平衡二分堆的使用及优先级队列的实现
Priority queues are a kind of queue in which the elements are dequeued in priority order.
They are a mutable data abstraction: enqueues and dequeues are destructive. Each element has a priority, an element of a totally ordered set (usually a number) More important things come out first, even if they were added later Our convention: smaller number = higher priority There is no (fast) operation to find out whether an arbitrary element is in the queue Useful for event-based simulators (with priority = simulated time), real-time games, searching, routing, compression via Huffman coding(Turn?on?JavaScript?to?see?code?examples)
There are many ways to implement this signature. For example, we could implement it as a linked list, with Repeat #2 as necessary.
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Example: inserting 4 into previous tree.
/* * @(#)PriorityQueue.java1.16 06/04/21 * * Copyright 2006 Sun Microsystems, Inc. All rights reserved. * SUN PROPRIETARY/CONFIDENTIAL. Use is subject to license terms. */package java.util;/** * An unbounded priority {@linkplain Queue queue} based on a priority heap. * The elements of the priority queue are ordered according to their * {@linkplain Comparable natural ordering}, or by a {@link Comparator} * provided at queue construction time, depending on which constructor is * used. A priority queue does not permit {@code null} elements. * A priority queue relying on natural ordering also does not permit * insertion of non-comparable objects (doing so may result in * {@code ClassCastException}). * * <p>The <em>head</em> of this queue is the <em>least</em> element * with respect to the specified ordering. If multiple elements are * tied for least value, the head is one of those elements -- ties are * broken arbitrarily. The queue retrieval operations {@code poll}, * {@code remove}, {@code peek}, and {@code element} access the * element at the head of the queue. * * <p>A priority queue is unbounded, but has an internal * <i>capacity</i> governing the size of an array used to store the * elements on the queue. It is always at least as large as the queue * size. As elements are added to a priority queue, its capacity * grows automatically. The details of the growth policy are not * specified. * * <p>This class and its iterator implement all of the * <em>optional</em> methods of the {@link Collection} and {@link * Iterator} interfaces. The Iterator provided in method {@link * #iterator()} is <em>not</em> guaranteed to traverse the elements of * the priority queue in any particular order. If you need ordered * traversal, consider using {@code Arrays.sort(pq.toArray())}. * * <p> <strong>Note that this implementation is not synchronized.</strong> * Multiple threads should not access a {@code PriorityQueue} * instance concurrently if any of the threads modifies the queue. * Instead, use the thread-safe {@link * java.util.concurrent.PriorityBlockingQueue} class. * * <p>Implementation note: this implementation provides * O(log(n)) time for the enqueing and dequeing methods * ({@code offer}, {@code poll}, {@code remove()} and {@code add}); * linear time for the {@code remove(Object)} and {@code contains(Object)} * methods; and constant time for the retrieval methods * ({@code peek}, {@code element}, and {@code size}). * * <p>This class is a member of the * <a href="{@docRoot}/../technotes/guides/collections/index.html"> * Java Collections Framework</a>. * * @since 1.5 * @version 1.16, 04/21/06 * @author Josh Bloch, Doug Lea * @param <E> the type of elements held in this collection */public class PriorityQueue<E> extends AbstractQueue<E> implements java.io.Serializable { private static final long serialVersionUID = -7720805057305804111L; private static final int DEFAULT_INITIAL_CAPACITY = 11; /** * Priority queue represented as a balanced binary heap: the two * children of queue[n] are queue[2*n+1] and queue[2*(n+1)]. The * priority queue is ordered by comparator, or by the elements' * natural ordering, if comparator is null: For each node n in the * heap and each descendant d of n, n <= d. The element with the * lowest value is in queue[0], assuming the queue is nonempty. */ private transient Object[] queue; /** * The number of elements in the priority queue. */ private int size = 0; /** * The comparator, or null if priority queue uses elements' * natural ordering. */ private final Comparator<? super E> comparator; /** * The number of times this priority queue has been * <i>structurally modified</i>. See AbstractList for gory details. */ private transient int modCount = 0; /** * Creates a {@code PriorityQueue} with the default initial * capacity (11) that orders