Priority Queues

© 2004 Goodrich, Tamassia

Priority Queues

1
Priority Queue ADT (§ 7.1.3)
A priority queue stores a
collection of entries
Each entry is a pair
(key, value)
Main method...
Total Order Relations (§ 7.1.1)
Keys in a priority
queue can be
arbitrary objects
on which an order
is defined
Two distinc...
Entry ADT (§ 7.1.2)
An entry in a priority
queue is simply a keyvalue pair
Priority queues store
entries to allow for
effi...
Comparator ADT (§ 7.1.2)
A comparator encapsulates
the action of comparing two
objects according to a given
total order re...
Example Comparator
Lexicographic comparison of 2-D
points:
/** Comparator for 2D points under the
standard lexicographic o...
Priority Queue Sorting (§ 7.1.4)
We can use a priority
queue to sort a set of
comparable elements
1.

2.

Insert the eleme...
Sequence-based Priority Queue
Implementation with an
unsorted list

Implementation with a
sorted list

4

1

5

2

3

1

P...
Selection-Sort
Selection-sort is the variation of PQ-sort where the
priority queue is implemented with an unsorted
sequenc...
Selection-Sort Example
Sequence S

Priority Queue P

Input:

(7,4,8,2,5,3,9)

()

Phase 1
(a)
(b)
..
.
(g)

(4,8,2,5,3,9)
...
Insertion-Sort
Insertion-sort is the variation of PQ-sort where the
priority queue is implemented with a sorted
sequence
R...
Insertion-Sort Example
Input:

Sequence S
(7,4,8,2,5,3,9)

Phase 1
(a)
(b)
(c)
(d)
(e)
(f)
(g)

(4,8,2,5,3,9)
(8,2,5,3,9)
...
In-place Insertion-sort
Instead of using an
external data structure,
we can implement
selection-sort and
insertion-sort in...
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Priority queues

  1. 1. Priority Queues © 2004 Goodrich, Tamassia Priority Queues 1
  2. 2. Priority Queue ADT (§ 7.1.3) 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 © 2004 Goodrich, Tamassia Priority Queues Additional methods   min() returns, but does not remove, an entry with smallest key size(), isEmpty() Applications:    Standby flyers Auctions Stock market 2
  3. 3. Total Order Relations (§ 7.1.1) 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 © 2004 Goodrich, Tamassia 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 Queues 3
  4. 4. Entry ADT (§ 7.1.2) An entry in a priority queue is simply a keyvalue 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 © 2004 Goodrich, Tamassia As a Java interface: /** * Interface for a key-value * pair entry **/ public interface Entry { public Object key(); public Object value(); } Priority Queues 4
  5. 5. Comparator ADT (§ 7.1.2) 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 © 2004 Goodrich, Tamassia The primary method of the Comparator ADT:  Priority Queues 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. 5
  6. 6. Example Comparator Lexicographic comparison of 2-D 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); } } © 2004 Goodrich, Tamassia 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 Queues 6
  7. 7. Priority Queue Sorting (§ 7.1.4) We can use a priority queue to sort a set of comparable elements 1. 2. Insert the elements one by one with a series of insert 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 © 2004 Goodrich, Tamassia 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) Priority Queues 7
  8. 8. Sequence-based Priority Queue Implementation with an unsorted list Implementation with a sorted list 4 1 5 2 3 1 Performance:   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 © 2004 Goodrich, Tamassia Priority Queues 2 3 4 5 Performance:   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 8
  9. 9. 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: 1. 2. Inserting the elements into the priority queue with n insert 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 © 2004 Goodrich, Tamassia Priority Queues 9
  10. 10. Selection-Sort Example Sequence S Priority Queue P Input: (7,4,8,2,5,3,9) () 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) () © 2004 Goodrich, Tamassia Priority Queues (7,4,8,2,5,3,9) 10
  11. 11. 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: 1. Inserting the elements into the priority queue with n insert operations takes time proportional to 1 + 2 + …+ n 2. 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 © 2004 Goodrich, Tamassia Priority Queues 11
  12. 12. Insertion-Sort Example Input: Sequence S (7,4,8,2,5,3,9) 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) .. . () © 2004 Goodrich, Tamassia Priority queue P () Priority Queues 12
  13. 13. In-place Insertion-sort Instead of using an external data structure, we can implement selection-sort and insertion-sort in-place 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 swaps instead of modifying the sequence © 2004 Goodrich, Tamassia Priority Queues 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 13
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