Priority Queues

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Priority Queues Pauline Wilcox and Brian
Palmer Department of Computer Science
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Aims

• Summary of rest of term 2
• Introduce priority queue structure
• The priority queue as an ADT
• Example applications
• Sorting
• Implementation Issues

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The Priority Queue
P5
P2
P1
P9
P25

• We call it a priority queue - but its not FIFO
• Items in queue have PRIORITY
• Elements are removed from priority queue in
  either increasing or decreasing priority
• Min Priority Queue
• Max Priority Queue

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The Priority Queue
P5
P2
P1
P9
P25

• Consider situation where we have a computer whose
  services we are selling
• Users need different amounts of time
• Maximise earnings by min priority queue of users
• i.e. when machine becomes free, the user who
  needs least time gets the machine the average
  delay is minimised

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The Priority Queue
P5
P2
P1
P9
P25

• Consider situation where users are willing to pay
  more to secure access - they are in effect
  bidding against each other
• Maximise earnings by max priority queue of users
• i.e. when machine becomes free, the user who is
  willing to pay most gets the machine

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The Priority Queue

• Common data structure in computer science
• Responsible for scheduling jobs
• Unix (linux) can allocate processes a priority
• Time allocated to process is based on priority of
  job
• Priority of jobs in print scheduler

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Max Priority Queue ADT

• AbstractDataType MaxPriorityQueue
• instances
• finite collection of zero or more elements each
  has a priority
• operations
• isEmpty() Return true if the queue is empty
  else false
• size() Return number of elements in the queue
• getMax() Return the element with the maximum
  priority
• put(x) Add element x into the priority queue
• removeMax() Remove the element with the
  maximum priority from the priority and return it

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Max Priority Queue Interface

• public interface MaxPriorityQueue
• public boolean isEmpty()
• public int size()
• public Comparable getMax()
• public void put(Comparable theObject)
• public Comparable remove()

It may be desirable to include a method to allow
priority of elements to be changed (increased or
decreased) If we dont do this some items may
never be accessed!
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Application

• Sorting
• Use element key as priority
• Put elements to be sorted into a priority queue
• Extract elements in priority order
• if a min priority queue is used, elements are
  extracted in ascending order of priority (or key)
• if a max priority queue is used, elements are
  extracted in descending order of priority (or
  key)

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Sorting Example

• Sort five elements whose keys are 6, 8, 2, 4, 1
  using a max priority queue
• Step 1
• Put the five elements into a max priority queue
• Step 2
• Do five remove max operations placing removed
  elements into the sorted array from right to left

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After Putting Into Max Priority Queue

• Max Priority Queue
• Sorted Array

6
8
4
1
2
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After First Remove Max Operation

• Max Priority Queue
• Sorted Array

4
6
1
2
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After Second Remove Max Operation

• Max Priority Queue
• Sorted Array

4
1
2
6
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After Third Remove Max Operation

• Max Priority Queue
• Sorted Array

1
2
6
4
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After Fourth Remove Max Operation

• Max Priority Queue
• Sorted Array

1
6
4
2
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After Fifth Remove Max Operation

• Max Priority Queue
• Sorted Array

6
4
2
1
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Complexity Of Sorting

• Sort n elements
• n put operations gt O(n log n) time
• n remove max operations gt O(n log n) time
• total time is O(n log n)
• Compare
• with O(n2) for sort methods like the bubblesort
• Heap Sort
• Uses a max priority queue that is implemented as
  a heap
• Initial put operations are replaced by a heap
  initialization step that takes O(n) time

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Implementation Options

• Priority queue can be regarded as a linear list
• Unordered linear list
• Time complexities
• Ordered linear list
• Time complexities

put operation is O(1) removeMax is O(n)
put operation is O(n) removeMax is O(1)
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Implementation Options

• Priority queue can be regarded as a heap
• isEmpty, size, and get gt O(1) time
• put and remove gt O(log n) time
• where n is the size of the priority queue
• Were going to look at the heap
• A complete binary tree with values at its nodes
  arranged in a particular way (the priority!)

i.e. this is better than linear list option on
average
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Min Tree Definition

• Each tree node has a value
• Value in any node is the minimum value in the
  subtree for which that node is the root
• Equivalently, no descendant has a smaller value

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4
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Max Tree Example

• Root has maximum value

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4
9
8
4
2
7
3
1
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Heap Definitions

• Complete binary tree
• Min tree
• Complete binary tree
• Max tree

Min Heap
Max Heap
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Min Heap With 9 Nodes

• Complete binary tree with 9 nodes

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Min Heap With 9 Nodes

• Complete binary tree with 9 nodes that is also a
  min tree

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Max Heap With 9 Nodes

• Complete binary tree with 9 nodes that is also a
  max tree.

Since a heap is a complete binary tree, the
height of an n node heap is log2 (n1)
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A Heap Is Efficiently Represented As An Array
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Putting An Element Into A Max Heap

• Complete binary tree with 10 nodes

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Putting An Element Into A Max Heap

• New element is 5

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Putting An Element Into A Max Heap

• New element is 20

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Putting An Element Into A Max Heap

• New element is 20

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Putting An Element Into A Max Heap

• New element is 20

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Putting An Element Into A Max Heap

• New element is 20

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Putting An Element Into A Max Heap

• Complete binary tree with 11 nodes

20
9
7
8
6
2
6

7
5
1
7
7
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Putting An Element Into A Max Heap

• New element is 15

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2
6

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Putting An Element Into A Max Heap

• New element is 15

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Putting An Element Into A Max Heap

• New element is 15

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7
15
6
2
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9
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8
7
5
1
7
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Complexity is O(log n), where n is heap size
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Removing The Max Element

• Max element is in the root

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Removing The Max Element

• After max element is removed - root is left empty

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Removing The Max Element

• Heap will have 10 nodes, reinsert last leaf node
  (8) into the heap

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Removing The Max Element

• Reinsert 8 into the heap

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Removing The Max Element

• Reinsert 8 into the heap

15
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5
1
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Removing The Max Element

• Reinsert 8 into the heap

15
7
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Removing The Max Element

• Max element is 15

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Removing The Max Element

• After max element is removed

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Removing The Max Element

• Heap with 9 nodes - reinsert last leaf node (7)
  into the heap

7
9
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2
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5
1
7
7
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Removing The Max Element

• Reinsert 7

7
9
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2
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8
5
1
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Removing The Max Element

• Reinsert 7

9
7
6
2
6
8
5
1
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Removing The Max Element

• Reinsert 7

9
8
7
6
2
6
7
5
1
Complexity is again O(log n)