Heaps

Heaps

Priority Queues

04/20/23 Data Structure: Heaps 2

Outline

Binary heaps

Binomial queues

Leftist heaps

Binary Heaps


Heap order propertyFor every root node N, the key in the parent of N is smaller than or equal to the key in N.

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The smallest value is in the root.


Operations on priority queues• Insert

– Put an element into the queue

• DeleteMin– Find the minimal element– Return it– Remove it from the queue


Binary Heaps

• A binary heap is an implementation of a priority queue.

• A binary heap is a complete binary tree with heap order property.


Complete Binary TreeA binary tree is a complete binary tree if

• every level in the tree is completely filled,

• except the leaf level, which is filled from left to right.

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Height is in log2 n,

where n is the

number of nodes.


Array implementation of complete binary tree

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Class BinaryHeap public class BinaryHeap{ public BinaryHeap( )

{ this( DEFAULT_CAPACITY );}public BinaryHeap( int capacity ){ currentSize = 0;

array = new Comparable[ capacity + 1 ];}…private static final int DEFAULT_CAPACITY = 100;private int currentSize; // Number of elements in heapprivate Comparable [ ] array; // The heap array

}


percolateUp

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Method percolateUp

private void percolateUp( int hole )

{ Comparable x = array[hole];

while (hole > 1 && x.compareTo( array[ hole / 2 ] ) < 0)

{ array[ hole ] = array[ hole/2 ];

hole = hole/2;

}

array[ hole ] = x;

}


Method percolateUpprivate void percolateUp( int hole ){ while (hole>1 && array[hole].compareTo( array[ hole/2 ])<0)

{ swap(hole, hole/2); hole = hole/2;

}}

private void swap( int p1, int p2 ){ Comparable x = array[p1];

array[p1] = array[p2];array[p2] = x;

}


PercolateDown

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Method percolateDownprivate void percolateDown( int hole ){ int child;

while( hole * 2 <= currentSize){ child = hole * 2;

if(child != currentSize&&array[ child+1].compareTo( array[child ])<0)

child++; // choose the smaller childif( array[ child ].compareTo( array[ hole ] ) < 0 )

swap( hole, child );else

break; hole = child;

}}


Method percolateDownprivate void percolateDown( int hole ){ int child;

Comparable tmp = array[ hole ]; // save the value of the nodewhile ( hole * 2 <= currentSize ){ child = hole * 2;

if(child != currentSize&&array[ child+1].compareTo( array[child ])<0)

child++; // choose the smaller childif( array[ child ].compareTo( tmp ) < 0 ){ array[ hole ] = array[ child ]; // move child up

hole = child; // move hole down}else

break; }array[ hole ] = tmp; // put the value in the hole

}


Insertion

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Method insert

public void insert( Comparable x ) throws Overflow

{ if( isFull( ) ) throw new Overflow( );

array[++currentSize]=x;

percolateUp(currentSize);

}


DeleteMin

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Method deleteMinpublic Comparable findMin( ){ if( isEmpty( ) ) return null;

return array[ 1 ];}

public Comparable deleteMin( ){ if( isEmpty( ) ) return null;

Comparable minItem = findMin( );array[ 1 ] = array[ currentSize--];percolateDown( 1 );return minItem;

}


Method buildHeapprivate void buildHeap( )

{ for( int i = currentSize / 2; i > 0; i-- )

percolateDown( i );

}

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Method decreaseKey

public void decreaseKey(int p, Comparable d) throws outOfRange

{ if( p>currentSize ) throw new outOfRange();array[ p ] = array[ p ] - d;percolateUp( p );

}


Method increaseKey

public void increaseKey(int p, Comparable d) throws outOfRange

{ if( p>currentSize ) throw new outOfRange();array[ p ] = array[ p ] + d;percolateDown( p );

}

Binomial Queues


Binomial TreeA binomial tree is defined recursively as follow:

• A binomial tree of height 0, denoted by B0, is a one-node tree.

• A binomial tree of height k, denoted by Bk, is formed by attaching a binomial tree Bk-1 to the root of another binomial tree Bk-1.


Property of Binomial Trees• A binomial tree of height k has 2k nodes.

• The number of nodes at depth d is the binomial coefficient k

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Structure• A binomial queue is a collection of heap-

ordered trees, each of which is a binomial tree.

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Binomial Nodesclass BinomialNode{ BinomialNode ( Comparable theElement ){ this ( theElement, null, null );}BinomialNode ( Comparable theElement,

BinomialNode lt, BinomialNode nt ){ element = theElement;

Child = lt; nextSibling = nt;}Comparable element;BinomialNode Child; BinomialNode nextSibling;

}


Examples: binomial nodes

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Child

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Binomial Queuespublic class BinomialQueue{ public BinomialQueue( ){ theTrees = new BinomialNode[ MAX_TREES ];

makeEmpty( ); }...

public void makeEmpty( ){ currentSize = 0;

for( int i=0; i < theTrees.length; i++ )theTrees[ i ] = null; }

private static final int MAX_TREES = 14;private int currentSize; private BinomialNode [ ] theTrees; private int capacity( ){ return 2*theTrees.length - 1; }

}


Method combineTrees

private static BinomialNode combineTrees( BinomialNode t1,BinomialNode t2 )

