binary heap

In computer science, a heap is a specialized tree-based data structure that satisfies the heap property: If A is a parentnode of B then the key of node A is ordered with respect to the key of node B with the same ordering applying across the heap. Either the keys of parent nodes are always greater than or equal to those of the children and the highest key is in the root node (this kind of heap is called max heap) or the keys of parent nodes are less than or equal to those of the children and the lowest key is in the root node (min heap). Heaps are crucial in several efficient graph algorithms such as Dijkstra's algorithm, and in the sorting algorithm heapsort. A common implementation of a heap is the binary heap, in which the tree is a complete binary tree (see figure).

Heaps are usually implemented in an array, and do not require pointers between elements.

Full and almost full binary heaps may be represented in a very space-efficient way using an array alone. The first (or last) element will contain the root. The next two elements of the array contain its children. The next four contain the four children of the two child nodes, etc. Thus the children of the node at position n would be at positions 2n and 2n+1 in a one-based array, or 2n+1 and 2n+2 in a zero-based array. This allows moving up or down the tree by doing simple index computations. Balancing a heap is done by swapping elements which are out of order. As we can build a heap from an array without requiring extra memory (for the nodes, for example), heapsort can be used to sort an array in-place.

The operations commonly performed with a heap are:

• create-heap: create an empty heap
• heapify: create a heap out of given array of elements
• find-max or find-min: find the maximum item of a max-heap or a minimum item of a min-heap, respectively (aka, peek)
• delete-max or delete-min: removing the root node of a max- or min-heap, respectively
• increase-key or decrease-key: updating a key within a max- or min-heap, respectively
• insert: adding a new key to the heap
• merge: joining two heaps to form a valid new heap containing all the elements of both.
• meld(h1,h2): Return the heap formed by taking the union of the item-disjoint heaps h1 and h2. Melding destroys h1 and h2.
• size: return the number of items in the heap.
• isEmpty(): returns true if the heap is empty, false otherwise.
• buildHeap(list): builds a new heap from a list of keys.
• ExtractMin() [or ExtractMax()]: Returns the node of minimum value from a min heap [or maximum value from a max heap] after removing it from the heap
• Union(): Creates a new heap by joining two heaps given as input.
• Shift-up: Move a node up in the tree, as long as needed (depending on the heap condition: min-heap or max-heap)
• Shift-down: Move a node down in the tree, similar to Shift-up

Different types of heaps implement the operations in different ways, but notably, insertion is often done by adding the new element at the end of the heap in the first available free space. This will tend to violate the heap property, and so the elements are then reordered until the heap property has been reestablished. Construction of a binary (or d-ary) heap out of a given array of elements may be performed faster than a sequence of consecutive insertions into an originally empty heap using the classic Floyd's algorithm, with the worst-case number of comparisons equal to 2N − 2s₂(N) − e₂(N) (for a binary heap), wheres₂(N) is the sum of all digits of the binary representation of N and e₂(N) is the exponent of 2 in the prime factorization of N

（reference from：https://en.wikipedia.org/wiki/Heap_(data_structure)）

Binary heap

There are several types of heaps, but in the current article we are going to discuss the binary heap. For short, let's call it just "heap". It is used to implement priority queue ADTand in the heapsort algorithm. Heap is a complete binary tree, which answers to the heap property.

http://www.algolist.net/Data_structures/Binary_heap

Implementation in java

路径：commons-collections-3.2.1-src/src/java/org/apache/commons/collections/BinaryHeap.java

/**
 *  Licensed to the Apache Software Foundation (ASF) under one or more
 *  contributor license agreements.  See the NOTICE file distributed with
 *  this work for additional information regarding copyright ownership.
 *  The ASF licenses this file to You under the Apache License, Version 2.0
 *  (the "License"); you may not use this file except in compliance with
 *  the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 *  Unless required by applicable law or agreed to in writing, software
 *  distributed under the License is distributed on an "AS IS" BASIS,
 *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  See the License for the specific language governing permissions and
 *  limitations under the License.
 */
package org.apache.commons.collections;

import java.util.AbstractCollection;
import java.util.Comparator;
import java.util.Iterator;
import java.util.NoSuchElementException;

