第三十二篇玩转数据结构—

第三十二篇玩转数据结构——AVL树（AVL Tree）

1.. 平衡二叉树
平衡二叉树要求，对于任意一个节点，左子树和右子树的高度差不能超过1。
平衡二叉树的高度和节点数量之间的关系也是O(logn)
为二叉树标注节点高度并计算平衡因子
AVL树是一棵平衡二叉树
2.. 实现AVL树的业务逻辑
import java.util.ArrayList;

public class AVLTree<K extends Comparable<K>, V> {

    private class Node {
        public K key;
        public V value;
        public Node left;
        public Node right;
        public int height;

        // 构造函数
        public Node(K key, V value) {
            this.key = key;
            this.value = value;
            left = null;
            right = null;
            height = 1;
        }
    }

    private Node root;
    private int size;

    // 构造函数
    public AVLTree() {
        root = null;
        size = 0;
    }

    // 实现getSize方法
    public int getSize() {
        return size;
    }

    // 实现isEmpty方法
    public boolean isEmpty() {
        return size == 0;
    }

    // 判断该二叉树是否为二分搜索树
    public boolean isBST() {
        ArrayList<K> keys = new ArrayList<>();
        inOrder(root, keys);
        for (int i = 1; i < keys.size(); i++) {
            if (keys.get(i - 1).compareTo(keys.get(i)) > 0) {
                return false;
            }
        }
        return true;
    }

    private void inOrder(Node node, ArrayList<K> keys) {

        if (node == null) {
            return;
        }
        inOrder(node.left, keys);
        keys.add(node.key);
        inOrder(node.right, keys);
    }

    // 判断二叉树是否为平衡二叉树
    public boolean isBalanced() {
        return isBalanced(root);
    }

    // 判断以node为根的二叉树是否为平衡二叉树
    private boolean isBalanced(Node node) {

        if (node == null) {
            return true;
        }
        int balanceFactor = getBalanceFactor(node);
        if (Math.abs(balanceFactor) > 1) {
            return false;
        }
        return isBalanced(node.left) && isBalanced(node.right);
    }

    // 返回节点node的高度值
    private int getHeight(Node node) {
        if (node == null) {
            return 0;
        }
        return node.height;
    }

    // 返回节点node的平衡因子
    private int getBalanceFactor(Node node) {
        if (node == null) {
            return 0;
        }
        return getHeight(node.left) - getHeight(node.right);
    }

    // 对节点y进行向右旋转操作，返回旋转后新的根节点x
    //        y                              x
    //       /                            /   
    //      x   T4     向右旋转 (y)        z     y
    //     /        - - - - - - - ->    /    / 
    //    z   T3                       T1  T2 T3 T4
    //   / 
    // T1   T2
    private Node rightRotate(Node y) {

        Node x = y.left;
        Node T3 = x.right;

        // 向右旋转
        x.right = y;
        y.left = T3;

        // 更新height
        y.height = Math.max(getHeight(y.left), getHeight(y.right)) + 1;
        x.height = Math.max(getHeight(x.left), getHeight(x.right)) + 1;

        return x;
    }

    // 对节点y进行向左旋转操作，返回旋转后新的根节点x
    //    y                             x
    //  /                            /   
    // T1   x      向左旋转 (y)       y     z
    //     /    - - - - - - - ->   /    / 
    //   T2  z                     T1 T2 T3 T4
    //      / 
    //     T3 T4
    private Node leftRotate(Node y) {

        Node x = y.right;
        Node T2 = x.left;

        // 向左旋转
        x.left = y;
        y.right = T2;

        //更新height
        y.height = Math.max(getHeight(y.left), getHeight(y.right)) + 1;
        x.height = Math.max(getHeight(x.left), getHeight(x.right)) + 1;

        return x;
    }

    // 实现add方法
    public void add(K key, V value) {
        root = add(root, key, value);
    }

    // 向以node为根节点的二分搜索树中插入元素(key, value)，递归算法
    // 返回插入新元素后的二分搜索树的根
    private Node add(Node node, K key, V value) {

        if (node == null) {
            size++;
            return new Node(key, value);
        }

        if (key.compareTo(node.key) < 0) {
            node.left = add(node.left, key, value);
        } else if (key.compareTo(node.key) > 0) {
            node.right = add(node.right, key, value);
        } else {
            node.value = value;
        }

        // 更新height值
        node.height = 1 + Math.max(getHeight(node.left), getHeight(node.right));

        // 计算平衡因子
        int balanceFactor = getBalanceFactor(node);

        // 平衡维护
        // LL
        if (balanceFactor > 1 && getBalanceFactor(node.left) >= 0) {
            return rightRotate(node);
        }
        // RR
        if (balanceFactor < -1 && getBalanceFactor(node.right) <= 0) {
            return leftRotate(node);
        }

