ch8 - Chapter 8 Priority Queues Objectives Priority Queue...

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Unformatted text preview: Chapter 8: Priority Queues Objectives: Priority Queue ADT Comparator design pattern Heaps Priority Queue Implementation with List and Heap Adaptable Priority Queues Sorting: – – – – Priority Queue­sort Selection­sort Insert­sort Heap­ort CSC311: Data Structures CSC311: Data Structures 1 Priority Queue ADT Priority A priority queue stores a collection of entries Each entry is a pair (key, value) Main methods of the Priority Queue ADT – insert(k, x) inserts an entry with key k and value x – removeMin() removes and returns the entry with smallest key Additional methods – min() returns, but does not remove, an entry with smallest key – size(), isEmpty() Applications: – – – Standby flyers Auctions Stock market Total Order Relations Total Keys in a priority queue can be arbitrary objects on which an order is defined Two distinct entries 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 Entry ADT Entry An entry in a priority queue is simply a key­ value pair Priority queues store entries to allow for efficient insertion and removal based on keys Methods: – key(): returns the key for this entry – value(): returns the value associated with this entry As a Java interface: /** * Interface for a key­value * pair entry **/ public interface Entry { public Object key(); public Object value(); } Comparator ADT Comparator A comparator encapsulates the action of comparing two objects according to a given total order relation A generic priority queue uses an auxiliary comparator The comparator is external to the keys being compared When the priority queue needs to compare two keys, it uses its comparator The primary method of the Comparator ADT: – compare(x, y): Returns an integer i such that i < 0 if a < b, i = 0 if a = b, and i > 0 if a > b; an error occurs if a and b cannot be compared. Example Comparator Example Lexicographic comparison of 2­D points: /** Comparator for 2D points under the standard lexicographic order. */ public class Lexicographic implements Comparator { int xa, ya, xb, yb; public int compare(Object a, Object b) throws ClassCastException { xa = ((Point2D) a).getX(); ya = ((Point2D) a).getY(); xb = ((Point2D) b).getX(); yb = ((Point2D) b).getY(); if (xa != xb) return (xb ­ xa); else return (yb ­ ya); } } Point objects: /** Class representing a point in the plane with integer coordinates */ public class Point2D { protected int xc, yc; // coordinates public Point2D(int x, int y) { xc = x; yc = y; } public int getX() { return xc; } public int getY() { return yc; } } Priority Queue Sorting Priority We can use a priority queue to sort a set of comparable elements 1. Insert the elements one by one with a series of insert operations 2. 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.removeFirst () P.insert (e, 0) while ¬P.isEmpty() e ← P.removeMin().key() S.insertLast(e) Sequence-based Priority Queue Sequence-based Implementation with an unsorted list 4 5 2 Performance: 3 1 – insert takes O(1) time since we can insert the item at the beginning or end of the sequence – removeMin and min take O(n) time since we have to traverse the entire sequence to find the smallest key Implementation with a sorted list 1 2 3 Performance: 4 5 – insert takes O(n) time since we have to find the place where to insert the item – removeMin and min take O(1) time, since the smallest key is at the beginning Selection-Sort Selection-Sort Selection­sort is the variation of PQ­sort where the priority queue is implemented with an unsorted sequence Running time of Selection­sort: 1. Inserting the elements into the priority queue with n insert operations takes O(n) time time 2. Removing the elements in sorted order from the priority queue with n removeMin operations takes time proportional to 1 + 2 + …+ n Selection­sort runs in O(n2) time time Selection-Sort Example Selection-Sort Input: Sequence S (7,4,8,2,5,3,9) Priority Queue P () Phase 1 (a) (b) .. . (g) (4,8,2,5,3,9) (8,2,5,3,9) .. .. . . () (7) (7,4) Phase 2 (a) (b) (c) (d) (e) (f) (g) (2) (2,3) (2,3,4) (2,3,4,5) (2,3,4,5,7) (2,3,4,5,7,8) (2,3,4,5,7,8,9) (7,4,8,5,3,9) (7,4,8,5,9) (7,8,5,9) (7,8,9) (8,9) (9) () (7,4,8,2,5,3,9) Insertion-Sort Insertion-Sort Insertion­sort is the variation of PQ­sort where the priority queue is implemented with a sorted sequence Running time of Insertion­sort: 1. 2. Inserting the elements into the priority queue with n insert 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 time Insertion­sort runs in O(n2) time time Insertion-Sort Example Insertion-Sort Input: Sequence S (7,4,8,2,5,3,9) Priority queue P () Phase 1 (a) (b) (c) (d) (e) (f) (g) (4,8,2,5,3,9) (8,2,5,3,9) (2,5,3,9) (5,3,9) (3,9) (9) () (7) (4,7) (4,7,8) (2,4,7,8) (2,4,5,7,8) (2,3,4,5,7,8) (2,3,4,5,7,8,9) Phase 2 (a) (b) .. . (g) (2) (2,3) .. . (2,3,4,5,7,8,9) (3,4,5,7,8,9) (4,5,7,8,9) .. . () In-place Insertion-sort In-place Instead of using an external data structure, we can implement selection­sort and insertion­sort in­place A portion of the input sequence itself serves as the priority queue For in­place insertion­sort – We keep sorted the initial portion of the sequence – We can use swaps instead of modifying the sequence 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 1 2 3 4 5 1 2 3 4 5 Heaps Heaps A heap is a binary tree storing keys at its nodes and satisfying the following properties: – Heap­Order: 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 1, i 2 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 node of depth h 2 5 9 6 7 last node Height of a Heap Height Theorem: A heap storing n keys has height O(log n) keys has height (log Proof: (we apply the complete binary tree property) – Let h be the height of a heap storing n keys keys 0, – Since there are 2i keys at depth i = 0, … , h − 1 and at least one key and at least one key h−1 at depth h, we have n ≥ 1 + 2 + 4 + … + 2 + 1 log – Thus, n ≥ 2h , i.e., h ≤ log n , i.e., depth keys 0 1 1 2 h−1 2h−1 h 1 Heaps and Priority Queues Heaps 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 Insertion 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 – Restore the heap­order property (discussed next) 2 5 9 6 z 7 insertion node 2 5 9 6 7 z 1 Upheap Upheap After the insertion of a new key k, the heap­order property may be violated Algorithm upheap restores the heap­order 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 (log (log 2 1 5 9 1 7 z 6 5 9 2 7 z 6 Removal from a Heap Removal 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 – Remove w – Restore the heap­order property (discussed next) 2 5 9 6 7 w last node 7 5 w 9 new last node 6 Downheap Downheap After replacing the root key with the key k of the last node, the heap­order property may be violated Algorithm downheap restores the heap­order 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 (log (log 7 5 9 w 5 6 7 9 w 6 Updating the Last Node Updating The insertion node can be found by traversing a path of O(log n) (log nodes 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 Consider a priority queue with n items implemented by means of a heap – the space used is O(n) – methods insert and removeMin take O(log n) (log time time – methods size, isEmpty, and min take time O(1) (1) time time Using a heap­based priority queue, we can sort a sequence of n elements in O(n log n) time time The resulting algorithm is called heap­sort Heap­sort is much faster than quadratic sorting algorithms, such as insertion­sort and selection­sort Vector-based Heap Implementation 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 cell of at rank 0 is not used Operation insert corresponds to inserting at rank n + 1 Operation removeMin corresponds to removing at rank 1 Yields in­place heap­sort 2 5 6 9 7 2 0 5 6 9 7 1 2 3 4 5 Merging Two Heaps Merging 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 We can construct a heap storing n given keys in using a bottom­up construction with log n log 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 Example Example 16 15 4 25 16 12 6 5 15 4 7 23 11 12 6 20 27 7 23 20 Example (contd.) Example 25 16 5 15 4 15 16 11 12 6 4 25 5 27 9 23 6 12 11 20 23 9 27 20 Example (contd.) Example 7 8 15 16 4 25 5 6 12 11 23 9 4 5 25 20 6 15 16 27 7 8 12 11 23 9 27 20 Example (end) Example 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 23 9 27 20 Analysis Analysis We visualize the worst­case 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, bottom­up heap construction runs in O(n) time time Bottom­up heap construction is faster than n successive insertions and speeds up the first phase of heap­sort Adaptable Adaptable Priority Queues Priority 3a 5g CSC311: Data Structures CSC311: Data Structures 4e 31 Motivating Example Motivating Suppose we have an online trading system where orders to purchase and sell a given stock are stored in two priority queues (one for sell orders and one for buy orders) as (p,s) entries: The key, p, of an order is the price The value, s, for an entry is the number of shares A buy order (p, s) is executed when a sell order (p’, s’) with price p’<p is added (the execution is complete if s’>s) – A sell order (p, s) is executed when a buy order (p’, s’) with price p’>p is added (the execution is complete if s’>s) – – – What if someone wishes to cancel their order before it executes? What if someone wishes to update the price or number of shares for their order? Methods of the Adaptable Priority Queue ADT Priority remove(e): Remove from P and return remove entry e. replaceKey(e,k): Replace with k and return the key of entry e of P; an error condition occurs if k is invalid (that is, k cannot be compared with other keys). replaceValue(e,x): Replace with x and return the value of entry e of P. Example Example Operation insert(5,A) insert(3,B) insert(7,C) min() (5,A),(7,C) key(e2) remove(e1) (7,C) replaceKey(e2,9) (9,B) replaceValue(e3,D) Output e1 e2 e3 e2 3 e1 3 C P (5,A) (3,B),(5,A) (3,B),(5,A),(7,C) (3,B), (3,B),(5,A),(7,C) (3,B), (7,C), (7,D),(9,B) Locating Entries Locating In order to implement the operations remove(k), replaceKey(e), and replaceValue(k), we need fast ways of locating an entry e in a priority queue. We can always just search the entire data structure to find an entry e, but there are better ways for locating entries. Location-Aware Entries Location-Aware A locator­aware entry identifies and tracks the location of its (key, value) object within a data structure Intuitive notion: – Coat claim check – Valet claim ticket – Reservation number Main idea: – Since entries are created and returned from the data structure itself, it can return location­aware entries, thereby making future updates easier List Implementation List A location­aware list entry is an object storing – – – key value position (or rank) of the item in the list In turn, the position (or array cell) stores the entry Back pointers (or ranks) are updated during swaps nodes/positions header 2c 4c 5c 8c entries trailer Heap Implementation Heap A location­aware heap entry is an object storing – – – key value position of the entry in the underlying heap In turn, each heap position stores an entry Back pointers are updated during entry swaps 2d 4a 8g 6b 5e 9c Performance Performance Using location­aware entries we can achieve the following running times (times better than those achievable without location­aware entries are highlighted in red): Method Unsorted List size, isEmpty O(1) insert O(1) min O(n) removeMin O(n) remove O(1) replaceKey O(1) replaceValue O(1) Sorted List O(1) O(n) O(1) O(1) O(1) O(n) O(1) Heap O(1) O(log n) (log O(1) O(log n) (log O(log n) (log O(log n) (log O(1) ...
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This note was uploaded on 01/02/2012 for the course COMPUTER 101 taught by Professor Dr.kahan during the Spring '11 term at Akademia Ekonomiczna w Krakowie.

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