八、优先队列、堆排序

优先队列

一种常见的数据结构，需要支持两种操作：删除最大(最小)元素和插入元素。这种数据类型叫做优先队列。

API

MaxPQ()//创建一个优先队列MaxPQ(int max)//创建一个最大容量为max的优先队列MaxPQ(key[] a)//用a[]中的元素创建一个优先队列void Insert()//向优先队列中插入一个元素key max()//向优先队列中插入一个元素key delMax()//删除并返回最大元素boolean isEmpty()//返回队列是否为空int size()//返回优先队列中的元素个数

问题：输入N个字符串，每个字符串都对应着一个整数，你的任务就是从中找出最大的（或者最小的）M个整数（及其关联的字符串）。这些输入可能是金融事务，例如Transaction类。在某些应用场景中，输入量可能非常巨大，甚至可以任务输入是无限的。解决这个问题的一种方法是将输入排序然后从中找出M个最大的元素，但是我们已经说明了输入将会很庞大，另一种方法就是将每个新的输入和已知的M个最大的元素比价，但除非M较小，否则这种比较的代价会非常高昂。只要能够有效地实现insert()和delMin()就能解决这个任务

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初级实现

1. 数组实现（无序）:删的时才找最大的元素
2. 有序数组实现：insert的之后就排序
3. 链接表示法：基于链表的下压栈，可以选择修改Push或者Pop来实现功能
   对比（使用堆是比较理想的，下面将会讨论）：
   

堆得定义

定义：当一棵二叉树的每个结点都大于等于它的两个子节点时，它称为堆有序的

相应地，在堆有序的二叉树中，每个结点都小于等于它的父节点。从任意结点向上，我们都能得到一列非递减的元素；从任意结点向下，我们都能得到一列非递增的元素。特别的：

二叉堆表示法：
二叉堆：就是堆有序的完全二叉树，元素在数组中按照层级存储（一层一层的放入数组中，不用数组的第一个元素）。下面简称堆

堆中：位置K的结点的父节点的位置为k/2，子节点的位置分别是2k和2k+1

一个结论：一棵大小为N的完全二叉树的高度为lgN

堆的算法

堆的有序化：就是使堆有序。一般会遇到两种情况：
当某个节点的优先级上升（或是在堆底加入一个新的元素时），我们需要由下至上的恢复堆的顺序（上浮, 和父节点比较，大就交换）。
相反，我们要由上至下恢复元素（下沉，和子节点中较大的元素交换）。

给出基于堆得有序优先队列代码(注意下沉和上浮操作，不难)：

public class MaxPQ<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    public MaxPQ(int initCapacity) {        pq = (Key[]) new Object[initCapacity + 1];        N = 0;    }    public MaxPQ() {        this(1);    }    public MaxPQ(int initCapacity, Comparator<Key> comparator) {        this.comparator = comparator;        pq = (Key[]) new Object[initCapacity + 1];        N = 0;    }    public MaxPQ(Comparator<Key> comparator) {        this(1, comparator);    }    public MaxPQ(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 isMaxHeap();    }    public boolean isEmpty() {        return N == 0;    }    public int size() {        return N;    }    public Key max() {        if (isEmpty()) throw new NoSuchElementException("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;    }    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 isMaxHeap();    }    public Key delMax() {        if (isEmpty()) throw new NoSuchElementException("Priority queue underflow");        Key max = pq[1];        exch(1, N--);        sink(1);        pq[N+1] = null;     // to avoid loiterig and help with garbage collection        if ((N > 0) && (N == (pq.length - 1) / 4)) resize(pq.length / 2);        assert isMaxHeap();        return max;    }    private void swim(int k) {        while (k > 1 && less(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 && less(j, j+1)) j++;            if (!less(k, j)) break;            exch(k, j);            k = j;        }    }    private boolean less(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 max heap?    private boolean isMaxHeap() {        return isMaxHeap(1);    }    // is subtree of pq[1..N] rooted at k a max heap?    private boolean isMaxHeap(int k) {        if (k > N) return true;        int left = 2*k, right = 2*k + 1;        if (left  <= N && less(k, left))  return false;        if (right <= N && less(k, right)) return false;        return isMaxHeap(left) && isMaxHeap(right);    }    public Iterator<Key> iterator() {        return new HeapIterator();    }    private class HeapIterator implements Iterator<Key> {        // create a new pq        private MaxPQ<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 MaxPQ<Key>(size());            else                    copy = new MaxPQ<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.delMax();        }    }    public static void main(String[] args) {        MaxPQ<String> pq = new MaxPQ<String>();        while (!StdIn.isEmpty()) {            String item = StdIn.readString();            if (!item.equals("-")) pq.insert(item);            else if (!pq.isEmpty()) StdOut.print(pq.delMax() + " ");        }        StdOut.println("(" + pq.size() + " left on pq)");    }}

