堆数据结构总结

一、最小堆（最大堆）

package com.ldl.algorithms.Exercise;

/*************************************************************************
 *  Compilation:  javac MinPQ.java
 *  Execution:    java MinPQ < input.txt
 *  
 *  Generic min priority queue implementation with a binary heap.
 *  Can be used with a comparator instead of the natural order.
 *
 *  % java MinPQ < tinyPQ.txt
 *  E A E (6 left on pq)
 *
 *  We use a one-based array to simplify parent and child calculations.
 *
 *************************************************************************/

import java.util.Comparator;
import java.util.Iterator;
import java.util.NoSuchElementException;
import com.ldl.algorithms.StdIn;
import com.ldl.algorithms.StdOut;

/**
 *  The <tt>MinPQ</tt> class represents a priority queue of generic keys.
 *  It supports the usual <em>insert</em> and <em>delete-the-minimum</em>
 *  operations, along with methods for peeking at the maximum key,
 *  testing if the priority queue is empty, and iterating through
 *  the keys.
 *  <p>
 *  The <em>insert</em> and <em>delete-the-minimum</em> operations take
 *  logarithmic amortized time.
 *  The <em>min</em>, <em>size</em>, and <em>is-empty</em> operations take constant time.
 *  Construction takes time proportional to the specified capacity or the number of
 *  items used to initialize the data structure.
 *  <p>
 *  This implementation uses a binary heap.
 *  <p>
 *  For additional documentation, see <a href="http://algs4.cs.princeton.edu/24pq">Section 2.4</a> of
 *  <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
 */
public class MinPQ<Key> implements Iterable<Key> {
    private Key[] pq;                    // store items at indices 1 to N
    private int N;                       // number of items on priority queue
    private Comparator<Key> comparator;  // optional comparator

   /**
     * Create an empty priority queue with the given initial capacity.
     */
    public MinPQ(int initCapacity) {
        pq = (Key[]) new Object[initCapacity + 1];
        N = 0;
    }

   /**
     * Create an empty priority queue.
     */
    public MinPQ() { this(1); }

   /**
     * Create an empty priority queue with the given initial capacity,
     * using the given comparator.
     */
    public MinPQ(int initCapacity, Comparator<Key> comparator) {
        this.comparator = comparator;
        pq = (Key[]) new Object[initCapacity + 1];
        N = 0;
    }

   /**
     * Create an empty priority queue using the given comparator.
     */
    public MinPQ(Comparator<Key> comparator) { this(1, comparator); }

   /**
     * Create a priority queue with the given items.
     * Takes time proportional to the number of items using sink-based heap construction.
     */
    public MinPQ(Key[] keys) {
        N = keys.length;
        pq = (Key[]) new Object[keys.length + 1];
        for (int i = 0; i < N; i++)
            pq[i+1] = keys[i];
        for (int k = N/2; k >= 1; k--)
            sink(k);
        assert isMinHeap();
    }

   /**
     * Is the priority queue empty?
     */
    public boolean isEmpty() {
        return N == 0;
    }

   /**
     * Return the number of items on the priority queue.
     */
    public int size() {
        return N;
    }

   /**
     * Return the smallest key on the priority queue.
     * Throw an exception if no such key exists because the priority queue is empty.
     */
    public Key min() {
        if (isEmpty()) throw new RuntimeException("Priority queue underflow");
        return pq[1];
    }

    // helper function to double the size of the heap array
    private void resize(int capacity) {
        assert capacity > N;
        Key[] temp = (Key[]) new Object[capacity];
        for (int i = 1; i <= N; i++) temp[i] = pq[i];
        pq = temp;
    }

   /**
     * Add a new key to the priority queue.
     */
    public void insert(Key x) {
        // double size of array if necessary
        if (N == pq.length - 1) resize(2 * pq.length);

        // add x, and percolate it up to maintain heap invariant
        pq[++N] = x;
        swim(N);
        assert isMinHeap();
    }

   /**
     * Delete and return the smallest key on the priority queue.
     * Throw an exception if no such key exists because the priority queue is empty.
     */
    public Key delMin() {
        if (N == 0) throw new RuntimeException("Priority queue underflow");
        exch(1, N);
        Key min = pq[N--];
        sink(1);
        pq[N+1] = null;         // avoid loitering and help with garbage collection
        if ((N > 0) && (N == (pq.length - 1) / 4)) resize(pq.length  / 2);
        assert isMinHeap();
        return min;
    }