{ if( t1.element.compareTo( t2.element ) > 0 )return combineTrees( t2, t1 );

t2.nextSibling = t1.Child;t1.leftChild = t2;return t1;

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Method merge (1)public void merge( BinomialQueue rhs )

throws Overflow{ if( this == rhs ) return;if( currentSize+rhs.currentSize>capacity() )

throw new Overflow( );currentSize += rhs.currentSize;BinomialNode carry = null;for( int i=0,j=1; j<=currentSize; i++,j*=2 ){ BinomialNode t1 = theTrees[ i ];

BinomialNode t2 = rhs.theTrees[ i ];


Method merge (2)// No trees if (t1==null && t2==null && carry==null) {}// Only this if (t1!=null && t2==null && carry==null) {}// Only rhs if (t1==null && t2!=null && carry==null){ theTrees[i] = t2; rhs.theTrees[i] = null;}// Only carryif (t1==null && t2==null && carry!=null) { theTrees[ i ] = carry; carry = null; }// this & rhs if (t1!=null && t2==null && carry!=null){ carry = combineTrees( t1, t2 ); theTrees[i]=rhs.theTrees[i]=null; }


Method merge (3)// this and carryif (t1!=null && t2==null && carry!=null){ carry = combineTrees( t1, carry );theTrees[ i ] = null; }

// rhs and carryif (t1==null && t2!=null && carry!=null){ carry = combineTrees( t2, carry );rhs.theTrees[ i ] = null; }

// All threeif (t1!=null && t2!=null && carry!=null){ theTrees[ i ] = carry;carry = combineTrees( t1, t2 );rhs.theTrees[ i ] = null; }

}


Method merge (4)

for( int k=0; k < rhs.theTrees.length; k++ )rhs.theTrees[ k ] = null;

rhs.currentSize = 0;}


Method insert

public void insert( Comparable x ) throws Overflow

{ BinomialQueue oneItem= new BinomialQueue( );oneItem.currentSize = 1;oneItem.theTrees[0] = new BinomialNode( x );merge( oneItem );

}


Method deleteMin (2)for( int j = minIndex - 1; j >= 0; j-- ){ deletedQueue.theTrees[ j ] = deletedTree;

deletedTree = deletedTree.nextSibling;deletedQueue.theTrees[ j ].nextSibling =

null;}theTrees[ minIndex ] = null;

currentSize -= deletedQueue.currentSize + 1;try { merge( deletedQueue ); }catch( Overflow e ) { }return minItem;

}


Method deleteMinpublic Comparable deleteMin( ){ if( isEmpty( ) ) return null;int minIndex = findMinIndex( );Comparable minItem =

theTrees[minIndex].element;BinomialNode deletedTree =

theTrees[ minIndex ].child;BinomialQueue deletedQueue =

new BinomialQueue( );deletedQueue.currentSize =(1 << minIndex)-1;

Leftist Heaps


Null Path Length• The null path length of the

node X, npl(X), is the length of the shortest path from X to a node without 2 children.

• npl(X) = 1 + min (npl(Y1), npl(Y2)), where Y1 and Y2 are children of X.

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Leftist Trees• A leftist tree is a binary tree such that for

every node X in the tree npl(left) npl(right), where left and right are left child and right child of X.

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Examples of leftist trees

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Leftist Heap• A leftist heap is a leftist tree with heap order

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Leftist Tree Nodesclass LeftHeapNode{ LeftHeapNode ( Comparable theElement ){ this ( theElement, null, null ); }

LeftHeapNode ( Comparable theElement,LeftHeapNode lt, LeftHeapNode rt )

{ element = theElement; npl = 0;left = lt; right = rt; }

Comparable element; int npl; LeftHeapNode left; LeftHeapNode right;

}


Merge Leftist Heaps

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Merge Leftist Heaps

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Merge Leftist Heaps

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Method merge1private static LeftHeapNode merge1

( LeftHeapNode h1, LeftHeapNode h2 ){ if ( h1.left == null ) // Single node

h1.left = h2; // Other fields in h1 OKelse{ h1.right = merge ( h1.right, h2 );

if ( h1.left.npl < h1.right.npl )swapChildren ( h1 );h1.npl = h1.right.npl + 1;

}return h1;

}


Method mergepublic void merge( LeftistHeap rhs ){ if( this == rhs ) return;root = merge( root, rhs.root );rhs.root = null;

}private static LeftHeapNode merge( LeftHeapNode h1, LeftHeapNode h2 )

{ if( h1 == null ) return h2;if( h2 == null ) return h1;if( h1.element.compareTo( h2.element ) < 0 )

return merge1( h1, h2 );else

return merge1( h2, h1 );}


Methods insert, deleteMInpublic void insert( Comparable x ){ root=merge(new LeftHeapNode(x),root);}

public Comparable deleteMin ( ){ if ( isEmpty ( ) ) return null;Comparable minItem = root.element;root = merge ( root.left, root.right );return minItem;

}


Applications• Event simulation

• Merge sort


Event Simulation• Events have the scheduled time to happen.

• Advancing time by clock ticks is too slow.

• We need to find the “next” event.

• So, events are put into heaps with time as the element.

• We choose the next event from the root of the heapp.


Heap Sort

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Heap Sort

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Heaps

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