Chapter 13 Priority Queues. 2 Priority queue A stack is first in, last out A queue is first in,...

Chapter 13 Priority Queues

2

Priority queue

A stack is first in, last out A queue is first in, first out A priority queue is least-in-first-out

The “smallest” element is the first one removed (You could also define a largest-in-first-out priority queue)

The definition of “smallest” is up to the programmer (for example, you might define it by implementing Comparator or Comparable)

If there are several “smallest” elements, the implementer must decide which to remove first

Remove any “smallest” element (don’t care which) Remove the first one added

3

A priority queue ADT Here is one possible ADT:

PriorityQueue(): a constructor void add(Comparable o): inserts o into the priority queue Comparable removeLeast(): removes and returns the least

element Comparable getLeast(): returns (but does not remove) the

least element boolean isEmpty(): returns true iff empty int size(): returns the number of elements void clear(): discards all elements

4

Evaluating implementations When we choose a data structure, it is important to look

at usage patterns If we load an array once and do thousands of searches on it,

we want to make searching fast—so we would probably sort the array

If we load a huge array and expect to do only a few searches, we probably don’t want to spend time sorting the array

For almost all uses of a queue (including a priority queue), we eventually remove everything that we add

Hence, when we analyze a priority queue, neither “add” nor “remove” is more important—we need to look at the timing for “add + remove”

5

Array implementations A priority queue could be implemented as an unsorted

array (with a count of elements) Adding an element would take O(1) time (why?) Removing an element would take O(n) time (why?) Hence, adding and removing an element takes O(n) time This is an inefficient representation

A priority queue could be implemented as a sorted array (again, with a count of elements) Adding an element would take O(n) time (why?) Removing an element would take O(1) time (why?) Hence, adding and removing an element takes O(n) time Again, this is inefficient

6

Linked list implementations A priority queue could be implemented as an unsorted

linked list Adding an element would take O(1) time (why?) Removing an element would take O(n) time (why?)

A priority queue could be implemented as a sorted linked list Adding an element would take O(n) time (why?) Removing an element would take O(1) time (why?)

As with array representations, adding and removing an element takes O(n) time Again, these are inefficient implementations

7

Binary tree implementations

A priority queue could be represented as a (not necessarily balanced) binary search tree Insertion times would range from O(log n) to O(n) (why?) Removal times would range from O(log n) to O(n) (why?)

A priority queue could be represented as a balanced binary search tree Insertion and removal could destroy the balance We need an algorithm to rebalance the binary tree Good rebalancing algorithms require only O(log n) time,

but are complicated

8

Heap implementation A priority queue can be implemented as a heap In order to do this, we have to define the heap property

In Heapsort, a node has the heap property if it is at least as large as its children (for a MAX heap)

For a priority queue, we will define a node to have the heap property if it is as least as small as its children (since we are using smaller numbers to represent higher priorities) – i.e. a MIN heap

12

8 3Heapsort: Blue node has the MAX heap property

3

8 12Priority queue: Blue node has the MIN heap property

9

Array representation of a heap

Left child of node i is 2*i + 1, right child is 2*i + 2 Unless the computation yields a value larger than lastIndex, in

which case there is no such child Parent of node i is (i – 1)/2

Unless i == 0

12

1418

6

8

3

3 12 6 18 14 8 0 1 2 3 4 5 6 7 8 9 10 11 12

lastIndex = 5

10

Using the heap To add an element:

Increase lastIndex and put the new value there Reheap the newly added node

This is called up-heap bubbling or percolating up Up-heap bubbling requires O(log n) time

To remove an element: Remove the element at location 0 Move the element at location lastIndex to location 0, and decrement

lastIndex Reheap the new root node (the one now at location 0)

This is called down-heap bubbling or percolating down Down-heap bubbling requires O(log n) time

Thus, it requires O(log n) time to add and remove an element

11

Comments A priority queue is a data structure that is designed to

return elements in order of priority Efficiency is usually measured as the sum of the time

it takes to add and to remove an element Simple implementations take O(n) time Heap implementations take O(log n) time Balanced binary tree implementations take O(log n) time Binary tree implementations, without regard to balance, can

take O(n) (linear) time Thus, for any sort of heavy-duty use, heap or balanced

binary tree implementations are better

12

Java 5 java.util.PriorityQueue Java 5 finally has a PriorityQueue class, based on heaps

PriorityQueue<E> queue = new PriorityQueue<E>();

boolean add(E o) boolean remove(Object o)

boolean offer(E o) E peek() boolean poll()

void clear() int size()

Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3

13

Heaps

A heap is a binary tree with properties:1. It is complete

• Each level of tree completely filled• Except possibly bottom level (nodes in left most

positions)

2. It satisfies heap-order property• Data in each node >= data in children


14

Heaps

• Which of the following are MAX heaps?