/***
 * Binary heap implementation of <code>PriorityQueue</code>.
 * <p>
 * The <code>PriorityQueue</code> interface has now been replaced for most uses
 * by the <code>Buffer</code> interface. This class and the interface are
 * retained for backwards compatibility. The intended replacement is
 * {@link org.apache.commons.collections.buffer.PriorityBuffer PriorityBuffer}.
 * <p>
 * The removal order of a binary heap is based on either the natural sort
 * order of its elements or a specified {@link Comparator}.  The
 * {@link #pop()} method always returns the first element as determined
 * by the sort order.  (The <code>isMinHeap</code> flag in the constructors
 * can be used to reverse the sort order, in which case {@link #pop()}
 * will always remove the last element.)  The removal order is
 * <i>not</i> the same as the order of iteration; elements are
 * returned by the iterator in no particular order.
 * <p>
 * The {@link #insert(Object)} and {@link #pop()} operations perform
 * in logarithmic time.  The {@link #peek()} operation performs in constant
 * time.  All other operations perform in linear time or worse.
 * <p>
 * Note that this implementation is not synchronized.  Use SynchronizedPriorityQueue
 * to provide synchronized access to a <code>BinaryHeap</code>:
 *
 * <pre>
 * PriorityQueue heap = new SynchronizedPriorityQueue(new BinaryHeap());
 * </pre>
 *
 * @deprecated Replaced by PriorityBuffer in buffer subpackage.
 *  Due to be removed in v4.0.
 * @since Commons Collections 1.0
 * @version $Revision: 646777 $ $Date: 2008-04-10 13:33:15 +0100 (Thu, 10 Apr 2008) $
 *
 * @author Peter Donald
 * @author Ram Chidambaram
 * @author Michael A. Smith
 * @author Paul Jack
 * @author Stephen Colebourne
 */
public final class BinaryHeap extends AbstractCollection
        implements PriorityQueue, Buffer {

    /***
     * The default capacity for a binary heap.
     */
    private final static int DEFAULT_CAPACITY = 13;
    /***
     * The number of elements currently in this heap.
     */
    int m_size;  // package scoped for testing
    /***
     * The elements in this heap.
     */
    Object[] m_elements;  // package scoped for testing
    /***
     * If true, the first element as determined by the sort order will
     * be returned.  If false, the last element as determined by the
     * sort order will be returned.
     */
    boolean m_isMinHeap;  // package scoped for testing
    /***
     * The comparator used to order the elements
     */
    Comparator m_comparator;  // package scoped for testing

    /***
     * Constructs a new minimum binary heap.
     */
    public BinaryHeap() {
        this(DEFAULT_CAPACITY, true);
    }

    /***
     * Constructs a new <code>BinaryHeap</code> that will use the given
     * comparator to order its elements.
     *
     * @param comparator  the comparator used to order the elements, null
     *  means use natural order
     */
    public BinaryHeap(Comparator comparator) {
        this();
        m_comparator = comparator;
    }

    /***
     * Constructs a new minimum binary heap with the specified initial capacity.
     *
     * @param capacity  The initial capacity for the heap.  This value must
     *  be greater than zero.
     * @throws IllegalArgumentException
     *  if <code>capacity</code> is &lt;= <code>0</code>
     */
    public BinaryHeap(int capacity) {
        this(capacity, true);
    }

    /***
     * Constructs a new <code>BinaryHeap</code>.
     *
     * @param capacity  the initial capacity for the heap
     * @param comparator  the comparator used to order the elements, null
     *  means use natural order
     * @throws IllegalArgumentException
     *  if <code>capacity</code> is &lt;= <code>0</code>
     */
    public BinaryHeap(int capacity, Comparator comparator) {
        this(capacity);
        m_comparator = comparator;
    }

    /***
     * Constructs a new minimum or maximum binary heap
     *
     * @param isMinHeap  if <code>true</code> the heap is created as a
     * minimum heap; otherwise, the heap is created as a maximum heap
     */
    public BinaryHeap(boolean isMinHeap) {
        this(DEFAULT_CAPACITY, isMinHeap);
    }

    /***
     * Constructs a new <code>BinaryHeap</code>.
     *
     * @param isMinHeap  true to use the order imposed by the given
     *   comparator; false to reverse that order
     * @param comparator  the comparator used to order the elements, null
     *  means use natural order
     */
    public BinaryHeap(boolean isMinHeap, Comparator comparator) {
        this(isMinHeap);
        m_comparator = comparator;
    }

    /***
     * Constructs a new minimum or maximum binary heap with the specified
     * initial capacity.
     *
     * @param capacity the initial capacity for the heap.  This value must
     * be greater than zero.
     * @param isMinHeap if <code>true</code> the heap is created as a
     *  minimum heap; otherwise, the heap is created as a maximum heap.
     * @throws IllegalArgumentException
     *  if <code>capacity</code> is <code>&lt;= 0</code>
     */
    public BinaryHeap(int capacity, boolean isMinHeap) {
        if (capacity <= 0) {
            throw new IllegalArgumentException("invalid capacity");
        }
        m_isMinHeap = isMinHeap;

