算法（第4版） Chapter 2.4 优先队列

摘要：堆中位置的结点的父节点的位置为，子节点的位置分别是和一个结论一棵大小为的完全二叉树的高度为用数组堆实现的完全二叉树是很严格的，但它的灵活性足以使我们高效地实现优先队列。

Algorithms Fourth Edition
Written By Robert Sedgewick & Kevin Wayne
Translated By 谢路云
Chapter 2 Section 4 优先队列

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

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： 根结点是堆有序的二叉树中的最大结点。

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，因为0*2=0，递推关系不合适）。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⌋

用数组（堆）实现的完全二叉树是很严格的，但它的灵活性足以使我们高效地实现优先队列。

我们用数组pq[N+1]来表示大小为N的堆，我们不使用pq[0]。

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

private void sink(int k) {
    while (2 * k <= N) {
        int j = 2 * k;
        if (j < N && less(j, j + 1)) j++; //找到子节点中更大的那个
        if (!less(k, j)) break; //如果父结点比较大，则终止
        exch(k, j);//如果父结点比较小，则把子节点中更大的那个jiaohuanshanglai
        k = j;
    }
}

复杂度

插入：不超过lgN+1次比较

删除最大元素：不超过2lgN次比较

简易版

public class MaxPQ> {
    private Key[] pq; // heap-ordered complete binary tree
    private int N = 0; // in pq[1..N] with pq[0] unused

    public MaxPQ(int maxN) {
        pq = (Key[]) new Comparable[maxN + 1];
    }

    public boolean isEmpty() {
        return N == 0;
    }

    public int size() {
        return N;
    }

    public void insert(Key v) {
        pq[++N] = v; //添加到最后
        swim(N); //上浮
    }

    public Key delMax() {
        Key max = pq[1]; // Retrieve max key from top.最大的为根结点
        exch(1, N--); // Exchange with last item.和最后一个结点交换，并减小N
        pq[N + 1] = null; // Avoid loitering.删除原来的最后一位
        sink(1); // Restore heap property.下沉
        return max;
    }

    // See above
    private boolean less(int i, int j)
    private void exch(int i, int j)    
    private void swim(int k)    
    private void sink(int k)
}

完整版

添加resize功能

public class MaxPQ implements Iterable {
    private Key[] pq;                    // store items at indices 1 to N
    private int N;                       // number of items on priority queue
    private Comparator comparator;  // optional Comparator
    public MaxPQ(int initCapacity) {
        pq = (Key[]) new Object[initCapacity + 1];
        N = 0;
    }
    public MaxPQ() {
        this(1);
    }
    public MaxPQ(int initCapacity, Comparator comparator) {
        this.comparator = comparator;
        pq = (Key[]) new Object[initCapacity + 1];
        N = 0;
    }
    public MaxPQ(Comparator 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) 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 iterator() {
        return new HeapIterator();
    }

    private class HeapIterator implements Iterator {

        // create a new pq
        private MaxPQ 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(size());
            else                    copy = new MaxPQ(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 pq = new MaxPQ();
        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)");
    }

}

增加索引

增加change, contains, delete方法

简易版

public class IndexMinPQ> implements Iterable {
    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) {
        this.maxN = maxN;
        keys = (Key[]) new Comparable[maxN + 1];    // 存一发原来的数组
        pq   = new int[maxN + 1];    // 这是二叉树，比如1位置放的是想要记录的是keys[3]，但是记录了3，即pq[1]=3
        qp   = new int[maxN + 1];    // 反过来，keys[3]放在哪里了呢？放在了树的1位置, qp[3]=1
        for (int i = 0; i <= maxN; i++)
            qp[i] = -1;
    }
    
    public void insert(int i, Key key) {
        if (contains(i)) throw new IllegalArgumentException("index is already in the priority queue");
        N++;
        qp[i] = N; // i放到了树最后的位置N，通过原数组i找到树中的位置N
        pq[N] = i; // 树的最后位置N放了i，通过树中的位置N找到原数组i
        keys[i] = key; //具体是什么
        swim(N); //上浮
    }
    
    private void swim(int k)  {
        while (k > 1 && greater(k/2, k)) {
            exch(k, k/2); //在这里pq,qp都换好了
            k = k/2;
        }
    }
    
    private void exch(int i, int j) {
        int swap = pq[i];
        pq[i] = pq[j];
        pq[j] = swap;
        qp[pq[i]] = i; //因为是逆运算
        qp[pq[j]] = j;
    }    
    
    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[min] = null;    // to help with garbage collection
        pq[N+1] = -1;        // not needed
        return min; 
    }
    
    public void changeKey(int i, Key key) {//改的是原来的数组
        if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
        keys[i] = key;
        swim(qp[i]); //可能往上
        sink(qp[i]); //可能往下
    }  
}      

完整版

public class IndexMinPQ> implements Iterable {
    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];   
        pq   = new int[maxN + 1];
        qp   = new int[maxN + 1];                   
        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 iterator() { return new HeapIterator(); }

    private class HeapIterator implements Iterator {
        // create a new pq
        private IndexMinPQ 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(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();
        }
    }
}