its elements according to their * {@linkplain Comparable natural ordering}. */ public PriorityQueue() { this(DEFAULT_INITIAL_CAPACITY, null); } /** * Creates a {@code PriorityQueue} with the specified initial * capacity that orders its elements according to their * {@linkplain Comparable natural ordering}. * * @param initialCapacity the initial capacity for this priority queue * @throws IllegalArgumentException if {@code initialCapacity} is less * than 1 */ public PriorityQueue(int initialCapacity) { this(initialCapacity, null); } /** * Creates a {@code PriorityQueue} with the specified initial capacity * that orders its elements according to the specified comparator. * * @param initialCapacity the initial capacity for this priority queue * @param comparator the comparator that will be used to order this * priority queue. If {@code null}, the {@linkplain Comparable * natural ordering} of the elements will be used. * @throws IllegalArgumentException if {@code initialCapacity} is * less than 1 */ public PriorityQueue(int initialCapacity, Comparator<? super E> comparator) { // Note: This restriction of at least one is not actually needed, // but continues for 1.5 compatibility if (initialCapacity < 1) throw new IllegalArgumentException(); this.queue = new Object[initialCapacity]; this.comparator = comparator; } /** * Creates a {@code PriorityQueue} containing the elements in the * specified collection. If the specified collection is an instance of * a {@link SortedSet} or is another {@code PriorityQueue}, this * priority queue will be ordered according to the same ordering. * Otherwise, this priority queue will be ordered according to the * {@linkplain Comparable natural ordering} of its elements. * * @param c the collection whose elements are to be placed * into this priority queue * @throws ClassCastException if elements of the specified collection * cannot be compared to one another according to the priority * queue's ordering * @throws NullPointerException if the specified collection or any * of its elements are null */ public PriorityQueue(Collection<? extends E> c) { initFromCollection(c); if (c instanceof SortedSet) comparator = (Comparator<? super E>) ((SortedSet<? extends E>)c).comparator(); else if (c instanceof PriorityQueue) comparator = (Comparator<? super E>) ((PriorityQueue<? extends E>)c).comparator(); else { comparator = null; heapify(); } } /** * Creates a {@code PriorityQueue} containing the elements in the * specified priority queue. This priority queue will be * ordered according to the same ordering as the given priority * queue. * * @param c the priority queue whose elements are to be placed * into this priority queue * @throws ClassCastException if elements of {@code c} cannot be * compared to one another according to {@code c}'s * ordering * @throws NullPointerException if the specified priority queue or any * of its elements are null */ public PriorityQueue(PriorityQueue<? extends E> c) { comparator = (Comparator<? super E>)c.comparator(); initFromCollection(c); } /** * Creates a {@code PriorityQueue} containing the elements in the * specified sorted set. This priority queue will be ordered * according to the same ordering as the given sorted set. * * @param c the sorted set whose elements are to be placed * into this priority queue * @throws ClassCastException if elements of the specified sorted * set cannot be compared to one another according to the * sorted set's ordering * @throws NullPointerException if the specified sorted set or any * of its elements are null */ public PriorityQueue(SortedSet<? extends E> c) { comparator = (Comparator<? super E>)c.comparator(); initFromCollection(c); } /** * Initializes queue array with elements from the given Collection. * * @param c the collection */ private void initFromCollection(Collection<? extends E> c) { Object[] a = c.toArray(); // If c.toArray incorrectly doesn't return Object[], copy it. if (a.getClass() != Object[].class) a = Arrays.copyOf(a, a.length, Object[].class); queue = a; size = a.length; } /** * Increases the capacity of the array. * * @param minCapacity the desired minimum capacity */ private void grow(int minCapacity) { if (minCapacity < 0) // overflow throw new OutOfMemoryError();int oldCapacity = queue.length; // Double size if small; else grow by 50% int newCapacity = ((oldCapacity < 64)? ((oldCapacity + 1) * 2): ((oldCapacity / 2) * 3)); if (newCapacity < 0) // overflow newCapacity = Integer.MAX_VALUE; if (newCapacity < minCapacity) newCapacity = minCapacity; queue = Arrays.copyOf(queue, newCapacity); } /** * Inserts the specified element into this priority queue. * * @return {@code true} (as specified by {@link