   /***********************************************************************
    * Helper functions to restore the heap invariant.
    **********************************************************************/

    private void swim(int k) {
        while (k > 1 && greater(k/2, k)) {
            exch(k, k/2);
            k = k/2;
        }
    }

    private void sink(int k) {
        while (2*k <= N) {
            int j = 2*k;
            if (j < N && greater(j, j+1)) j++;
            if (!greater(k, j)) break;
            exch(k, j);
            k = j;
        }
    }

   /***********************************************************************
    * Helper functions for compares and swaps.
    **********************************************************************/
    private boolean greater(int i, int j) {
        if (comparator == null) {
            return ((Comparable<Key>) pq[i]).compareTo(pq[j]) > 0;
        }
        else {
            return comparator.compare(pq[i], pq[j]) > 0;
        }
    }

    private void exch(int i, int j) {
        Key swap = pq[i];
        pq[i] = pq[j];
        pq[j] = swap;
    }

    // is pq[1..N] a min heap?
    private boolean isMinHeap() {
        return isMinHeap(1);
    }

    // is subtree of pq[1..N] rooted at k a min heap?
    private boolean isMinHeap(int k) {
        if (k > N) return true;
        int left = 2*k, right = 2*k + 1;
        if (left  <= N && greater(k, left))  return false;
        if (right <= N && greater(k, right)) return false;
        return isMinHeap(left) && isMinHeap(right);
    }


   /***********************************************************************
    * Iterators
    **********************************************************************/

   /**
     * Return an iterator that iterates over all of the keys on the priority queue
     * in ascending order.
     * <p>
     * The iterator doesn't implement <tt>remove()</tt> since it's optional.
     */
    public Iterator<Key> iterator() { return new HeapIterator(); }

    private class HeapIterator implements Iterator<Key> {
        // create a new pq
        private MinPQ<Key> copy;

        // add all items to copy of heap
        // takes linear time since already in heap order so no keys move
        public HeapIterator() {
            if (comparator == null) copy = new MinPQ<Key>(size());
            else                    copy = new MinPQ<Key>(size(), comparator);
            for (int i = 1; i <= N; i++)
                copy.insert(pq[i]);
        }

        public boolean hasNext()  { return !copy.isEmpty();                     }
        public void remove()      { throw new UnsupportedOperationException();  }

        public Key next() {
            if (!hasNext()) throw new NoSuchElementException();
            return copy.delMin();
        }
    }

   /**
     * A test client.
     */
    public static void main(String[] args) {
        MinPQ<String> pq = new MinPQ<String>();
        while (!StdIn.isEmpty()) {
            String item = StdIn.readString();
            if (!item.equals("-")) pq.insert(item);
            else if (!pq.isEmpty()) StdOut.print(pq.delMin() + " ");
        }
        StdOut.println("(" + pq.size() + " left on pq)");
    }

}

二、索引最小堆（索引最大堆）

package com.ldl.algorithms.Exercise;

/*************************************************************************
 *  Compilation:  javac IndexMinPQ.java
 *  Execution:    java IndexMinPQ
 *
 *  Indexed PQ implementation using a binary heap.
 *
 *********************************************************************/

import java.util.Iterator;
import java.util.NoSuchElementException;
import com.ldl.algorithms.StdOut;

public class IndexMinPQ<Key extends Comparable<Key>> implements Iterable<Integer> {
    private int N;           // number of elements on PQ
    private int[] pq;        // binary heap using 1-based indexing
    private int[] qp;        // inverse of pq - qp[pq[i]] = pq[qp[i]] = i
    private Key[] keys;      // keys[i] = priority of i

    public IndexMinPQ(int NMAX) {
        keys = (Key[]) new Comparable[NMAX + 1];    // make this of length NMAX??
        pq   = new int[NMAX + 1];
        qp   = new int[NMAX + 1];                   // make this of length NMAX??
        for (int i = 0; i <= NMAX; i++) qp[i] = -1;
    }

    // is the priority queue empty?
    public boolean isEmpty() { return N == 0; }

    // is k an index on the priority queue?
    public boolean contains(int k) {
        return qp[k] != -1;
    }

    // number of keys in the priority queue
    public int size() {
        return N;
    }

    // associate key with index k
    public void insert(int k, Key key) {
        if (contains(k)) throw new NoSuchElementException("item is already in pq");
        N++;
        qp[k] = N;
        pq[N] = k;
        keys[k] = key;
        swim(N);
    }

    // return the index associated with a minimal key
    public int minIndex() { 
        if (N == 0) throw new NoSuchElementException("Priority queue underflow");
        return pq[1];        
    }

