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Priority QueuesPriority Queues
Priority: some property of an object that allows it to be
prioritized with respect to other objects of the same type
Min Priority Queue: homogeneous collection of Comparables
with the following operations (duplicates are allowed). Smaller
value means higher priority.
void insert (Comparable x)
void deleteMin( )
void deleteMin ( Comparable min)
Comparable findMin( )
Construct from a set of initial values
boolean isEmpty( )
boolean isFull( )
void makeEmpty( )
www.ThesisScientist.comPriority Queue Applications
The shorter document on the printer queue, the
higher its priority.
Jobs queue within an operating system:
Users’ tasks are given priorities. System priority
The time an event “happens” is its priority.
Sorting (heap sort)
An elements “value” is its priority.
Use a sorted list. Sorted by priority upon
findMin( ) list.front( )
insert( ) list.insert( )
deleteMin( ) list.erase( list.begin( ) )
Use ordinary BST
findMin( ) tree.findMin( )
insert( ) tree.insert( )
deleteMin( ) tree.delete( tree.findMin( ) )
Use balanced BST
guaranteed O(lg n) for RedBlack
www.ThesisScientist.comMin Binary Heap
A min binary heap is a complete binary tree with
the further property that at every node neither
child is smaller than the value in that node (or
equivalently, both children are at least as large
as that node).
This property is called a partial ordering.
As a result of this partial ordering, every path
from the root to a leaf visits nodes in a non
What other properties of the Min Binary Heap
result from this property
www.ThesisScientist.comMin Binary Heap Performance
Performance (n is the number of elements in
construction O( n )
findMin O( 1 )
insert O( lg n )
deleteMin O( lg n )
Heap efficiency results, in part, from the
Conceptually a complete binary tree
Implementation in an array/vector (in level order)
with the root at index 1
www.ThesisScientist.comMin Binary Heap Properties
For a node at index i
its left child is at index 2i
its right child is at index 2i+1
its parent is at index i/2
No pointer storage
Fast computation of 2i and i/2 by bit shifting
i 1 = 2i
i 1 = i/2
www.ThesisScientist.comHeap is a Complete Binary Tree
www.ThesisScientist.comWhich satisfies the properties of a Heap
www.ThesisScientist.comMin BinaryHeap Definition
BinaryHeapAnyType extends Comparable super AnyType
public BinaryHeap( ) / See online code /
public BinaryHeap( int capacity ) / See online code /
public BinaryHeap( AnyType items )/ Figure 6.14 /
public void insert( AnyType x ) / Figure 6.8 /
public AnyType findMin( ) / TBD /
public AnyType deleteMin( ) / Figure 6.12 /
public boolean isEmpty( ) / See online code /
public void makeEmpty( ) / See online code /
private static final int DEFAULTCAPACITY = 10;
private int currentSize; // Number of elements in heap
private AnyType array; // The heap array
private void percolateDown( int hole )/ Figure 6.12 /
private void buildHeap( ) / Figure 6.14 /
private void enlargeArray(int newSize)/ code online /
www.ThesisScientist.comMin BinaryHeap Implementation
public AnyType findMin( )
if ( isEmpty( ) ) throw Underflow( );
CBT property (heap shape):
Easy, just insert new element at “the end” of the array
Min heap order
Could be wrong after insertion if new element is smaller
than its ancestors
Continuously swap the new element with its parent until
parent is not greater than it
Called sift up or percolate up
Performance of insert is O( lg n ) in the worst
case because the height of a CBT is O( lg n )
www.ThesisScientist.comMin BinaryHeap Insert (cont.)
Insert into the priority queue, maintaining heap order.
Duplicates are allowed.
param x the item to insert.
public void insert( AnyType x )
if( currentSize == array.length 1 )
enlargeArray( array.length 2 + 1 );
// Percolate up
int hole = ++currentSize;
for( ; hole 1 x.compareTo(arrayhole/2) 0; hole/=2 )
array hole = array hole / 2 ;
array hole = x;
Remove min element (the root)
Maintain heap shape
Maintain min heap order
To maintain heap shape, actual node
removed is “last one” in the array
Replace root value with value from last node and
delete last node
Siftdown the new root value
Continually exchange value with the smaller child until
no child is smaller.
www.ThesisScientist.comMin BinaryHeap Deletion(cont.)
Remove the smallest item from the priority queue.
return the smallest item, or throw
UnderflowException, if empty.
public AnyType deleteMin( )
if( isEmpty( ) )
throw new UnderflowException( );
AnyType minItem = findMin( );
array 1 = array currentSize ;
percolateDown( 1 );
Internal method to percolate down in the heap.
param hole the index at which the percolate begins.
private void percolateDown( int hole )
AnyType tmp = array hole ;
for( ; hole 2 = currentSize; hole = child )
child = hole 2;
if( child = currentSize
array child + 1 .compareTo( array child ) 0 )
if( array child .compareTo( tmp ) 0 )
array hole = array child ;
array hole = tmp;
www.ThesisScientist.comConstructing a Min BinaryHeap
A BH can be constructed in O(n) time.
Suppose we are given an array of objects in
an arbitrary order. Since it’s an array with no
holes, it’s already a CBT. It can be put into
heap order in O(n) time.
