优先队列(堆)--二叉堆学习笔记

1.结构性质：

           堆是一棵被完全填满的二叉数。一棵高为h的完全二叉树有2h到22+1-1个节点，这意味着完全二叉数的高是[log N],显然是O(log N).因为完全二叉数很有规律，所以可以用一个数组来表示二叉堆而不需要链。

           对于树组的任意一位置i上的元素，其左儿子在位置2i上，右儿子在2i+1上，它的父亲节点则在[i/2]上。

2.堆序性质：

          在堆中，对于每一个节点X，X的父亲中的关键字小于等于X中的关键字，根接点除外。

3.构架示例：

    // BinaryHeap class
    //
    // CONSTRUCTION: with optional capacity (that defaults to 100)
    //
    // ******************PUBLIC OPERATIONS*********************
    // void insert( x )       --> Insert x
    // Comparable deleteMin( )--> Return and remove smallest item
    // Comparable findMin( )  --> Return smallest item
    // boolean isEmpty( )     --> Return true if empty; else false
    // boolean isFull( )      --> Return true if full; else false
    // void makeEmpty( )      --> Remove all items
    // ******************ERRORS********************************
    // Throws Overflow if capacity exceeded

    /**
     * Implements a binary heap.
     * Note that all "matching" is based on the compareTo method.
     * @author Mark Allen Weiss
     */
    public class BinaryHeap
    {
        /**
         * Construct the binary heap.
         */
        public BinaryHeap( )
        {
            this( DEFAULT_CAPACITY );
        }

        /**
         * Construct the binary heap.
         * @param capacity the capacity of the binary heap.
         */
        public BinaryHeap( int capacity )
        {
            currentSize = 0;
            array = new Comparable[ capacity + 1 ];
        }

        /**
         * Insert into the priority queue, maintaining heap order.
         * Duplicates are allowed.
         * @param x the item to insert.
         * @exception Overflow if container is full.
         */
        public void insert( Comparable x ) throws Overflow
        {
            if( isFull( ) )
                throw new Overflow( );

                // Percolate up
            int hole = ++currentSize;
            for( ; hole > 1 && x.compareTo( array[ hole / 2 ] ) < 0; hole /= 2 )
                array[ hole ] = array[ hole / 2 ];
            array[ hole ] = x;
        }

        /**
         * Find the smallest item in the priority queue.
         * @return the smallest item, or null, if empty.
         */
        public Comparable findMin( )
        {
            if( isEmpty( ) )
                return null;
            return array[ 1 ];
        }

        /**
         * Remove the smallest item from the priority queue.
         * @return the smallest item, or null, if empty.
         */
        public Comparable deleteMin( )
        {
            if( isEmpty( ) )
                return null;

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

            return minItem;
        }

        /**
         * 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 );
        }

        /**
         * Test if the priority queue is logically empty.
         * @return true if empty, false otherwise.
         */
        public boolean isEmpty( )
        {
            return currentSize == 0;
        }

        /**
         * Test if the priority queue is logically full.
         * @return true if full, false otherwise.
         */
        public boolean isFull( )
        {
            return currentSize == array.length - 1;
        }

        /**
         * Make the priority queue logically empty.
         */
        public void makeEmpty( )
        {
            currentSize = 0;
        }

        private static final int DEFAULT_CAPACITY = 100;

        private int currentSize;      // Number of elements in heap
        private Comparable [ ] array; // The heap array

        /**
         * Internal method to percolate down in the heap.
         * @param hole the index at which the percolate begins.
         */
        private void percolateDown( int hole )
        {
/* 1*/      int child;
/* 2*/      Comparable tmp = array[ hole ];

/* 3*/      for( ; hole * 2 <= currentSize; hole = child )
            {
/* 4*/          child = hole * 2;
/* 5*/          if( child != currentSize &&
/* 6*/                  array[ child + 1 ].compareTo( array[ child ] ) < 0 )
/* 7*/              child++;
/* 8*/          if( array[ child ].compareTo( tmp ) < 0 )
/* 9*/              array[ hole ] = array[ child ];
                else
/*10*/              break;
            }
/*11*/      array[ hole ] = tmp;
        }

            // Test program
        public static void main( String [ ] args )
        {
            int numItems = 10000;
            BinaryHeap h = new BinaryHeap( numItems );
            int i = 37;

            try
            {
                for( i = 37; i != 0; i = ( i + 37 ) % numItems )
                    h.insert( new MyInteger( i ) );
                for( i = 1; i < numItems; i++ )
                    if( ((MyInteger)( h.deleteMin( ) )).intValue( ) != i )
                        System.out.println( "Oops! " + i );

                for( i = 37; i != 0; i = ( i + 37 ) % numItems )
                    h.insert( new MyInteger( i ) );
                h.insert( new MyInteger( 0 ) );
                i = 9999999;
                h.insert( new MyInteger( i ) );
                for( i = 1; i <= numItems; i++ )
                    if( ((MyInteger)( h.deleteMin( ) )).intValue( ) != i )
                        System.out.println( "Oops! " + i + " " );
            }
            catch( Overflow e )
              { System.out.println( "Overflow (expected)! " + i  ); }
        }
    }