Chapter 7

Chapter 7: Priority Queues

A priority queue is an ADT for storing a collection of prioritized elements that supports arbitrary element insertion but supports removal of elements in order of priority; that is, the element with first priority can be removed at any time.
Examples:

Standby passenger for a fully booked flight - priority here is measured in terms of the fare paid, frequent-flyer status, and check-in time
In any operating system, each process is assumed a priority and the scheduler will always choose a process with higher priority over one of lower priority.

The priority is typically a numerical value - the smallest numerical value should have first (highest) priority.

7.1 The Priority Queue Abstract Data Type

7.1.1 Keys, Priorities, and Total Order Relations

We define a key to be an object that is assigned to an element as a specific attribute for that element and that can be used to identify, rank, or weight that element.

Comparing Keys with Total Orders
PriorityQueues.pdf 4
PriorityQueues.pdf 3
Example 7.1: (read)

7.1.2 Sorting with a Priority Queue

PriorityQueues.pdf 7
Algorithm PriorityQueue(S,P):
Input: A sequence S storing n elements, on which a total order relation is defined, and a priority queue P, that compares keys using the same total order relation
Output: The sequence S sorted by the total order relation
while !S.isEmpty() do
   e <- S.removeFirst()   {remove an element e from S}
   P.insertItem(e,e)         {the key is the element itself}
while !P.isEmpty() do
   e <- P.minElement()    {get a smallest element from P}
   P.removeMin()               {remove this element from P}
   S.insertLast(e)            {add the element at the end of S}

[Picture]
The running time of the algorithm is determined by the running times of operations insertItem(e,e), minElement(), and removeMin(), which do depend how P is implemented.

7.1.3 Functions of a Priority Queue

PriorityQueues.pdf 3

Simplicity of the Priority Queue ADT

Example 7.2:

Operation	Output	Priority Queue
insertItem(5,A)	-	{ (5,A) }
insertItem(9,C)	-	{ (5,A), (9,C) }
insertItem(3,B)	-	{ (3,B), (5,A), (9,C) }
insertItem(7,D)	-	{ (3,B), (5,A), (7,D), (9,C) }
minElement()	B	{ (3,B), (5,A), (7,D), (9,C) }
minKey()	3	{ (3,B), (5,A), (7,D), (9,C) }
removeMin()	-	{ (5,A), (7,D), (9,C) }
size()	3	{ (5,A), (7,D), (9,C) }
minElement()	A	{ (5,A), (7,D), (9,C) }
removeMin()	-	{ (7,D), (9,C) }
removeMin()	-	{ (9,C) }
removeMin()	-	{ }
removeMin()	ERROR	{ }
isEmpty()	true	{ }

There are still two important issues that we have left undetermined to this point:

How do we keep track of the associations between elements and their keys?
How do we compare keys so as to determine a smallest key?

7.1.4 Compositions and Comparators

The Composition Pattern
The composition pattern defines a single object that is a composition of other objects. A pair (k,e) is the simplest composition, for it composes two objects, k and e, into a single pair object.
html-7.2 (Item)

The Comparator Pattern
Lexicographical less than operator:

bool operator<(const Point& p1, const Point& p2)
{ if (p1.getX() == p2.getX()) return p1.getY() < p2.getY();
  else return p1.getX() < p1.getX();
}

Assume that comp is a compatison class object. comp(a,b) returns integer i, such that

i < 0 if a < b,
i = 0 if a = b, and
i > 0 if a > b.

The approach that we will use is to overload "()" operator (to create so called function object).
PriorityQueues.pdf 5, 6
html-7.3 (Compare)

Using Comparator Objects
PriorityQueues.pdf 6
html-7.4 (Generic)

7.2 Implementing a Priority Queue with a Sequence

7.2.1 Implementation with an Unsorted Sequence

Add the new pair object p = (k,e) at the end of sequence S, by executing function insertLast(p) on S - takes O(1) time, independent of whether the sequence is implemented using an array or a linked list.
To perform operations minElement(), minKey() or removeMin() on P, we must inspect all the elements of sequence S to find an element p = (k,e) of S with minimal k - takes O(n) time.

7.2.2 Implementation with a Sorted Sequence

We can implement functions minElement()or minKey() on P simply by accessing the first element of the sequence with the first() function of S. We can implement removeMin() function of P as S.remove(S.first()). O(1)
The function insertItem(k,e) of P requires that we scan through the sequence S to find the appropriate position to insert the element and key. O(n)

html-7.5 (SSPQ1)
html-7.6 (SSPQ2)

Comparing the Two Implementations
PriorityQueues.pdf 8

7.2.3 Selection-Sort and Insertion-Sort

Selection-Sort
PriorityQueues.pdf 9

Insertion-Sort
PriorityQueues.pdf 10

7.3 Heaps

7.3.1 The Heap Data Structure

heap.pdf 10

Proposition 7.5: A heap T storing n keys has height h = [log(n + 1)].
heap.pdf 11

7.3.2 Implementing a Priority Queue with a Heap

heap.pdf 12

The Vector Representation of a Heap
heap.pdf 19

Insertion
heap.pdf 13

Up-Heap Bubbling after an Insertion
heap.pdf 14

Removal
heap.pdf 15

Down-Heap Bubbling after a Removal
heap.pdf 16

Analysis

7.3.3 C++ Implementation

html-7.7 (HeapTree)
html-7.8 (HPQ1)
html-7.9 (HPQ2)

7.3.4 Heap-Sort

heap.pdf 18

Implementing Heap-Sort In-Place