Create the array and store n elements in it in
arbitrary order. O(n) to copy all the objects.
Heapify the array starting in the “middle” and
working your way up towards the root
for (int index = n/2 ; index 0; index)
percolateDown( index );
www.ThesisScientist.comConstructing a Min BinaryHeap(cont.)
//Construct the binary heap given an array of items.
public BinaryHeap( AnyType items )
currentSize = items.length;
array = (AnyType) new Comparable (currentSize + 2)11/10 ;
int i = 1;
for( AnyType item : items )
array i++ = item;
// Establish heap order property from an arbitrary
// arrangement of items. Runs in linear time.
private void buildHeap( )
for( int i = currentSize / 2; i 0; i )
percolateDown( i );
www.ThesisScientist.comPerformance of Construction
A CBT has 2 nodes on level h1.
On level hl, at most 1 swap is needed per node.
On level h2, at most 2 swaps are needed.
On level 0, at most h swaps are needed.
Number of swaps = S
h h1 h2 0
= 2 0 + 2 1 + 2 2 + … + 2 h
h h h
i i i
2 (hi)h 2 i2
i0 i0 i0
= h(2 1) ((h1)2 +2)
= 2 (h(h1))h2
= 2 h2
www.ThesisScientist.comPerformance of Construction (cont.)
But 2 h2 = O(2 )
But n = 1 + 2 + 4 + … + 2 =
Therefore, n = O(2 )
So S = O(n)
A heap of n nodes can be built in O(n) time.
Given n values we can sort them in place in O(n log n) time
Insert values into array O(n)
repeatedly delete min O(lg n), n times
Using a min heap, this code sorts in reverse (high down to
With a max heap, it sorts in normal (low up to high) order.
Given an unsorted array A of size n
for (i = n1; i = 1; i)
x = findMin( );
Ai+1 = x;
MinBinary heaps support insert, findMin,
deleteMin, and construct efficiently.
They do not efficiently support the meld or
merge operation in which 2 BHs are merged
into one. If H and H are of size n and n ,
1 2 1 2
then the merge is in O(n + n ) .
www.ThesisScientist.comLeftist Min Heap
findMin O( 1 )
deleteMin O( lg n )
insert O( lg n )
construct O( n )
merge O( lg n )
The null path length, npl(X), of a node, X, is defined as the length of
the shortest path from X to a node without two children (a nonfull
Note that npl(NULL) = 1.
A Leftist Tree is a binary tree in which at each node X, the null path
length of X’s right child is not larger than the null path length of the X’s
left child .
I.E. the length of the path from X’s right child to its nearest nonfull node
is not larger than the length of the path from X’s left child to its nearest
An important property of leftist trees:
At every node, the shortest path to a nonfull node is along the
“Proof”: Suppose this was not true. Then, at some node the path on
the left would be shorter than the path on the right, violating the
leftist tree definition.
www.ThesisScientist.comLeftist Min Heap
A leftist min heap is a leftist tree in which
the values in the nodes obey heap order (the
tree is partially ordered).
Since a LMH is not necessarily a CBT we do
not implement it in an array. An explicit tree
implementation is used.
findMin return root value, same as MBH
deleteMin implemented using meld operation
insert implemented using meld operation
construct implemented using meld operation
// Merge rhs into the priority queue.
// rhs becomes empty. rhs must be different from this.
// param rhs the other leftist heap.
public void merge( LeftistHeapAnyType rhs )
if( this == rhs ) return; // Avoid aliasing problems
root = merge( root, rhs.root );
rhs.root = null;
// Internal method to merge two roots.
// Deals with deviant cases and calls recursive merge1.
private NodeAnyType merge(NodeAnyType h1, NodeAnyType h2 )
if( h1 == null ) return h2;
if( h2 == null ) return h1;
if( h1.element.compareTo( h2.element ) 0 )
return merge1( h1, h2 );
return merge1( h2, h1 );
Internal method to merge two roots.
Assumes trees are not empty, and h1's root contains smallest item.
private NodeAnyType merge1( NodeAnyType h1, NodeAnyType h2 )
if( h1.left == null ) // Single node
h1.left = h2; // Other fields in h1 already accurate
h1.right = merge( h1.right, h2 );
if( h1.left.npl h1.right.npl )
swapChildren( h1 );
h1.npl = h1.right.npl + 1;
Performance: O( lg n )
The rightmost path of each tree has at most
lg(n+1) nodes. So O( lg n ) nodes will be
Show the steps needed to merge the Leftist
Heaps below. The final result is shown on the
17 12 10 15
19 20 30 25
www.ThesisScientist.comStudent Exercise Final Result
www.ThesisScientist.comMin Leftist Heap Operations
Other operations implemented using Merge( )
Make item into a 1node LH, X
Merge(left subtree, right subtree)
construct from N items
Make N LHs from the N values, one element in each
Merge each in
one at a time (simple, but slow)
use queue and build pairwise (complex but faster)
Make n leftist heaps, H ….H each with one data
Instantiate QueueLeftistHeap q;
for (i = 1; i = n; i++)
Leftist Heap h = q.dequeue( );
while ( q.isEmpty( ) )
q.enqueue( merge( h, q.dequeue( ) ) );
h = q.dequeue( );