算法（第4版） Chapter 2.4 優(yōu)先隊列

Turbo 發(fā)布于2019-08-14 17:17 / 3590人閱讀

摘要：堆中位置的結(jié)點的父節(jié)點的位置為，子節(jié)點的位置分別是和一個結(jié)論一棵大小為的完全二叉樹的高度為用數(shù)組堆實現(xiàn)的完全二叉樹是很嚴格的，但它的靈活性足以使我們高效地實現(xiàn)優(yōu)先隊列。

Algorithms Fourth Edition
Written By Robert Sedgewick & Kevin Wayne
Translated By 謝路云
Chapter 2 Section 4 優(yōu)先隊列

優(yōu)先隊列 優(yōu)先隊列API

N個數(shù)找到最大M個元素的時間成本

不同數(shù)據(jù)結(jié)構(gòu)下的時間成本

堆 堆的定義

定義：當一棵二叉樹的每個結(jié)點都大于等于它的兩個子節(jié)點時，它稱為堆有序

相應地，在堆有序的二叉樹中，每個結(jié)點都小于等于它的父節(jié)點。從任意結(jié)點向上，我們都能得到一列非遞減的元素；從任意結(jié)點向下，我們都能得到一列非遞增的元素。特別的：根結(jié)點是堆有序的二叉樹中的最大結(jié)點。

二叉堆表示法

二叉堆：就是堆有序的完全二叉樹，元素在數(shù)組中按照層級存儲（一層一層的放入數(shù)組中，不用數(shù)組的第一個元素，因為0*2=0，遞推關(guān)系不合適）。下面簡稱堆。

堆中：位置K的結(jié)點的父節(jié)點的位置為 ?k/2? ，子節(jié)點的位置分別是 2k 和 2k+1

一個結(jié)論：一棵大小為N的完全二叉樹的高度為 ?lgN?

用數(shù)組（堆）實現(xiàn)的完全二叉樹是很嚴格的，但它的靈活性足以使我們高效地實現(xiàn)優(yōu)先隊列。

堆的算法

我們用數(shù)組pq[N+1]來表示大小為N的堆，我們不使用pq[0]。

上浮（由下至上的堆有序）

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

下沉（由上至下的堆有序）

private void sink(int k) {
    while (2 * k <= N) {
        int j = 2 * k;
        if (j < N && less(j, j + 1)) j++; //找到子節(jié)點中更大的那個
        if (!less(k, j)) break; //如果父結(jié)點比較大，則終止
        exch(k, j);//如果父結(jié)點比較小，則把子節(jié)點中更大的那個jiaohuanshanglai
        k = j;
    }
}

MaxPQ 代碼

復雜度

插入：不超過lgN+1次比較

刪除最大元素：不超過2lgN次比較

簡易版

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.最大的為根結(jié)點
        exch(1, N--); // Exchange with last item.和最后一個結(jié)點交換，并減小N
        pq[N + 1] = null; // Avoid loitering.刪除原來的最后一位
        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)");
    }

}

索引優(yōu)先隊列

增加索引

增加change, contains, delete方法

索引優(yōu)先隊列API

各方法的時間成本

IndexMinPQ 代碼

簡易版

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];    // 存一發(fā)原來的數(shù)組
        pq   = new int[maxN + 1];    // 這是二叉樹，比如1位置放的是想要記錄的是keys[3]，但是記錄了3，即pq[1]=3
        qp   = new int[maxN + 1];    // 反過來，keys[3]放在哪里了呢？放在了樹的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放到了樹最后的位置N，通過原數(shù)組i找到樹中的位置N
        pq[N] = i; // 樹的最后位置N放了i，通過樹中的位置N找到原數(shù)組i
        keys[i] = key; //具體是什么
        swim(N); //上浮
    }
    
    private void swim(int k)  {
        while (k > 1 && greater(k/2, k)) {
            exch(k, k/2); //在這里pq,qp都換好了
            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; //因為是逆運算
        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) {//改的是原來的數(shù)組
        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();
        }
    }
}