【墨鳌】【数据结构】【AVL树】

AVL Tree

• 在 Binary Search Tree 现有属性之上，依赖于可以其二分查找的特性，进行树高的调整优化
• 在每个节点多维护一个子树高度（height）的信息
• 每次 insert/remove 时，按照限制条件，动态旋转，以满足任意节点的平衡因子的绝对值 \(<=1\)

节点属性

• key - 可以比较的对象类型
• left，right - 左右儿子节点
• height - 子树高度

    public class AVLNode<Key> {
        public Key key;
        public AVLNode<Key> left, right;
        int height;
        ...
    }

AVLTree 核心部分

• root - 根节点
• insert - 插入 key
• remove - 移除 key
• balance - 姿态调整

姿态调整

LL型姿态：X右旋

RR型姿态：X左旋

LR型姿态：Y左旋 再 X右旋

RL型姿态：Y右旋 再 X左旋

插入操作

• 普通二分查找到元素需要插入的位置，添加新的结点
• 重新计算新的 height，以及平衡因子
• 遇到不平衡直接进行姿态调整

删除操作

• 相较于插入会复杂一点，核心在于找删除元素的前驱/后继结点
• 同插入，先搜索到相应位置
• 如果，为 leaf 结点了，直接删除，不影响树的平衡
• 如果，为中间结点，有前驱（或者后继），选择其一，交换其 key 到当前结点
  • 删除前驱/后继（此时必然是删除 leaf 结点），然后根据新 height 调整姿态即可

测试与使用

• 结合上一篇文章
• 与 BST 对比测试

package trees;


import trees.objects.AVLTree;
import trees.objects.BinaryTree;

import java.util.Arrays;
import java.util.List;


public class Tests {
    public static void main(String[] args) {
        AVLTree<Integer> avl=new AVLTree<>();
        BinaryTree<Integer,Integer> bst=new BinaryTree<>();
        List<Integer> keys = Arrays.asList(1, 19, 30, 36, 50, 89, 101, 40, 90, 105, 103);
        for (int key : keys) {
            bst.put(key,key);
            avl.insert(key);
            System.out.println("BST:");
            bst.printTree();
            System.out.println("AVL Tree:");
            avl.printTree();
            System.out.println("**************************************");
        }
    }
}

完整Java代码实现

点击此处，访问GitHub代码仓库

public class AVLTree<T extends Comparable<T>> {

    private static final int MAX_HEIGHT_DIFFERENCE = 1;

    private AVLNode<T> root;

    public class AVLNode<Key> {
        public Key key;
        public AVLNode<Key> left, right;

        int height;

        public AVLNode(Key key, AVLNode<Key> left, AVLNode<Key> right) {
            this.key = key;
            this.left = left;
            this.right = right;
            this.height = 1;
        }
    }

    public AVLTree() {
        root = null;
    }

    public AVLTree(T... keys) {
        if (keys == null || keys.length < 1) {
            throw new NullPointerException();
        }

        root = new AVLNode<>(keys[0], null, null);
        for (int i = 1; i < keys.length && keys[i] != null; i++) {
            root = insert(root, keys[i]);
        }
    }

    public T find(T key) {
        if (key == null || root == null) {
            return null;
        }
        return find(root, key, key.compareTo(root.key));
    }

    private T find(AVLNode<T> node, T key, int cmp) {
        if (node == null) {
            return null;
        }

        if (cmp == 0) {
            return node.key;
        }

        return find(
                (node = cmp > 0 ? node.right : node.left),
                key,
                node == null ? 0 : key.compareTo(node.key));
    }

    public void insert(T key) {
        if (key == null) {
            throw new NullPointerException();
        }
        root = insert(root, key);
    }

    private AVLNode<T> insert(AVLNode<T> node, T key) {
        if (node == null) {
            return new AVLNode<>(key, null, null);
        }

        int cmp = key.compareTo(node.key);
        if (cmp == 0) {
            return node;
        }
        if (cmp < 0) {
            node.left = insert(node.left, key);
        } else {
            node.right = insert(node.right, key);
        }