    // return a minimal key
    public Key minKey() { 
        if (N == 0) throw new NoSuchElementException("Priority queue underflow");
        return keys[pq[1]];        
    }

    // delete a minimal key and returns its associated index
    public int delMin() { 
        if (N == 0) throw new NoSuchElementException("Priority queue underflow");
        int min = pq[1];        
        exch(1, N--); 
        sink(1);
        qp[min] = -1;            // delete
        keys[pq[N+1]] = null;    // to help with garbage collection
        pq[N+1] = -1;            // not needed
        return min; 
    }

    // return key associated with index k
    public Key keyOf(int k) {
        if (!contains(k)) throw new NoSuchElementException("item is not in pq");
        else return keys[k];
    }

    // change the key associated with index k
    public void change(int k, Key key) {
        changeKey(k, key);
    }

    // change the key associated with index k
    public void changeKey(int k, Key key) {
        if (!contains(k)) throw new NoSuchElementException("item is not in pq");
        keys[k] = key;
        swim(qp[k]);
        sink(qp[k]);
    }

    // decrease the key associated with index k
    public void decreaseKey(int k, Key key) {
        if (!contains(k)) throw new NoSuchElementException("item is not in pq");
        if (keys[k].compareTo(key) <= 0) throw new RuntimeException("illegal decrease");
        keys[k] = key;
        swim(qp[k]);
    }

    // increase the key associated with index k
    public void increaseKey(int k, Key key) {
        if (!contains(k)) throw new NoSuchElementException("item is not in pq");
        if (keys[k].compareTo(key) >= 0) throw new RuntimeException("illegal decrease");
        keys[k] = key;
        sink(qp[k]);
    }

    // delete the key associated with index k
    public void delete(int k) {
        if (!contains(k)) throw new NoSuchElementException("item is not in pq");
        int index = qp[k];
        exch(index, N--);
        swim(index);
        sink(index);
        keys[k] = null;
        qp[k] = -1;
    }


   /**************************************************************
    * General helper functions
    **************************************************************/
    private boolean greater(int i, int j) {
        return keys[pq[i]].compareTo(keys[pq[j]]) > 0;
    }

    private void exch(int i, int j) {
        int swap = pq[i]; pq[i] = pq[j]; pq[j] = swap;
        qp[pq[i]] = i; qp[pq[j]] = j;
    }


   /**************************************************************
    * Heap helper functions
    **************************************************************/
    private void swim(int k)  {
        while (k > 1 && greater(k/2, k)) {
            exch(k, k/2);
            k = k/2;
        }
    }

    private void sink(int k) {
        while (2*k <= N) {
            int j = 2*k;
            if (j < N && greater(j, j+1)) j++;
            if (!greater(k, j)) break;
            exch(k, j);
            k = j;
        }
    }


   /***********************************************************************
    * Iterators
    **********************************************************************/

   /**
     * Return an iterator that iterates over all of the elements on the
     * priority queue in ascending order.
     * <p>
     * The iterator doesn't implement <tt>remove()</tt> since it's optional.
     */
    public Iterator<Integer> iterator() { return new HeapIterator(); }

    private class HeapIterator implements Iterator<Integer> {
        // create a new pq
        private IndexMinPQ<Key> copy;

        // add all elements to copy of heap
        // takes linear time since already in heap order so no keys move
        public HeapIterator() {
            copy = new IndexMinPQ<Key>(pq.length - 1);
            for (int i = 1; i <= N; i++)
                copy.insert(pq[i], keys[pq[i]]);
        }

        public boolean hasNext()  { return !copy.isEmpty();                     }
        public void remove()      { throw new UnsupportedOperationException();  }

        public Integer next() {
            if (!hasNext()) throw new NoSuchElementException();
            return copy.delMin();
        }
    }


    public static void main(String[] args) {
        // insert a bunch of strings
        String[] strings = { "it", "was", "the", "best", "of", "times", "it", "was", "the", "worst" };

        IndexMinPQ<String> pq = new IndexMinPQ<String>(strings.length);
        for (int i = 0; i < strings.length; i++) {
            pq.insert(i, strings[i]);
        }

        // delete and print each key
        while (!pq.isEmpty()) {
            int i = pq.delMin();
            StdOut.println(i + " " + strings[i]);
        }
        StdOut.println();

        // reinsert the same strings
        for (int i = 0; i < strings.length; i++) {
            pq.insert(i, strings[i]);
        }

