優先佇列

一種常見的資料結構，需要支援兩種操作：刪除最大(最小)元素和插入元素。這種資料型別叫做優先佇列。

API

MaxPQ()//建立一個優先佇列
MaxPQ(int max)//建立一個最大容量為max的優先佇列
MaxPQ(key[] a)//用a[]中的元素建立一個優先佇列
void Insert()//向優先佇列中插入一個元素
key max()//向優先佇列中插入一個元素
key delMax()//刪除並返回最大元素
boolean isEmpty()//返回佇列是否為空
int size()//返回優先佇列中的元素個數

問題：輸入N個字串，每個字串都對應著一個整數，你的任務就是從中找出最大的（或者最小的）M個整數（及其關聯的字串）。這些輸入可能是金融事務，例如

初級實現

1. 陣列實現（無序）:刪的時才找最大的元素
2. 有序陣列實現：insert的之後就排序
3. 連結表示法：基於連結串列的下壓棧，可以選擇修改Push或者Pop來實現功能
   對比（使用堆是比較理想的，下面將會討論）：
   

堆得定義

定義：當一棵二叉樹的每個結點都大於等於它的兩個子節點時，它稱為堆有序的

相應地，在堆有序的二叉樹中，每個結點都小於等於它的父節點。從任意結點向上，我們都能得到一列非遞減的元素；從任意結點向下，我們都能得到一列非遞增的元素。特別的：

二叉堆表示法：
二叉堆：就是堆有序的完全二叉樹，元素在陣列中按照層級儲存（一層一層的放入陣列中，不用陣列的第一個元素）。下面簡稱堆

堆中：位置K的結點的父節點的位置為k/2，子節點的位置分別是2k和2k+1

一個結論：一棵大小為N的完全二叉樹的高度為lgN

堆的演算法

堆的有序化：就是使堆有序。一般會遇到兩種情況：
當某個節點的優先順序上升（或是在堆底加入一個新的元素時），我們需要由下至上的恢復堆的順序（上浮, 和父節點比較，大就交換）。
相反，我們要由上至下恢復元素（下沉，和子節點中較大的元素交換）。

給出基於堆得有序優先佇列程式碼(注意下沉和上浮操作，不難)：

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)");
    }

}

索引優先佇列

能引用已經進入優先佇列中的元素。多了change、delete等方法，能夠將索引為K的元素設為傳進來的item以及刪除索引位置的key。程式碼（這裡採用的是MinPQ）：
注意理解這個的陣列pq和keys的作用（使用qp將不斷變化的pq陣列的元素和索引掛鉤，keys存值，很聰明的想法）

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();
        }
    }
}

索引優先佇列的使用案例：

多項歸併問題：將多個有序的輸入流歸併成一個有序（按照優先順序）的輸入流。程式碼：

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); 
    } 
} 

堆排序

可以把任意優先佇列變成一種排序方法。將所有元素插入一個查詢最小元素的優先佇列，然後再重複呼叫刪除最小元素的操作來講它們按順序刪去。用無序陣列實現優先佇列這麼做相當於進行一次插入排序。下面討論堆排序。
堆的構造：從中間點到左邊掃描陣列（如下圖中的5開始），並呼叫sink函式（不要從左到右，因為後半元素都是葉子節點，還呼叫sink效率不高。）


程式碼：

public class HeapSort {


    public static void sort(int[] a){
        int N = a.length;
        int[] keys = new int[N+1];
        //注意，堆的資料結構是從1開始的，0不用
        for (int i = 1; i < keys.length; i++) {
            keys[i] = a[i-1];
        }
//      //構造堆,使得堆是有序的
        for(int k = N/2;k>=1;k--) sink(keys,k,N);
        //排序，相當於毀掉堆
        while(N>1){
        exch(keys,1,N--);
        sink(keys,1,N);
        }
        //重新寫回陣列
        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));
    }
}