Collection#add}) * @throws ClassCastException if the specified element cannot be * compared with elements currently in this priority queue * according to the priority queue's ordering * @throws NullPointerException if the specified element is null */ public boolean add(E e) { return offer(e); } /** * Inserts the specified element into this priority queue. * * @return {@code true} (as specified by {@link Queue#offer}) * @throws ClassCastException if the specified element cannot be * compared with elements currently in this priority queue * according to the priority queue's ordering * @throws NullPointerException if the specified element is null */ public boolean offer(E e) { if (e == null) throw new NullPointerException(); modCount++; int i = size; if (i >= queue.length) grow(i + 1); size = i + 1; if (i == 0) queue[0] = e; else siftUp(i, e); return true; } public E peek() { if (size == 0) return null; return (E) queue[0]; } private int indexOf(Object o) {if (o != null) { for (int i = 0; i < size; i++) if (o.equals(queue[i])) return i; } return -1; } /** * Removes a single instance of the specified element from this queue, * if it is present. More formally, removes an element {@code e} such * that {@code o.equals(e)}, if this queue contains one or more such * elements. Returns {@code true} if and only if this queue contained * the specified element (or equivalently, if this queue changed as a * result of the call). * * @param o element to be removed from this queue, if present * @return {@code true} if this queue changed as a result of the call */ public boolean remove(Object o) {int i = indexOf(o);if (i == -1) return false;else { removeAt(i); return true;} } /** * Version of remove using reference equality, not equals. * Needed by iterator.remove. * * @param o element to be removed from this queue, if present * @return {@code true} if removed */ boolean removeEq(Object o) {for (int i = 0; i < size; i++) { if (o == queue[i]) { removeAt(i); return true; } } return false; } /** * Returns {@code true} if this queue contains the specified element. * More formally, returns {@code true} if and only if this queue contains * at least one element {@code e} such that {@code o.equals(e)}. * * @param o object to be checked for containment in this queue * @return {@code true} if this queue contains the specified element */ public boolean contains(Object o) {return indexOf(o) != -1; } /** * Returns an array containing all of the elements in this queue. * The elements are in no particular order. * * <p>The returned array will be "safe" in that no references to it are * maintained by this queue. (In other words, this method must allocate * a new array). The caller is thus free to modify the returned array. * * <p>This method acts as bridge between array-based and collection-based * APIs. * * @return an array containing all of the elements in this queue */ public Object[] toArray() { return Arrays.copyOf(queue, size); } /** * Returns an array containing all of the elements in this queue; the * runtime type of the returned array is that of the specified array. * The returned array elements are in no particular order. * If the queue fits in the specified array, it is returned therein. * Otherwise, a new array is allocated with the runtime type of the * specified array and the size of this queue. * * <p>If the queue fits in the specified array with room to spare * (i.e., the array has more elements than the queue), the element in * the array immediately following the end of the collection is set to * {@code null}. * * <p>Like the {@link #toArray()} method, this method acts as bridge between * array-based and collection-based APIs. Further, this method allows * precise control over the runtime type of the output array, and may, * under certain circumstances, be used to save allocation costs. * * <p>Suppose <tt>x</tt> is a queue known to contain only strings. * The following code can be used to dump the queue into a newly * allocated array of <tt>String</tt>: * * <pre> * String[] y = x.toArray(new String[0]);</pre> * * Note that <tt>toArray(new Object[0])</tt> is identical in function to * <tt>toArray()</tt>. * * @param a the array into which the elements of the queue are to * be stored, if it is big enough; otherwise, a new array of the * same runtime type is allocated for this purpose. * @return an array containing all of the elements in this queue * @throws ArrayStoreException if the runtime type of the specified array * is not a supertype of the runtime type of every element in * this queue * @throws NullPointerException if the specified array is null */ public <T> T[] toArray(T[] a) { if (a.length < size) // Make a new array