CMSC 341Priority QueuesPriority Queue ApplicationsPossible ImplementationsMin Binary HeapMin Binary Heap PerformanceMin Binary Heap PropertiesHeap is a Complete Binary TreeWhich satisfies the properties of a Heap?Min BinaryHeap DefinitionMin BinaryHeap ImplementationInsert OperationMin BinaryHeap Insert (cont.)Insert 14Deletion OperationMin BinaryHeap Deletion(cont.)MinBinaryHeap percolateDown(cont.)deleteMindeleteMin (cont.)Constructing a Min BinaryHeapConstructing a Min BinaryHeap(cont.)Performance of ConstructionPerformance of Construction (cont.)Heap SortLimitationsLeftist Min HeapLeftist TreeSlide 28MergeMerge (cont.)Slide 31Slide 32Slide 33Slide 34Student ExerciseStudent Exercise Final ResultMin Leftist Heap OperationsLH ConstructCMSC 341Binary HeapsPriority Queues8/3/2007UMBC CSMC 341 PQueue2Priority 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( )8/3/2007UMBC CSMC 341 PQueue3Priority Queue ApplicationsPrinter management: The shorter document on the printer queue, the higher its priority.Jobs queue within an operating system:Users’ tasks are given priorities. System priority high.SimulationsThe time an event “happens” is its priority.Sorting (heap sort)An elements “value” is its priority.8/3/2007UMBC CSMC 341 PQueue4Possible ImplementationsUse a sorted list. Sorted by priority upon insertion.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 Red-Black8/3/2007UMBC CSMC 341 PQueue5Min 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-decreasing order.What other properties of the Min Binary Heap result from this property?8/3/2007UMBC CSMC 341 PQueue6Min Binary Heap PerformancePerformance (n is the number of elements in the heap)construction O( n )findMin O( 1 )insert O( lg n )deleteMin O( lg n )Heap efficiency results, in part, from the implementationConceptually a complete binary treeImplementation in an array/vector (in level order) with the root at index 18/3/2007UMBC CSMC 341 PQueue7Min 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 shiftingi << 1 = 2ii >> 1 = i/28/3/2007UMBC CSMC 341 PQueue8Heap is a Complete Binary Tree8/3/2007UMBC CSMC 341 PQueue9Which satisfies the properties of a Heap?8/3/2007UMBC CSMC 341 PQueue10Min BinaryHeap Definition public class BinaryHeap<AnyType 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 DEFAULT_CAPACITY = 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 */}}8/3/2007UMBC CSMC 341 PQueue11Min BinaryHeap Implementationpublic AnyType findMin( ){if ( isEmpty( ) ) throw Underflow( ); return array[1];}8/3/2007UMBC CSMC 341 PQueue12Insert OperationMust maintain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 )8/3/2007UMBC CSMC 341 PQueue13Min 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(array[hole/2]) < 0; hole/=2 ) array[ hole ] = array[ hole / 2 ]; array[ hole ] = x;}8/3/2007UMBC CSMC 341 PQueue14Insert 148/3/2007UMBC CSMC 341 PQueue15Deletion OperationSteps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Sift-down the new root value Continually exchange value with the smaller child until no child is smaller.8/3/2007UMBC CSMC 341 PQueue16Min 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 ); return minItem; }8/3/2007UMBC CSMC 341 PQueue17MinBinaryHeap percolateDown(cont.) /** * Internal method to percolate down in the heap. * @param hole the index at which the percolate begins. */private void percolateDown( int hole ){ int child; AnyType