Transcript Heaps and Priority Queues

Chapter 13 Priority Queues
Priority queue
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A stack is first in, last out
A queue is first in, first out
A priority queue is least-in-first-out
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The “smallest” element is the first one removed
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(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
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Remove any “smallest” element (don’t care which)
Remove the first one added
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A priority queue ADT
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Here is one possible ADT:
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PriorityQueue(): a constructor
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void add(Comparable o): inserts o into the priority queue
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Comparable removeLeast(): removes and returns the least
element
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Comparable getLeast(): returns (but does not remove) the
least element
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boolean isEmpty(): returns true iff empty
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int size(): returns the number of elements
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void clear(): discards all elements
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Evaluating implementations
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When we choose a data structure, it is important to look
at usage patterns
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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”
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Array implementations
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A priority queue could be implemented as an unsorted
array (with a count of elements)
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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)
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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
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Linked list implementations
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A priority queue could be implemented as an unsorted
linked list
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A priority queue could be implemented as a sorted
linked list
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Adding an element would take O(1) time (why?)
Removing an element would take O(n) time (why?)
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
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Again, these are inefficient implementations
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Binary tree implementations
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A priority queue could be represented as a (not
necessarily balanced) binary search tree
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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
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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
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Heap implementation
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A priority queue can be implemented as a heap
In order to do this, we have to define the heap property
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In Heapsort, a node has the heap property if it is at least as large as its
children (for a MAX heap)
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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
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12
8
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Heapsort: Blue node
has the MAX heap
property
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Priority queue: Blue node
has the MIN heap
property
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Array representation of a heap
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0
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14
1
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8
3
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lastIndex = 5
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10
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3 12 6 18 14 8
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Left child of node i is 2*i + 1, right child is 2*i + 2
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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
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Unless i == 0
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Using the heap
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To add an element:
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Increase lastIndex and put the new value there
Reheap the newly added node
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This is called up-heap bubbling or percolating up
Up-heap bubbling requires O(log n) time
To remove an element:
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Remove the element at location 0
Move the element at location lastIndex to location 0, and decrement
lastIndex
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Reheap the new root node (the one now at location 0)
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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
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Comments
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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
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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
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Java 5 java.util.PriorityQueue
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Java 5 finally has a PriorityQueue class, based on heaps
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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()
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Heaps
A heap is a binary tree with properties:
1. It is complete
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Each level of tree completely filled
Except possibly bottom level (nodes in left most
positions)
2. It satisfies heap-order property
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Data in each node >= data in children
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Heaps
• Which of the following are MAX heaps?
A
B
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C
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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)
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Implementing a Heap
• Note the placement of the nodes in the
array (note the array cells start at 1 not 0,
unlike our implementation)
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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]
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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]
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Basic Heap Operations
• Delete max item
– Max item is the root, replace with last node in
tree
Result called a
semiheap
– Then interchange root with larger of two children
– Continue this with the resulting sub-tree(s)
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Percolate Down Algorithm
(for an array starting at 1—your handout has pseudocode for 0
based array)
1. Set c = 2 * r
2. 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 = c
iii. Set c = 2 * c
else
Terminate repetition
End while
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
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Basic Heap Operations
• 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
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Percolate Up
for 0-based array
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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
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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
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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 subtree
in myArray[r] , … myArray[n]
End for
• Puts largest element at root
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
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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
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
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Heapsort
• Again swap root with rightmost leaf
• Continue this process with shrinking sublist
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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]
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
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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
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
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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
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–
–
–
–
–
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Constructor
Insert
Find, remove smallest/largest (priority) element
Replace
Change priority
Delete an item
Join two priority queues into a larger one
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
Education, Inc. All rights reserved. 0-13-140909-3
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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
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
Education, Inc. All rights reserved. 0-13-140909-3
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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
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
Education, Inc. All rights reserved. 0-13-140909-3
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OSsim Animation
• Trace Table
Minute Task
Dequeue'd
Wait
Time
num
Arrivals
Tasks
Enqueue'd
1
-
2
new Task(0,1)
new Task(1,1)
-
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
Education, Inc. All rights reserved. 0-13-140909-3
PQ
(0,1)
(1,1)
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OSsim Animation
• Trace Table
Minute Task
Dequeue'd
Wait
Time
num
Arrivals
Tasks
Enqueue'd
1
-
-
2
2
(0,1)
1
2
new Task(0,1)
new Task(1,1)
new Task(0,2)
new Task(0,2)
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
Education, Inc. All rights reserved. 0-13-140909-3
PQ
(0,2)
(0,2)
(1,1)
33
OSsim Animation
• Trace Table
Minute Task
Dequeue'd
Wait
Time
num
Arrivals
Tasks
Enqueue'd
1
-
-
2
2
(0,1)
1
2
3
(0,2)
1
1
new Task(0,1)
new Task(1,1)
new Task(0,2)
new Task(0,2)
new Task(1,3)
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
Education, Inc. All rights reserved. 0-13-140909-3
PQ
(0,2)
(1,1)
(1,3)
34
OSsim Animation
• Trace Table
Minute Task
Dequeue'd
Wait
Time
num
Arrivals
Tasks
Enqueue'd
1
-
-
2
2
(0,1)
1
2
3
(0,2)
1
1
new Task(0,1)
new Task(1,1)
new Task(0,2)
new Task(0,2)
new Task(1,3)
4
(0,2)
1
0
-
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
Education, Inc. All rights reserved. 0-13-140909-3
PQ
(1,1)
(1,3)
35
OSsim Animation
• Trace Table
PQ
(1,3)
Minute Task
Dequeue'd
Wait
Time
num
Arrivals
Tasks
Enqueue'd
1
-
-
2
2
(0,1)
1
2
3
(0,2)
1
1
new Task(0,1)
new Task(1,1)
new Task(0,2)
new Task(0,2)
new Task(1,3)
4
(0,2)
1
0
-
5
(1,1)
4
0
-
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
Education, Inc. All rights reserved. 0-13-140909-3
36