of a's runtime type, but my contents: return (T[]) Arrays.copyOf(queue, size, a.getClass());System.arraycopy(queue, 0, a, 0, size); if (a.length > size) a[size] = null; return a; } /** * Returns an iterator over the elements in this queue. The iterator * does not return the elements in any particular order. * * @return an iterator over the elements in this queue */ public Iterator<E> iterator() { return new Itr(); } private final class Itr implements Iterator<E> { /** * Index (into queue array) of element to be returned by * subsequent call to next. */ private int cursor = 0; /** * Index of element returned by most recent call to next, * unless that element came from the forgetMeNot list. * Set to -1 if element is deleted by a call to remove. */ private int lastRet = -1; /** * A queue of elements that were moved from the unvisited portion of * the heap into the visited portion as a result of "unlucky" element * removals during the iteration. (Unlucky element removals are those * that require a siftup instead of a siftdown.) We must visit all of * the elements in this list to complete the iteration. We do this * after we've completed the "normal" iteration. * * We expect that most iterations, even those involving removals, * will not need to store elements in this field. */ private ArrayDeque<E> forgetMeNot = null; /** * Element returned by the most recent call to next iff that * element was drawn from the forgetMeNot list. */ private E lastRetElt = null; /** * The modCount value that the iterator believes that the backing * Queue should have. If this expectation is violated, the iterator * has detected concurrent modification. */ private int expectedModCount = modCount; public boolean hasNext() { return cursor < size || (forgetMeNot != null && !forgetMeNot.isEmpty()); } public E next() { if (expectedModCount != modCount) throw new ConcurrentModificationException(); if (cursor < size) return (E) queue[lastRet = cursor++]; if (forgetMeNot != null) { lastRet = -1; lastRetElt = forgetMeNot.poll(); if (lastRetElt != null) return lastRetElt; } throw new NoSuchElementException(); } public void remove() { if (expectedModCount != modCount) throw new ConcurrentModificationException(); if (lastRet != -1) { E moved = PriorityQueue.this.removeAt(lastRet); lastRet = -1; if (moved == null) cursor--; else { if (forgetMeNot == null) forgetMeNot = new ArrayDeque<E>(); forgetMeNot.add(moved); } } else if (lastRetElt != null) { PriorityQueue.this.removeEq(lastRetElt); lastRetElt = null; } else { throw new IllegalStateException(); } expectedModCount = modCount; } } public int size() { return size; } /** * Removes all of the elements from this priority queue. * The queue will be empty after this call returns. */ public void clear() { modCount++; for (int i = 0; i < size; i++) queue[i] = null; size = 0; } public E poll() { if (size == 0) return null; int s = --size; modCount++; E result = (E) queue[0]; E x = (E) queue[s]; queue[s] = null; if (s != 0) siftDown(0, x); return result; } /** * Removes the ith element from queue. * * Normally this method leaves the elements at up to i-1, * inclusive, untouched. Under these circumstances, it returns * null. Occasionally, in order to maintain the heap invariant, * it must swap a later element of the list with one earlier than * i. Under these circumstances, this method returns the element * that was previously at the end of the list and is now at some * position before i. This fact is used by iterator.remove so as to * avoid missing traversing elements. */ private E removeAt(int i) { assert i >= 0 && i < size; modCount++; int s = --size; if (s == i) // removed last element queue[i] = null; else { E moved = (E) queue[s]; queue[s] = null; siftDown(i, moved); if (queue[i] == moved) { siftUp(i, moved); if (queue[i] != moved) return moved; } } return null; } /** * Inserts item x at position k, maintaining heap invariant by * promoting x up the tree until it is greater than or equal to * its parent, or is the root. * * To simplify and speed up coercions and comparisons. the * Comparable and Comparator versions are separated into different * methods that are otherwise identical. (Similarly for siftDown.) * * @param k the position to fill * @param x the item to insert */ private void siftUp(int k, E x) { if (comparator != null) siftUpUsingComparator(k, x); else siftUpComparable(k, x); } private void siftUpComparable(int k, E x) { Comparable<? super E> key = (Comparable<? super E>) x; while (k > 0) { int parent = (k - 1) >>> 1; Object e = queue[parent]; if (key.compareTo((E) e) >= 0) break; queue[k] = e; k = parent; } queue[k] = key; } private void siftUpUsingComparator(int k, E x) { while (k > 0) { int parent = (k - 1) >>> 1; Object e = queue[parent]; if (comparator.compare(x, (E) e) >= 0) break; queue[k] = e; k = parent; } queue[k] = x; } /** * Inserts item x at position k, maintaining heap invariant by * demoting x down the tree repeatedly until it is less than or * equal to its children or is a leaf. * * @param k the position to fill * @param x the item to insert */ private void siftDown(int k, E x) { if (comparator != null) siftDownUsingComparator(k, x); else siftDownComparable(k, x); } private void siftDownComparable(int k, E x) { Comparable<? super E> key = (Comparable<? super E>)x; int half = size >>> 1; // loop while a non-leaf while (k < half) { int child = (k << 1) + 1; // assume left child is least Object c = queue[child]; int right = child + 1; if (right < size && ((Comparable<? super E>) c).compareTo((E) queue[right]) > 0) c = queue[child = right]; if (key.compareTo((E) c) <= 0) break; queue[k] = c; k = child; } queue[k] = key; } private void siftDownUsingComparator(int k, E x) { int half = size >>> 1; while (k < half) { int child = (k << 1) + 1; Object c = queue[child]; int right = child + 1; if (right < size && comparator.compare((E) c, (E) queue[right]) > 0) c = queue[child = right]; if (comparator.compare(x, (E) c) <= 0) break; queue[k] = c; k = child; } queue[k] = x; } /** * Establishes the heap invariant (described above) in the entire tree, * assuming nothing about the order of the elements prior to the call. */ private void heapify() { for (int i = (size >>> 1) - 1; i >= 0; i--) siftDown(i, (E) queue[i]); } /** * Returns the comparator used to order the elements in this * queue, or {@code null} if this queue is sorted according to * the {@linkplain Comparable natural ordering} of its elements. * * @return the comparator used to order this queue, or * {@code null} if this queue is sorted according to the * natural ordering of its elements */ public Comparator<? super E> comparator() { return comparator; } /** * Saves the state of the instance to a stream (that * is, serializes it). * * @serialData The length of the array backing the instance is * emitted (int), followed by all of its elements * (each an {@code Object}) in the proper order. * @param s the stream */ private void writeObject(java.io.ObjectOutputStream s) throws java.io.IOException{ // Write out element count, and any hidden stuff s.defaultWriteObject(); // Write out array length, for compatibility with 1.5 version s.writeInt(Math.max(2, size + 1)); // Write out all elements in the "proper order". for (int i = 0; i < size; i++) s.writeObject(queue[i]); } /** * Reconstitutes the {@code PriorityQueue} instance from a stream * (that is, deserializes it). * * @param s the stream */ private void readObject(java.io.ObjectInputStream s) throws java.io.IOException, ClassNotFoundException { // Read in size, and any hidden stuff s.defaultReadObject(); // Read in (and discard) array length s.readInt();queue = new Object[size]; // Read in all elements. for (int i = 0; i < size; i++) queue[i] = s.readObject();// Elements are guaranteed to be in "proper order", but the// spec has never explained what that might be.heapify(); }}?
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// BinaryHeap class//// CONSTRUCTION: empty or with initial array.//// ******************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// void makeEmpty( ) --> Remove all items// ******************ERRORS********************************// Throws UnderflowException for findMin and deleteMin when empty/** * Implements a binary heap. * Note that all "matching" is based on the compareTo method. * @author Mark Allen Weiss */public class BinaryHeap implements PriorityQueue { /** * Construct the binary heap. */ public BinaryHeap( ) { currentSize = 0; array = new Comparable[ DEFAULT_CAPACITY + 1 ]; } /** * Construct the binary heap from an array. * @param items the inital items in the binary heap. */ public BinaryHeap( Comparable [ ] items ) { currentSize = items.length; array = new Comparable[ items.length + 1 ]; for( int i = 0; i < items.length; i++ ) array[ i + 1 ] = items[ i ]; buildHeap( ); } /** * Insert into the priority queue. * Duplicates are allowed. * @param x the item to insert. * @return null, signifying that decreaseKey cannot be used. */ public PriorityQueue.Position insert( Comparable x ) { if( currentSize + 1 == array.length ) doubleArray( ); // Percolate up int hole = ++currentSize; array[ 0 ] = x; for( ; x.compareTo( array[ hole / 2 ] ) < 0; hole /= 2 ) array[ hole ] = array[ hole / 2 ]; array[ hole ] = x; return null; } /** * @throws UnsupportedOperationException because no Positions are returned * by the insert method for