !P. is. Empty() e P. min. Element() P. remove. Min() S. insert. Last(e)

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 • Performance: • insert. Item takes O(1) time since we can insert the item at the beginning or end of the sequence • remove. Min, min. Key and min. Element take O(n) time since we have to traverse the entire sequence to find the smallest key • 1 Store the items of the priority queue in a sequence, sorted by key 2 3 4 5 • Performance: • insert. Item takes O(n) time since we have to find the place where to insert the item • remove. Min, min. Key and min. Element take O(1) time since the smallest key is at the beginning of the sequence

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 insert. Item operations takes O(n) time • Removing the elements in sorted order from the priority queue with n remove. Min operations takes time proportional to 1 + 2 + …+ n • Selection-sort runs in O(n 2) time

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

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 insert. Item operations takes time proportional to 1 + 2 + …+ n • • Removing the elements in sorted order from the priority queue with a series of n remove. Min operations takes O(n) time Insertion-sort runs in O(n 2) time

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

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 swap. Elements instead of modifying the sequence 5 4 2 3 1 4 5 2 3 1 2 4 5 3 1 2 3 4 5

Heaps and Priority Queues 2 5 9 6 7

What is a heap? • A heap is a binary tree storing • The last node of a heap is keys at its internal nodes and the rightmost internal satisfying the following node of depth h - 1 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 leaf nodes 2 5 9 6 7 last node

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 + 1 depth keys 0 1 1 2 h-2 2 h-2 h-1 1

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.

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)

Insertion into a Heap • Method insert. Item 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 6 7 z 1

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 2 7 z 6

Removal from a Heap • Method remove. Min 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 6 w 7 last node 7 5 9 w 6

Downheap • After replacing the root key with the key k of the last node, the heaporder property may be violated • Algorithm downheap restores the heap property by swapping key k with 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 9 w 5 6 7 9 w 6

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

Heap-Sort • Consider a priority queue with n items implemented by means of a heap • the space used is O(n) • methods insert. Item and remove. Min take O(log n) time • methods size, is. Empty, min. Key, and min. Element 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

Exercise: Heap-Sort • Heap-sort is the variation of PQ-sort where the priority queue is implemented with a heap (first n insert. Items, then n remove. Mins) 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)

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 2 i+1 • the right child is at rank 2 i +2 • parent is • Links between nodes are not explicitly stored • The leaves are not represented • Operation insert. Item corresponds to inserting at rank n • Operation remove. Min corresponds to removing at rank n-1 • Yields in-place heap-sort 2 5 6 9 7 0 1 2 3 4

Priority Queue Sort Summary • PQ-Sort consists of n insertions followed by n remove. Min ops Insert Remove. Min PQ-Sort Total Insertion Sort (ordered sequence) O(n) O(1) O(n 2) Selection Sort (unordered sequence) O(1) O(n 2) O(logn) O(n logn) Heap Sort (binary heap, O(logn) vector-based implementation)

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 heap-order property 3 8 2 5 4 6 7 3 8 2 5 4 6 2 3 8 4 5 7 6

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 2 i -1 keys are merged into heaps with 2 i+1 -1 keys 2 i -1 2 i+1 -1

Example 16 15 4 12 6 9 23 20

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

Locators 3 a 1 g 4 e

Locators • 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

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 • replace. Key(l, k): replaces the key of the item with locator l • replace. Element(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(), replace. Key(l, k), replace. Element(l, o)

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 4 e 8 c

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