        //+1 as 0 is noop
        m_elements = new Object[capacity + 1];
    }

    /***
     * Constructs a new <code>BinaryHeap</code>.
     *
     * @param capacity  the initial capacity for the heap
     * @param isMinHeap  true to use the order imposed by the given
     *   comparator; false to reverse that order
     * @param comparator  the comparator used to order the elements, null
     *  means use natural order
     * @throws IllegalArgumentException
     *  if <code>capacity</code> is <code>&lt;= 0</code>
     */
    public BinaryHeap(int capacity, boolean isMinHeap, Comparator comparator) {
        this(capacity, isMinHeap);
        m_comparator = comparator;
    }

    //-----------------------------------------------------------------------
    /***
     * Clears all elements from queue.
     */
    public void clear() {
        m_elements = new Object[m_elements.length];  // for gc
        m_size = 0;
    }

    /***
     * Tests if queue is empty.
     *
     * @return <code>true</code> if queue is empty; <code>false</code>
     *  otherwise.
     */
    public boolean isEmpty() {
        return m_size == 0;
    }

    /***
     * Tests if queue is full.
     *
     * @return <code>true</code> if queue is full; <code>false</code>
     *  otherwise.
     */
    public boolean isFull() {
        //+1 as element 0 is noop
        return m_elements.length == m_size + 1;
    }

    /***
     * Inserts an element into queue.
     *
     * @param element  the element to be inserted
     */
    public void insert(Object element) {
        if (isFull()) {
            grow();
        }
        //percolate element to it's place in tree
        if (m_isMinHeap) {
            percolateUpMinHeap(element);
        } else {
            percolateUpMaxHeap(element);
        }
    }

    /***
     * Returns the element on top of heap but don't remove it.
     *
     * @return the element at top of heap
     * @throws NoSuchElementException  if <code>isEmpty() == true</code>
     */
    public Object peek() throws NoSuchElementException {
        if (isEmpty()) {
            throw new NoSuchElementException();
        } else {
            return m_elements[1];
        }
    }

    /***
     * Returns the element on top of heap and remove it.
     *
     * @return the element at top of heap
     * @throws NoSuchElementException  if <code>isEmpty() == true</code>
     */
    public Object pop() throws NoSuchElementException {
        final Object result = peek();
        m_elements[1] = m_elements[m_size--];

        // set the unused element to 'null' so that the garbage collector
        // can free the object if not used anywhere else.(remove reference)
        m_elements[m_size + 1] = null;

        if (m_size != 0) {
            // percolate top element to it's place in tree
            if (m_isMinHeap) {
                percolateDownMinHeap(1);
            } else {
                percolateDownMaxHeap(1);
            }
        }

        return result;
    }

    /***
     * Percolates element down heap from the position given by the index.
     * <p>
     * Assumes it is a minimum heap.
     *
     * @param index the index for the element
     */
    protected void percolateDownMinHeap(final int index) {
        final Object element = m_elements[index];
        int hole = index;

        while ((hole * 2) <= m_size) {
            int child = hole * 2;

            // if we have a right child and that child can not be percolated
            // up then move onto other child
            if (child != m_size && compare(m_elements[child + 1], m_elements[child]) < 0) {
                child++;
            }

            // if we found resting place of bubble then terminate search
            if (compare(m_elements[child], element) >= 0) {
                break;
            }

            m_elements[hole] = m_elements[child];
            hole = child;
        }

        m_elements[hole] = element;
    }

    /***
     * Percolates element down heap from the position given by the index.
     * <p>
     * Assumes it is a maximum heap.
     *
     * @param index the index of the element
     */
    protected void percolateDownMaxHeap(final int index) {
        final Object element = m_elements[index];
        int hole = index;

        while ((hole * 2) <= m_size) {
            int child = hole * 2;

            // if we have a right child and that child can not be percolated
            // up then move onto other child
            if (child != m_size && compare(m_elements[child + 1], m_elements[child]) > 0) {
                child++;
            }

            // if we found resting place of bubble then terminate search
            if (compare(m_elements[child], element) <= 0) {
                break;
            }

            m_elements[hole] = m_elements[child];
            hole = child;
        }

        m_elements[hole] = element;
    }

    /***
     * Percolates element up heap from the position given by the index.
     * <p>
     * Assumes it is a minimum heap.
     *
     * @param index the index of the element to be percolated up
     */
    protected void percolateUpMinHeap(final int index) {
        int hole = index;
        Object element = m_elements[hole];
        while (hole > 1 && compare(element, m_elements[hole / 2]) < 0) {
            // save element that is being pushed down
            // as the element "bubble" is percolated up
            final int next = hole / 2;
            m_elements[hole] = m_elements[next];
            hole = next;
        }
        m_elements[hole] = element;
    }

    /***
     * Percolates a new element up heap from the bottom.
     * <p>
     * Assumes it is a minimum heap.
     *
     * @param element the element
     */
    protected void percolateUpMinHeap(final Object element) {
        m_elements[++m_size] = element;
        percolateUpMinHeap(m_size);
    }