        if (Math.abs(height(node.left) - height(node.right)) > MAX_HEIGHT_DIFFERENCE) {
            node = balance(node);
        }
        refreshHeight(node);
        return node;
    }

    private int height(AVLNode<T> node) {
        if (node == null) {
            return 0;
        }
        return node.height;
    }

    private void refreshHeight(AVLNode<T> node) {
        node.height = Math.max(height(node.left), height(node.right)) + 1;
    }

    private AVLNode<T> balance(AVLNode<T> node) {
        AVLNode<T> node1, node2;
        // ll
        if (height(node.left) > height(node.right) &&
                height(node.left.left) >= height(node.left.right)) {
            node1 = node.left;
            node.left = node1.right;
            node1.right = node;

            refreshHeight(node);
            return node1;
        }
        // lr
        if (height(node.left) > height(node.right) &&
                height(node.left.right) > height(node.left.left)) {
            node1 = node.left;
            node2 = node.left.right;
            node.left = node2.right;
            node1.right = node2.left;
            node2.left = node1;
            node2.right = node;

            refreshHeight(node);
            refreshHeight(node1);
            return node2;
        }
        // rr
        if (height(node.right) > height(node.left) &&
                height(node.right.right) >= height(node.right.left)) {
            node1 = node.right;
            node.right = node1.left;
            node1.left = node;

            refreshHeight(node);
            return node1;
        }
        // rl
        if (height(node.right) > height(node.left) &&
                height(node.right.left) > height(node.right.right)) {
            node1 = node.right;
            node2 = node.right.left;
            node.right = node2.left;
            node1.left = node2.right;
            node2.left = node;
            node2.right = node1;

            refreshHeight(node);
            refreshHeight(node1);
            return node2;
        }
        return node;
    }

    public void remove(T key) {
        if (key == null) {
            throw new NullPointerException();
        }
        root = remove(root, key);
    }

    private AVLNode<T> remove(AVLNode<T> node, T key) {
        if (node == null) {
            return null;
        }

        int cmp = key.compareTo(node.key);
        if (cmp < 0) {
            node.left = remove(node.left, key);
        }
        if (cmp > 0) {
            node.right = remove(node.right, key);
        }
        if (cmp == 0) {
            if (node.left == null || node.right == null) {
                return node.left == null ? node.right : node.left;
            }
            T successorKey = successorOf(node).key;
            node = remove(node, successorKey);
            node.key = successorKey;
        }

        if (Math.abs(height(node.left) - height(node.right)) > MAX_HEIGHT_DIFFERENCE) {
            node = balance(node);
        }
        refreshHeight(node);
        return node;
    }

    private AVLNode<T> successorOf(AVLNode<T> node) {
        if (node == null) {
            throw new NullPointerException();
        }
        if (node.left == null || node.right == null) {
            return node.left == null ? node.right : node.left;
        }

        return height(node.left) > height(node.right) ?
                findMax(node.left, node.left.right, node.left.right == null) :
                findMin(node.right, node.right.left, node.right.left == null);
    }

    private AVLNode<T> findMax(AVLNode<T> node, AVLNode<T> right, boolean rightIsNull) {
        if (rightIsNull) {
            return node;
        }
        return findMax((node = right), node.right, node.right == null);
    }

    private AVLNode<T> findMin(AVLNode<T> node, AVLNode<T> left, boolean leftIsNull) {
        if (leftIsNull) {
            return node;
        }
        return findMin((node = left), node.left, node.left == null);
    }

    // -------------------------- Print Tree --------------------------

    private void printTree(AVLNode node, String prefix, boolean isLeft) {
        if (node == null) {
            System.out.println("Empty tree");
            return;
        }

        if (node.right != null) {
            printTree(node.right, prefix + (isLeft ? "│   " : "    "), false);
        }

        System.out.println(prefix + (isLeft ? "└── " : "┌── ") + node.key);

        if (node.left != null) {
            printTree(node.left, prefix + (isLeft ? "    " : "│   "), true);
        }
    }

    private void printTree(AVLNode node) {
        printTree(node, "", true);
    }

    public void printTree() {
        System.out.println(" >> START TO PRINT THE TREE:");
        printTree(root);
        System.out.println(" << END TO PRINT THE TREE");
    }
}