1.AVL树基本介绍
1.1名字由来
1.2AVL树属性
1.3平衡对比
1.4简单的继承结构
1.5添加导致的失衡
2.AVL树的四种旋转
通过旋转来解决添加或删除导致AVL树失衡
2.1LL-右旋转(单旋)
2.2RR-左旋转(单旋)
2.3RL-LL右旋转,RR左旋转(双旋)
2.4LR-RR左旋转,LL右旋转(双旋)
2.5旋转的别名
3.添加导致失衡
3.1添加之后的修复代码
@Override protected void afterAdd(Node<E> node) { while ((node = node.parent) != null) { if (isBalanced(node)) { // 更新高度 updateHeight(node); } else { // 恢复平衡 rebalance(node); // 整棵树恢复平衡 break; } } } /** * 恢复平衡 * @param grand 高度最低的那个不平衡节点 */ @SuppressWarnings("unused") private void rebalance2(Node<E> grand) { Node<E> parent = ((AVLNode<E>)grand).tallerChild(); Node<E> node = ((AVLNode<E>)parent).tallerChild(); if (parent.isLeftChild()) { // L if (node.isLeftChild()) { // LL rotateRight(grand); } else { // LR rotateLeft(parent); rotateRight(grand); } } else { // R if (node.isLeftChild()) { // RL rotateRight(parent); rotateLeft(grand); } else { // RR rotateLeft(grand); } } } /** * 恢复平衡 * @param grand 高度最低的那个不平衡节点 */ private void rebalance(Node<E> grand) { Node<E> parent = ((AVLNode<E>)grand).tallerChild(); Node<E> node = ((AVLNode<E>)parent).tallerChild(); if (parent.isLeftChild()) { // L if (node.isLeftChild()) { // LL rotate(grand, node, node.right, parent, parent.right, grand); } else { // LR rotate(grand, parent, node.left, node, node.right, grand); } } else { // R if (node.isLeftChild()) { // RL rotate(grand, grand, node.left, node, node.right, parent); } else { // RR rotate(grand, grand, parent.left, parent, node.left, node); } } }
旋转代码
private void rotateLeft(Node<E> grand) { Node<E> parent = grand.right; Node<E> child = parent.left; grand.right = child; parent.left = grand; afterRotate(grand, parent, child); } private void rotateRight(Node<E> grand) { Node<E> parent = grand.left; Node<E> child = parent.right; grand.left = child; parent.right = grand; afterRotate(grand, parent, child); } private void afterRotate(Node<E> grand, Node<E> parent, Node<E> child) { // 让parent称为子树的根节点 parent.parent = grand.parent; if (grand.isLeftChild()) { grand.parent.left = parent; } else if (grand.isRightChild()) { grand.parent.right = parent; } else { // grand是root节点 root = parent; } // 更新child的parent if (child != null) { child.parent = grand; } // 更新grand的parent grand.parent = parent; // 更新高度 updateHeight(grand); updateHeight(parent); }
3.2实例
实例2
3.3统一所有操作
3.4统一所有旋转操作
private void rotate( Node<E> r, // 子树的根节点 Node<E> b, Node<E> c, Node<E> d, Node<E> e, Node<E> f) { // 让d成为这棵子树的根节点 d.parent = r.parent; if (r.isLeftChild()) { r.parent.left = d; } else if (r.isRightChild()) { r.parent.right = d; } else { root = d; } //b-c b.right = c; if (c != null) { c.parent = b; } updateHeight(b); // e-f f.left = e; if (e != null) { e.parent = f; } updateHeight(f); // b-d-f d.left = b; d.right = f; b.parent = d; f.parent = d; updateHeight(d); } /** * 恢复平衡 * @param grand 高度最低的那个不平衡节点 */ private void rebalance(Node<E> grand) { Node<E> parent = ((AVLNode<E>)grand).tallerChild(); Node<E> node = ((AVLNode<E>)parent).tallerChild(); if (parent.isLeftChild()) { // L if (node.isLeftChild()) { // LL rotate(grand, node, node.right, parent, parent.right, grand); } else { // LR rotate(grand, parent, node.left, node, node.right, grand); } } else { // R if (node.isLeftChild()) { // RL rotate(grand, grand, node.left, node, node.right, parent); } else { // RR rotate(grand, grand, parent.left, parent, node.left, node); } } }
3.5独立AVLNode
4.删除导致失衡
4.1实例
4.2LL-右旋转(单旋)
4.3RR-左旋转(单旋)
4.4LR-RR左旋转，LL右旋转(双旋)
4.5RL-LL右旋转，RR左旋转(双旋)
4.5删除之后的修复
5.总结
6.AVL树代码总结