BinaryHeap. */ public void decreaseKey( PriorityQueue.Position p, Comparable newVal ) { throw new UnsupportedOperationException( "Cannot use decreaseKey for binary heap" ); } /** * Find the smallest item in the priority queue. * @return the smallest item. * @throws UnderflowException if empty. */ public Comparable findMin( ) { if( isEmpty( ) ) throw new UnderflowException( "Empty binary heap" ); return array[ 1 ]; } /** * Remove the smallest item from the priority queue. * @return the smallest item. * @throws UnderflowException if empty. */ public Comparable deleteMin( ) { 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; } /** * Returns size. * @return current size. */ public int size( ) { return currentSize; } /** * 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 ) { int child; Comparable tmp = array[ hole ]; for( ; hole * 2 <= currentSize; hole = child ) { child = hole * 2; if( child != currentSize && array[ child + 1 ].compareTo( array[ child ] ) < 0 ) child++; if( array[ child ].compareTo( tmp ) < 0 ) array[ hole ] = array[ child ]; else break; } array[ hole ] = tmp; } /** * Internal method to extend array. */ private void doubleArray( ) { Comparable [ ] newArray; newArray = new Comparable[ array.length * 2 ]; for( int i = 0; i < array.length; i++ ) newArray[ i ] = array[ i ]; array = newArray; } // Test program public static void main( String [ ] args ) { int numItems = 10000; BinaryHeap h1 = new BinaryHeap( ); Integer [ ] items = new Integer[ numItems - 1 ]; int i = 37; int j; for( i = 37, j = 0; i != 0; i = ( i + 37 ) % numItems, j++ ) { h1.insert( new Integer( i ) ); items[ j ] = new Integer( i ); } for( i = 1; i < numItems; i++ ) if( ((Integer)( h1.deleteMin( ) )).intValue( ) != i ) System.out.println( "Oops! " + i ); BinaryHeap h2 = new BinaryHeap( items ); for( i = 1; i < numItems; i++ ) if( ((Integer)( h2.deleteMin( ) )).intValue( ) != i ) System.out.println( "Oops! " + i ); }}// PriorityQueue interface//// ******************PUBLIC OPERATIONS*********************// Position insert( x ) --> Insert x// Comparable deleteMin( )--> Return and remove smallest item// Comparable findMin( ) --> Return smallest item// boolean isEmpty( ) --> Return true if empty; else false// void makeEmpty( ) --> Remove all items// int size( ) --> Return size// void decreaseKey( p, v)--> Decrease value in p to v// ******************ERRORS********************************// Throws UnderflowException for findMin and deleteMin when empty/** * PriorityQueue interface. * Some priority queues may support a decreaseKey operation, * but this is considered an advanced operation. If so, * a Position is returned by insert. * Note that all "matching" is based on the compareTo method. * @author Mark Allen Weiss */public interface PriorityQueue { /** * The Position interface represents a type that can * be used for the decreaseKey operation. */ public interface Position { /** * Returns the value stored at this position. * @return the value stored at this position. */ Comparable getValue( ); } /** * Insert into the priority queue, maintaining heap order. * Duplicates are allowed. * @param x the item to insert. * @return may return a Position useful for decreaseKey. */ Position insert( Comparable x ); /** * Find the smallest item in the priority queue. * @return the smallest item. * @throws UnderflowException if empty. */ Comparable findMin( ); /** * Remove the smallest item from the priority queue. * @return the smallest item. * @throws UnderflowException if empty. */ Comparable deleteMin( ); /** * Test if the priority queue is logically empty. * @return true if empty, false otherwise. */ boolean isEmpty( ); /** * Make the priority queue logically empty. */ void makeEmpty( ); /** * Returns the size. * @return current size. */ int size( ); /** * Change the value of the item stored in the pairing heap. * This is considered an advanced operation and might not * be supported by all priority queues. A priority queue * will signal its intention to not support decreaseKey by * having insert return null consistently. * @param p any non-null Position returned by insert. * @param newVal the new value, which must be smaller * than the currently stored value. * @throws IllegalArgumentException if p invalid. * @throws UnsupportedOperationException if appropriate. */ void decreaseKey( Position p, Comparable newVal );}/** * Exception class for access in empty containers * such as stacks, queues, and priority queues. * @author Mark Allen Weiss */public class UnderflowException extends RuntimeException { /** * Construct this exception object. * @param message the error message. */ public UnderflowException( String message ) { super( message ); }} ?