數據結構之基於堆的優先隊列

時間 2020-02-10

原文原文鏈接

優先隊列的最重要的操做：刪除最大元素（或最小）和插入元素。數據結構二叉堆可以很好的實現隊列的基本操做。
二叉堆的結點按照層級順序放入數組，用長度爲N+1的私有數組pq來表示一個大小爲N的堆（堆元素放在pq[1]至pq[N]之間，爲方便計數，未使用pq[0]），跟節點在位置1，它的子結點在位置2和3，以此類推。位置k的節點的父節點位置爲k/2，它的兩個子節點位置分別爲2k和2k+1。
當一顆二叉樹的每一個節點都大於等於它的兩個子節點時，稱爲大根堆。
當一顆二叉樹的每一個節點都小於等於它的兩個子節點時，稱爲小根堆。java

堆（大根堆）的有序化：數組

由下至上的堆有序化（上浮）：若是堆的有序狀態由於某個節點變得比它的父節點更大而被打破，就須要交換它和它的父節點來修復堆，以此類推，直到遇到一個更大的父節點。
由上至下的堆有序化（下沉）：若是堆的有序狀態由於某個節點變得比它的兩個子節點或者其中之一更小而被打破，須要將它和它的兩個子節點中的較大者交換來恢復堆，以此類推，直到它的子節點比它更小或者到達堆的底部。數據結構

基於堆的優先隊列的Java代碼實現：less

1.大根堆函數

package sort;

import edu.princeton.cs.algs4.StdIn;
import edu.princeton.cs.algs4.StdOut;
import java.util.Comparator;
import java.util.Iterator;
import java.util.NoSuchElementException;

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

    //初始化構造函數
    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);    //下沉
        assert isMaxHeap();
    }

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

    public int size() {
        return n;
    }

    //返回最大的元素
    public Key max() {
        if (isEmpty()) throw new NoSuchElementException("Priority queue underflow");
        return pq[1];
    }

    // 數組擴容
    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;
    }

    //插入元素
    public void insert(Key x) {

        if (n == pq.length - 1) resize(2 * pq.length);

        pq[++n] = x;
        swim(n);    //上浮
        assert isMaxHeap();
    }

    //刪除並返回此優先級隊列上的最大鍵。
    public Key delMax() {
        if (isEmpty()) throw new NoSuchElementException("Priority queue underflow");
        Key max = pq[1];
        exch(1, n--);
        sink(1);
        pq[n+1] = null;
        if ((n > 0) && (n == (pq.length - 1) / 4)) resize(pq.length / 2);
        assert isMaxHeap();
        return max;
    }

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

    //下沉
    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;
        }
    }

    //比較
    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;
        }
    }

    //交換
    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() {
        for (int i = 1; i <= n; i++) {
            if (pq[i] == null) return false;
        }
        for (int i = n+1; i < pq.length; i++) {
            if (pq[i] != null) return false;
        }
        if (pq[0] != null) return false;
        return isMaxHeapOrdered(1);
    }

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

    public Iterator<Key> iterator() {
        return new HeapIterator();
    }

    private class HeapIterator implements Iterator<Key> {

        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)");
    }

}

2.小根堆this

package sort;

import edu.princeton.cs.algs4.StdIn;
import edu.princeton.cs.algs4.StdOut;
import java.util.Comparator;
import java.util.Iterator;
import java.util.NoSuchElementException;

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

    public MinPQ(int initCapacity) {
        pq = (Key[]) new Object[initCapacity + 1];
        n = 0;
    }

    public MinPQ() {
        this(1);
    }

    public MinPQ(int initCapacity, Comparator<Key> comparator) {
        this.comparator = comparator;
        pq = (Key[]) new Object[initCapacity + 1];
        n = 0;
    }

    public MinPQ(Comparator<Key> comparator) {
        this(1, comparator);
    }

    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();
    }

    //判斷是否爲空
    public boolean isEmpty() {
        return n == 0;
    }

    //大小
    public int size() {
        return n;
    }

    //返回最小元素
    public Key min() {
        if (isEmpty()) throw new NoSuchElementException("Priority queue underflow");
        return pq[1];
    }

    // 數組擴容
    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;
    }

    //插入元素
    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();
    }

    //刪除最小元素
    public Key delMin() {
        if (isEmpty()) throw new NoSuchElementException("Priority queue underflow");
        Key min = 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 isMinHeap();
        return min;
    }

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

    //下沉
    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;
        }
    }

    //比較
    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;
        }
    }

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

    private boolean isMinHeap() {
        for (int i = 1; i <= n; i++) {
            if (pq[i] == null) return false;
        }
        for (int i = n+1; i < pq.length; i++) {
            if (pq[i] != null) return false;
        }
        if (pq[0] != null) return false;
        return isMinHeapOrdered(1);
    }

    private boolean isMinHeapOrdered(int k) {
        if (k > n) return true;
        int left = 2*k;
        int right = 2*k + 1;
        if (left  <= n && greater(k, left))  return false;
        if (right <= n && greater(k, right)) return false;
        return isMinHeapOrdered(left) && isMinHeapOrdered(right);
    }

    public Iterator<Key> iterator() {
        return new HeapIterator();
    }

    private class HeapIterator implements Iterator<Key> {
        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();
        }
    }

    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)");
    }

}