A B C


15

Implementing a Heap

• Use an array or vector• Number the nodes from top to bottom

– Number nodes on each row from left to right• Store data in ith node in ith location of array

(vector)


16

Implementing a Heap

• Note the placement of the nodes in the array (note the array cells start at 1 not 0, unlike our implementation)


17

Implementing a Heap(note array starts at 1 here)

• In an array implementation children of ith node are at myArray[2*i] and

myArray[2*i+1]• Parent of the ith node is at

mayArray[i/2]


18

Basic Heap Operations

• Constructor– Set mySize to 0, allocate array

• Empty– Check value of mySize

• Retrieve max item– Return root of the binary tree, myArray[1]


19


• Delete max item– Max item is the root, replace with last node in

tree

– Then interchange root with larger of two children– Continue this with the resulting sub-tree(s)

Result called a semiheap


20

Percolate Down Algorithm(for an array starting at 1—your handout has pseudocode for 0

based array)

1. Set c = 2 * r2. While r <= n do following

a. If c < n and myArray[c] < myArray[c + 1]Increment c by 1

b. If myArray[r] < myArray[c]i. Swap myArray[r] and myArray[c]ii. set r = ciii. Set c = 2 * c

else Terminate repetitionEnd while


21


• Insert an item– Amounts to a percolate up routine– Place new item at end of array

– Interchange with parent so long as it is greater than its parent

Percolate Upfor 0-based array

• PercolateUp(int leaf)• Set p = parent index of leaf• Set value = data at leaf index• While leaf > 0 AND value < parent value

– Change the leaf data to parent’s data– Set leaf index = parent index– Set p = new parent index of leaf

• Set data at final leaf position to value


22


23

Heapsort

• Given a list of numbers in an array– Stored in a complete binary tree

• Convert to a heap– Begin at last node not a leaf– Apply percolated down to this subtree– Continue


24

Heapsort

• Algorithm for converting a complete binary tree to a heap – called "heapify"For r = n/2 down to 1:

Apply percolate_down to the subtreein myArray[r] , … myArray[n]

End for• Puts largest element at root


25

Heapsort• Now swap element 1 (root of tree) with last

element

– This puts largest element in correct location• Use percolate down on remaining sublist

– Converts from semi-heap to heap


26

Heapsort

• Again swap root with rightmost leaf

• Continue this process with shrinking sublist


27

Heapsort Algorithm

1. Consider x as a complete binary tree, use heapify to convert this tree to a heap

2. for i = n down to 2:a. Interchange x[1] and x[i] (puts largest element at end)b. Apply percolate_down to convert binary tree corresponding to sublist in x[1] .. x[i-1]


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Heap Algorithms in STL

• Found in the <algorithm> library–make_heap() heapify–push_heap() insert–pop_heap() delete–sort_heap() heapsort

• Note program which illustrates these operations, Fig. 13.1


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Priority Queue• A collection of data elements

– Items stored in order by priority– Higher priority items removed ahead of lower

• Operations– Constructor– Insert– Find, remove smallest/largest (priority) element– Replace – Change priority– Delete an item– Join two priority queues into a larger one


30

Priority Queue

• Implementation possibilities– As a list (array, vector, linked list)– As an ordered list– Best is to use a heap

Basic operations have O(log2n) time

• Java priority queue class uses heap

OSsim.java

• Simulates a (very slow!) "operating system" • Each minute one task is processed• Each minute 0, 1 or 2 tasks arrive

– placed in PriorityQueue<Task> PQ– to be processed in a future minute


31

OSsim Animation• Trace Table


32

PQ

Minute Task Dequeue'd

Wait Time

numArrivals

Tasks Enqueue'd

1 - - 2 new Task(0,1)new Task(1,1)

(0,1)(1,1)



33

PQ


Wait Time

numArrivals

Tasks Enqueue'd

1

2

-

(0,1)

-

1

2

2

new Task(0,1)new Task(1,1)new Task(0,2)new Task(0,2)

(0,2)(0,2)(1,1)



34

PQ


Wait Time

numArrivals

Tasks Enqueue'd

1

2

3

-

(0,1)

(0,2)

-

1

1

2

2

1

new Task(0,1)new Task(1,1)new Task(0,2)new Task(0,2)new Task(1,3)

(0,2)(1,1)(1,3)



35

PQ


Wait Time

numArrivals

Tasks Enqueue'd

1

2

3

4

-

(0,1)

(0,2)

(0,2)

-

1

1

1

2

2

1

0


-

(1,1)(1,3)



36

PQ


Wait Time

numArrivals

Tasks Enqueue'd

1

2

3

4

5

-

(0,1)

(0,2)

(0,2)

(1,1)

-

1

1

1

4

2

2

1

0

0


-

-

(1,3)

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Chapter 13 Priority Queues. 2 Priority queue A stack is first in, last out A queue is first in,...

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