package com.mj.tree; import java.util.Comparator; public class AVLTree<E> extends BST<E> { public AVLTree() { this(null); } public AVLTree(Comparator<E> comparator) { super(comparator); } @Override protected void afterAdd(Node<E> node) { while ((node = node.parent) != null) { if (isBalanced(node)) { // 更新高度 updateHeight(node); } else { // 恢复平衡 rebalance(node); // 整棵树恢复平衡 break; } } } @Override protected void afterRemove(Node<E> node) { while ((node = node.parent) != null) { if (isBalanced(node)) { // 更新高度 updateHeight(node); } else { // 恢复平衡 rebalance(node); } } } @Override protected Node<E> createNode(E element, Node<E> parent) { return new AVLNode<>(element, parent); } /** * 恢复平衡 * @param grand 高度最低的那个不平衡节点 */ @SuppressWarnings("unused") private void rebalance2(Node<E> grand) { Node<E> parent = ((AVLNode<E>)grand).tallerChild(); Node<E> node = ((AVLNode<E>)parent).tallerChild(); if (parent.isLeftChild()) { // L if (node.isLeftChild()) { // LL rotateRight(grand); } else { // LR rotateLeft(parent); rotateRight(grand); } } else { // R if (node.isLeftChild()) { // RL rotateRight(parent); rotateLeft(grand); } else { // RR rotateLeft(grand); } } } /** * 恢复平衡 * @param grand 高度最低的那个不平衡节点 */ private void rebalance(Node<E> grand) { Node<E> parent = ((AVLNode<E>)grand).tallerChild(); Node<E> node = ((AVLNode<E>)parent).tallerChild(); if (parent.isLeftChild()) { // L if (node.isLeftChild()) { // LL rotate(grand, node, node.right, parent, parent.right, grand); } else { // LR rotate(grand, parent, node.left, node, node.right, grand); } } else { // R if (node.isLeftChild()) { // RL rotate(grand, grand, node.left, node, node.right, parent); } else { // RR rotate(grand, grand, parent.left, parent, node.left, node); } } } private void rotate( Node<E> r, // 子树的根节点 Node<E> b, Node<E> c, Node<E> d, Node<E> e, Node<E> f) { // 让d成为这棵子树的根节点 d.parent = r.parent; if (r.isLeftChild()) { r.parent.left = d; } else if (r.isRightChild()) { r.parent.right = d; } else { root = d; } //b-c b.right = c; if (c != null) { c.parent = b; } updateHeight(b); // e-f f.left = e; if (e != null) { e.parent = f; } updateHeight(f); // b-d-f d.left = b; d.right = f; b.parent = d; f.parent = d; updateHeight(d); } private void rotateLeft(Node<E> grand) { Node<E> parent = grand.right; Node<E> child = parent.left; grand.right = child; parent.left = grand; afterRotate(grand, parent, child); } private void rotateRight(Node<E> grand) { Node<E> parent = grand.left; Node<E> child = parent.right; grand.left = child; parent.right = grand; afterRotate(grand, parent, child); } private void afterRotate(Node<E> grand, Node<E> parent, Node<E> child) { // 让parent称为子树的根节点 parent.parent = grand.parent; if (grand.isLeftChild()) { grand.parent.left = parent; } else if (grand.isRightChild()) { grand.parent.right = parent; } else { // grand是root节点 root = parent; } // 更新child的parent if (child != null) { child.parent = grand; } // 更新grand的parent grand.parent = parent; // 更新高度 updateHeight(grand); updateHeight(parent); } private boolean isBalanced(Node<E> node) { return Math.abs(((AVLNode<E>)node).balanceFactor()) <= 1; } private void updateHeight(Node<E> node) { ((AVLNode<E>)node).updateHeight(); } private static class AVLNode<E> extends Node<E> { int height = 1; public AVLNode(E element, Node<E> parent) { super(element, parent); } public int balanceFactor() { int leftHeight = left == null ? 0 : ((AVLNode<E>)left).height; int rightHeight = right == null ? 0 : ((AVLNode<E>)right).height; return leftHeight - rightHeight; } public void updateHeight() { int leftHeight = left == null ? 0 : ((AVLNode<E>)left).height; int rightHeight = right == null ? 0 : ((AVLNode<E>)right).height; height = 1 + Math.max(leftHeight, rightHeight); } public Node<E> tallerChild() { int leftHeight = left == null ? 0 : ((AVLNode<E>)left).height; int rightHeight = right == null ? 0 : ((AVLNode<E>)right).height; if (leftHeight > rightHeight) return left; if (leftHeight < rightHeight) return right; return isLeftChild() ? left : right; } @Override public String toString() { String parentString = "null"; if (parent != null) { parentString = parent.element.toString(); } return element + "_p(" + parentString + ")_h(" + height + ")"; } } }