    /***
     * Percolates element up heap from from the position given by the index.
     * <p>
     * Assume it is a maximum heap.
     *
     * @param index the index of the element to be percolated up
     */
    protected void percolateUpMaxHeap(final int index) {
        int hole = index;
        Object element = m_elements[hole];

        while (hole > 1 && compare(element, m_elements[hole / 2]) > 0) {
            // save element that is being pushed down
            // as the element "bubble" is percolated up
            final int next = hole / 2;
            m_elements[hole] = m_elements[next];
            hole = next;
        }

        m_elements[hole] = element;
    }

    /***
     * Percolates a new element up heap from the bottom.
     * <p>
     * Assume it is a maximum heap.
     *
     * @param element the element
     */
    protected void percolateUpMaxHeap(final Object element) {
        m_elements[++m_size] = element;
        percolateUpMaxHeap(m_size);
    }

    /***
     * Compares two objects using the comparator if specified, or the
     * natural order otherwise.
     *
     * @param a  the first object
     * @param b  the second object
     * @return -ve if a less than b, 0 if they are equal, +ve if a greater than b
     */
    private int compare(Object a, Object b) {
        if (m_comparator != null) {
            return m_comparator.compare(a, b);
        } else {
            return ((Comparable) a).compareTo(b);
        }
    }

    /***
     * Increases the size of the heap to support additional elements
     */
    protected void grow() {
        final Object[] elements = new Object[m_elements.length * 2];
        System.arraycopy(m_elements, 0, elements, 0, m_elements.length);
        m_elements = elements;
    }

    /***
     * Returns a string representation of this heap.  The returned string
     * is similar to those produced by standard JDK collections.
     *
     * @return a string representation of this heap
     */
    public String toString() {
        final StringBuffer sb = new StringBuffer();

        sb.append("[ ");

        for (int i = 1; i < m_size + 1; i++) {
            if (i != 1) {
                sb.append(", ");
            }
            sb.append(m_elements[i]);
        }

        sb.append(" ]");

        return sb.toString();
    }

    /***
     * Returns an iterator over this heap's elements.
     *
     * @return an iterator over this heap's elements
     */
    public Iterator iterator() {
        return new Iterator() {

            private int index = 1;
            private int lastReturnedIndex = -1;

            public boolean hasNext() {
                return index <= m_size;
            }

            public Object next() {
                if (!hasNext()) throw new NoSuchElementException();
                lastReturnedIndex = index;
                index++;
                return m_elements[lastReturnedIndex];
            }

            public void remove() {
                if (lastReturnedIndex == -1) {
                    throw new IllegalStateException();
                }
                m_elements[ lastReturnedIndex ] = m_elements[ m_size ];
                m_elements[ m_size ] = null;
                m_size--;
                if( m_size != 0 && lastReturnedIndex <= m_size) {
                    int compareToParent = 0;
                    if (lastReturnedIndex > 1) {
                        compareToParent = compare(m_elements[lastReturnedIndex],
                            m_elements[lastReturnedIndex / 2]);
                    }
                    if (m_isMinHeap) {
                        if (lastReturnedIndex > 1 && compareToParent < 0) {
                            percolateUpMinHeap(lastReturnedIndex);
                        } else {
                            percolateDownMinHeap(lastReturnedIndex);
                        }
                    } else {  // max heap
                        if (lastReturnedIndex > 1 && compareToParent > 0) {
                            percolateUpMaxHeap(lastReturnedIndex);
                        } else {
                            percolateDownMaxHeap(lastReturnedIndex);
                        }
                    }
                }
                index--;
                lastReturnedIndex = -1;
            }

        };
    }

    /***
     * Adds an object to this heap. Same as {@link #insert(Object)}.
     *
     * @param object  the object to add
     * @return true, always
     */
    public boolean add(Object object) {
        insert(object);
        return true;
    }

    /***
     * Returns the priority element. Same as {@link #peek()}.
     *
     * @return the priority element
     * @throws BufferUnderflowException if this heap is empty
     */
    public Object get() {
        try {
            return peek();
        } catch (NoSuchElementException e) {
            throw new BufferUnderflowException();
        }
    }

    /***
     * Removes the priority element. Same as {@link #pop()}.
     *
     * @return the removed priority element
     * @throws BufferUnderflowException if this heap is empty
     */
    public Object remove() {
        try {
            return pop();
        } catch (NoSuchElementException e) {
            throw new BufferUnderflowException();
        }
    }

    /***
     * Returns the number of elements in this heap.
     *
     * @return the number of elements in this heap
     */
    public int size() {
        return m_size;
    }

}