        // LR
        if (balanceFactor > 1 && getBalanceFactor(node.left) < 0) {
            node.left = leftRotate(node.left);
            return rightRotate(node);
        }
        // RL
        if (balanceFactor < -1 && getBalanceFactor(node.right) > 0) {
            node.right = rightRotate(node.right);
            return leftRotate(node);
        }

        return node;
    }

    // 返回以node为根节点的二分搜索树中，key所在的节点
    private Node getNode(Node node, K key) {

        if (node == null)
            return null;

        if (key.compareTo(node.key) < 0) {
            return getNode(node.left, key);
        } else if (key.compareTo(node.key) > 0) {
            return getNode(node.right, key);
        } else {
            return node;
        }
    }

    public boolean contains(K key) {
        return getNode(root, key) != null;
    }

    public V get(K key) {

        Node node = getNode(root, key);
        return node == null ? null : node.value;
    }

    public void set(K key, V newValue) {
        Node node = getNode(root, key);
        if (node == null)
            throw new IllegalArgumentException(key + " doesn't exist!");

        node.value = newValue;
    }

    // 返回以node为根的二分搜索树的最小元素所在节点
    private Node minimum(Node node) {
        if (node.left == null) {
            return node;
        }
        return minimum(node.left);
    }

    // 实现remove方法
    // 删除二分搜索树中键为key的节点
    public V remove(K key) {
        Node node = getNode(root, key);

        if (node != null) {
            root = remove(root, key);
            return node.value;
        }
        return null;
    }

    // 删除以node为根节点的二分搜索树中键为key的节点，递归算法
    // 返回删除节点后新的二分搜索树的根
    private Node remove(Node node, K key) {
        if (node == null) {
            return null;
        }

        Node retNode;
        if (key.compareTo(node.key) < 0) {
            node.left = remove(node.left, key);
            retNode = node;
        } else if (key.compareTo(node.key) > 0) {
            node.right = remove(node.right, key);
            retNode = node;
        } else {
            // 待删除节点左子树为空的情况
            if (node.left == null) {
                Node rightNode = node.right;
                node.right = null;
                size--;
                retNode = rightNode;
                // 待删除节点右子树为空的情况
            } else if (node.right == null) {
                Node leftNode = node.left;
                node.left = null;
                size--;
                retNode = leftNode;
                // 待删除节点左右子树均不为空
                // 找到比待删除节点大的最小节点，即待删除节点右子树的最小节点
                // 用这个节点顶替待删除节点
            } else {
                Node successor = minimum(node.right);
                successor.right = remove(node.right, successor.key);  //这里进行了size--操作
                successor.left = node.left;
                node.left = null;
                node.right = null;
                retNode = successor;
            }
        }

        if (retNode == null) {
            return null;
        }

        // 更新height值
        retNode.height = 1 + Math.max(getHeight(retNode.left), getHeight(retNode.right));

        // 计算平衡因子
        int balanceFactor = getBalanceFactor(retNode);

        // 平衡维护
        // LL
        if (balanceFactor > 1 && getBalanceFactor(retNode.left) >= 0) {
            return rightRotate(retNode);
        }
        // RR
        if (balanceFactor < -1 && getBalanceFactor(retNode.right) <= 0) {
            return leftRotate(retNode);
        }

        // LR
        if (balanceFactor > 1 && getBalanceFactor(retNode.left) < 0) {
            node.left = leftRotate(retNode.left);
            return rightRotate(retNode);
        }
        // RL
        if (balanceFactor < -1 && getBalanceFactor(retNode.right) > 0) {
            node.right = rightRotate(retNode.right);
            return leftRotate(retNode);
        }

        return retNode;
    }

    // 打印测试
    public static void main(String[] args) {

        System.out.println("Pride and Prejudice");

        ArrayList<String> words = new ArrayList<>();

        if (FileOperation.readFile("pride-and-prejudice.txt", words)) {

            System.out.println("Total words: " + words.size());

            AVLTree<String, Integer> map = new AVLTree<>();
            for (String word : words) {
                if (map.contains(word)) {
                    map.set(word, map.get(word) + 1);
                } else {
                    map.add(word, 1);
                }
            }

            System.out.println("Total different words: " + map.getSize());
            System.out.println("Frequency of PRIDE: " + map.get("pride"));
            System.out.println("Frequency of PREJUDICE: " + map.get("prejudice"));

            System.out.println("is BST: " + map.isBST());

            System.out.println("is Balanced: " + map.isBalanced());
        }
    }
}
第三十二篇 玩转数据结构——AVL树（AVL Tree）

第三十二篇玩转数据结构——AVL树（AVL Tree）