        // print each key using the iterator
        for (int i : pq) {
            StdOut.println(i + " " + strings[i]);
        }
        while (!pq.isEmpty()) {
            pq.delMin();
        }

    }
}

三、堆排序

public class Heap {

    public static void sort(Comparable[] pq) {
        int N = pq.length;
        for (int k = N/2; k >= 1; k--)
            sink(pq, k, N);
        while (N > 1) {
            exch(pq, 1, N--);
            sink(pq, 1, N);
        }
    }

   /***********************************************************************
    * Helper functions to restore the heap invariant.
    **********************************************************************/

    private static void sink(Comparable[] pq, int k, int N) {
        while (2*k <= N) {
            int j = 2*k;
            if (j < N && less(pq, j, j+1)) j++;
            if (!less(pq, k, j)) break;
            exch(pq, k, j);
            k = j;
        }
    }

   /***********************************************************************
    * Helper functions for comparisons and swaps.
    * Indices are "off-by-one" to support 1-based indexing.
    **********************************************************************/
    private static boolean less(Comparable[] pq, int i, int j) {
        return pq[i-1].compareTo(pq[j-1]) < 0;
    }

    private static void exch(Object[] pq, int i, int j) {
        Object swap = pq[i-1];
        pq[i-1] = pq[j-1];
        pq[j-1] = swap;
    }

    // is v < w ?
    private static boolean less(Comparable v, Comparable w) {
        return (v.compareTo(w) < 0);
    }
        

   /***********************************************************************
    *  Check if array is sorted - useful for debugging
    ***********************************************************************/
    private static boolean isSorted(Comparable[] a) {
        for (int i = 1; i < a.length; i++)
            if (less(a[i], a[i-1])) return false;
        return true;
    }


    // print array to standard output
    private static void show(Comparable[] a) {
        for (int i = 0; i < a.length; i++) {
            StdOut.println(a[i]);
        }
    }

    // Read strings from standard input, sort them, and print.
    public static void main(String[] args) {
        String[] a = StdIn.readStrings();
        Heap.sort(a);
        show(a);
    }
}

四、最小最大堆（最大最小堆），即双端堆。见前面的博文。

五、用最小堆和最大堆实现中位数查找。见前面博文。

六、查找最大的M个元素

这里要用最小堆来实现。为什么呢？ 我们首先要创建一个容量为M的最小堆，当这个最小堆满时，容纳的是当前最大的M个元素，但是它们是以最小堆次序排列的，也就是堆顶为当面M个元素中的最小元素。当下一个新元素来到时，为了保证当前堆中的M个元素始终未当前最大的M个元素，我们应该要淘汰（删除）堆中的最小的那个元素，也就是最小堆的堆顶元素。所以，为了方便更新维护对，我们必须采用最小堆。

package com.ldl.algorithms.Exercise;

/*************************************************************************
 *  Compilation:  javac TopM.java
 *  Execution:    java TopM M < input.txt
 *  Dependencies: MinPQ.java Transaction.java StdIn.java StdOut.java
 *  Data files:   http://algs4.cs.princeton.edu/24pq/tinyBatch.txt
 * 
 *  Given an integer M from the command line and an input stream where
 *  each line contains a String and a long value, this MinPQ client
 *  prints the M lines whose numbers are the highest.
 * 
 *  % java TopM 5 < tinyBatch.txt 
 *  Thompson    2/27/2000  4747.08
 *  vonNeumann  2/12/1994  4732.35
 *  vonNeumann  1/11/1999  4409.74
 *  Hoare       8/18/1992  4381.21
 *  vonNeumann  3/26/2002  4121.85
 *
 *************************************************************************/

import com.ldl.algorithms.StdIn;
import com.ldl.algorithms.StdOut;

public class TopM {   

    // Print the top M lines in the input stream. 
    public static void main(String[] args) {
        int M = Integer.parseInt(args[0]); 
        MinPQ<Transaction> pq = new MinPQ<Transaction>(M+1); 

        while (StdIn.hasNextLine()) {
            // Create an entry from the next line and put on the PQ. 
            String line = StdIn.readLine();
            Transaction transaction = new Transaction(line);
            pq.insert(transaction); 

            // remove minimum if M+1 entries on the PQ
            if (pq.size() > M) 
                pq.delMin();
        }   // top M entries are on the PQ

        // print entries on PQ in reverse order
        Stack<Transaction> stack = new Stack<Transaction>();
        for (Transaction transaction : pq)
            stack.push(transaction);
        for (Transaction transaction : stack)
            StdOut.println(transaction);
    } 
}