索引优先队列

能引用已经进入优先队列中的元素。多了change、delete等方法，能够将索引为K的元素设为传进来的item以及删除索引位置的key。代码（这里采用的是MinPQ）：
注意理解这个的数组pq和keys的作用（使用qp将不断变化的pq数组的元素和索引挂钩，keys存值，很聪明的想法）

public class IndexMinPQ<Key extends Comparable<Key>> implements Iterable<Integer> {    private int maxN;        // maximum number of elements on PQ    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 maxN) {        if (maxN < 0) throw new IllegalArgumentException();        this.maxN = maxN;        keys = (Key[]) new Comparable[maxN + 1];    // make this of length maxN??        pq   = new int[maxN + 1];        qp   = new int[maxN + 1];                   // make this of length maxN??        for (int i = 0; i <= maxN; i++)            qp[i] = -1;    }    public boolean isEmpty() {        return N == 0;    }    public boolean contains(int i) {        if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();        return qp[i] != -1;    }    public int size() {        return N;    }    public void insert(int i, Key key) {        if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();        if (contains(i)) throw new IllegalArgumentException("index is already in the priority queue");        N++;        qp[i] = N;        pq[N] = i;        keys[i] = key;        swim(N);    }    public int minIndex() {         if (N == 0) throw new NoSuchElementException("Priority queue underflow");        return pq[1];            }    public Key minKey() {         if (N == 0) throw new NoSuchElementException("Priority queue underflow");        return keys[pq[1]];            }    public int delMin() {         if (N == 0) throw new NoSuchElementException("Priority queue underflow");        int min = pq[1];                exch(1, N--);         sink(1);        assert min == pq[N+1];        qp[min] = -1;        // delete        keys[min] = null;    // to help with garbage collection        pq[N+1] = -1;        // not needed        return min;     }    public Key keyOf(int i) {        if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");        else return keys[i];    }    public void changeKey(int i, Key key) {        if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");        keys[i] = key;        swim(qp[i]);        sink(qp[i]);    }    public void change(int i, Key key) {        changeKey(i, key);    }    public void decreaseKey(int i, Key key) {        if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");        if (keys[i].compareTo(key) <= 0)            throw new IllegalArgumentException("Calling decreaseKey() with given argument would not strictly decrease the key");        keys[i] = key;        swim(qp[i]);    }    public void increaseKey(int i, Key key) {        if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");        if (keys[i].compareTo(key) >= 0)            throw new IllegalArgumentException("Calling increaseKey() with given argument would not strictly increase the key");        keys[i] = key;        sink(qp[i]);    }    public void delete(int i) {        if (i < 0 || i >= maxN) throw new IndexOutOfBoundsException();        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");        int index = qp[i];        exch(index, N--);        swim(index);        sink(index);        keys[i] = null;        qp[i] = -1;    }    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;    }    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;        }    }    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();        }    }}

索引优先队列的使用案例：

多项归并问题：将多个有序的输入流归并成一个有序（按照优先级）的输入流。代码：

public class Multiway {     // This class should not be instantiated.    private Multiway() { }    // merge together the sorted input streams and write the sorted result to standard output    private static void merge(In[] streams) {         int N = streams.length;         IndexMinPQ<String> pq = new IndexMinPQ<String>(N);         for (int i = 0; i < N; i++)             if (!streams[i].isEmpty())                 pq.insert(i, streams[i].readString());         // Extract and print min and read next from its stream.         while (!pq.isEmpty()) {            StdOut.print(pq.minKey() + " ");             int i = pq.delMin();             if (!streams[i].isEmpty())                 pq.insert(i, streams[i].readString());         }        StdOut.println();    }     public static void main(String[] args) {         int N = args.length;         In[] streams = new In[N];         for (int i = 0; i < N; i++)             streams[i] = new In(args[i]);         merge(streams);     } } 

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堆排序

可以把任意优先队列变成一种排序方法。将所有元素插入一个查找最小元素的优先队列，然后再重复调用删除最小元素的操作来讲它们按顺序删去。用无序数组实现优先队列这么做相当于进行一次插入排序。下面讨论堆排序。
堆的构造：从中间点到左边扫描数组（如下图中的5开始），并调用sink函数（不要从左到右，因为后半元素都是叶子节点，还调用sink效率不高。）


代码：

public class HeapSort {    public static void sort(int[] a){        int N = a.length;        int[] keys = new int[N+1];        //注意，堆的数据结构是从1开始的，0不用        for (int i = 1; i < keys.length; i++) {            keys[i] = a[i-1];        }//      //构造堆,使得堆是有序的        for(int k = N/2;k>=1;k--) sink(keys,k,N);        //排序，相当于毁掉堆        while(N>1){        exch(keys,1,N--);        sink(keys,1,N);        }        //重新写回数组        for (int i = 0; i < a.length; i++) {            a[i] = keys[i+1];        }    }    private static void sink(int[] a, int k, int N) {        // TODO Auto-generated method stub        while(2*k<=N){            int j = 2*k;            if (j < N && less(a[j], a[j+1])) j++;            if (less(a[j], a[k])) break;            exch(a, k, j);            k = j;        }    }    private static boolean less(int k, int j) {        // TODO Auto-generated method stub        if (k<j) return true;        return false;    }    private static void exch(int[] a, int i, int n) {        // TODO Auto-generated method stub        int temp = a[i];        a[i] = a[n];        a[n] = temp;    }    public static void main(String[] args) {        int[] a = {2,4,7,8,2,1,0,9};        HeapSort.sort(a);        System.out.println(Arrays.toString(a));    }}