AVL樹(shù)
- 最早的自平衡的搜索樹(shù)結(jié)構(gòu)
-
對(duì)于任意一個(gè)節(jié)點(diǎn)，左子樹(shù)和右子樹(shù)的高度差不能超過(guò)一。
- 滿二叉樹(shù)(除了葉子節(jié)點(diǎn)之外，其他節(jié)點(diǎn)都有左右倆個(gè)子樹(shù))是平衡二叉樹(shù)。
- 完全二叉樹(shù)(可能有一個(gè)非葉子節(jié)點(diǎn)的右子樹(shù)是空，空缺的節(jié)點(diǎn)部分在整棵樹(shù)的右下部分，整顆樹(shù)的葉子節(jié)點(diǎn)最大的深度值和最小的深度值相差不超過(guò)一，所有的葉子節(jié)點(diǎn)要么在樹(shù)的最后一層，要么在樹(shù)的倒數(shù)第二層)是平衡二叉樹(shù)。
- 線段樹(shù)(空出來(lái)的部分不一定在整棵樹(shù)的右下角部分，整顆樹(shù)的葉子節(jié)點(diǎn)最大的深度值和最小的深度值相差不超過(guò)一)是平衡二叉樹(shù)。
平衡因子
二叉樹(shù)上節(jié)點(diǎn)的左子樹(shù)深度減去右子樹(shù)深度的值稱為平衡因子，那么平衡二叉樹(shù)上所有節(jié)點(diǎn)的平衡因子只可能是-1，0，1。只要二叉樹(shù)上的有一個(gè)節(jié)點(diǎn)的平衡因子的絕對(duì)值大于1，則該二叉樹(shù)就是不平衡的。
代碼示例
創(chuàng)建平衡二叉樹(shù)
public class AVLTree <K extends Comparable<K>, V> {
private class Node{
public K key;
public V value;
public Node left, right;
public int height; //記錄當(dāng)前節(jié)點(diǎn)所處的高度值
public Node(K key, V value){
this.key = key;
this.value = value;
left = null;
right = null;
height = 1;
}
}
private Node root;
private int size;
public AVLTree(){
root = null;
size = 0;
}
public int getSize() {
return size;
}
public boolean isEmpty() {
return size == 0;
}
private Node getNode(Node node, K key){
if (node == null)
return null;
if (key.compareTo(node.key) == 0)
return node;
else if (key.compareTo(node.key) < 0)
return getNode(node.left, key);
else
return getNode(node.right, key);
}
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;
}
}
獲取高度
//獲得節(jié)點(diǎn)node的高度
private int getHeight(Node node) {
if (node == null)
return 0;
return node.height;
}
獲取平衡因子
//獲取節(jié)點(diǎn)node的平衡因子
private int getBalanceFactor(Node node) {
if (node == null)
return 0;
return getHeight(node.left) - getHeight(node.right);
}
判斷是否是二叉樹(shù)
//判斷該二叉樹(shù)是否是一顆二分搜索樹(shù)
public boolean isBST() {
ArrayList<K> keys = new ArrayList<>();
inOrder(root, keys);
for (int i = 0; i < keys.size(); I++)
if (keys.get(i-1).compareTo(keys.get(i)) > 0)
return false;
return true;
}
//中序遍歷
private void inOrder(Node node, ArrayList<K> keys){
if (node == null)
return;
inOrder(node.left, keys);
keys.add(node.key);
inOrder(node.right, keys);
}
判斷是否是平衡二叉樹(shù)
//判斷該二叉樹(shù)是否是一顆平衡二叉樹(shù)
public boolean isBalanced() {
return isBalanced(root);
}
//判斷以node為根的二叉樹(shù)是否是一顆平衡二叉樹(shù)
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);
}
如何維護(hù)平衡，當(dāng)添加新元素時(shí)可能會(huì)破壞平衡
- 右旋轉(zhuǎn) RR
時(shí)機(jī)
右旋轉(zhuǎn)
右旋轉(zhuǎn)
右旋轉(zhuǎn)
右旋轉(zhuǎn)代碼
//右旋轉(zhuǎn)
// 對(duì)節(jié)點(diǎn)y進(jìn)行向右旋轉(zhuǎn)操作，返回旋轉(zhuǎn)后新的根節(jié)點(diǎn)x
// y x
// / \ / \
// x T4 向右旋轉(zhuǎn)(y) z y
// / \ -------------> / \ / \
// z T3 T1 T2 T3 T4
// / \
// T1 T2
private Node rightRotate(Node y) {
Node x = y.left;
Node T3 = x.right;
//向右旋轉(zhuǎn)過(guò)程
x.right = y;
y.left = T3;
//更新節(jié)點(diǎn)height值 先更新y 后更新x
y.height = 1 + Math.max(getHeight(y.left), getHeight(y.right));
x.height = 1 + Math.max(getHeight(x.left), getHeight(x.right));
return x;
}
- 左旋轉(zhuǎn) LL
左旋轉(zhuǎn)
左旋轉(zhuǎn)
左旋轉(zhuǎn)代碼
//左旋轉(zhuǎn)
// 對(duì)節(jié)點(diǎn)y進(jìn)行向左旋轉(zhuǎn)操作，返回旋轉(zhuǎn)后新的根節(jié)點(diǎn)x
// y x
// / \ / \
// T1 x 向左旋轉(zhuǎn)(y) y z
// / \ -------------> / \ / \
// T2 z T1 T2 T3 T4
// / \
// T3 T4
private Node leftRotate(Node y) {
Node x = y.right;
Node T2 = x.left;
//向左旋轉(zhuǎn)過(guò)程
x.left = y;
y.right = T2;
y.height = 1 + Math.max(getHeight(y.left), getHeight(y.right));
x.height = 1 + Math.max(getHeight(x.left), getHeight(x.right));
return x;
}
- LR
LR
先左旋
再右旋
LR代碼示例
//LR
if (banlanceFactor > 1 && getBalanceFactor(node.left) < 0){
node.left = leftRotate(node.left);
return rightRotate(node);
}
- RL
RL
先右旋
在左旋
RL代碼示例
//RL
if (banlanceFactor < -1 && getBalanceFactor(node.right) > 0){
node.right = rightRotate(node.right);
return leftRotate(node);
}
添加一個(gè)元素
//向二分搜索樹(shù)種添加新元素(key, value)
public void add(K key, V value) {
root = add(root, key, value);
}
//向以node為根的二分搜索樹(shù)中插入元素(key, value),遞歸算法
//返回插入新節(jié)點(diǎn)后二分搜索樹(shù)的根
private Node add(Node node, K key, V value) {
if (node == null) {
size ++;
return new Node(key, value);
}
//如果相等 則不作處理
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
node.height = 1 + Math.max(getHeight(node.left), getHeight(node.right));
//計(jì)算平衡因子
int banlanceFactor = getBalanceFactor(node);
if (Math.abs(banlanceFactor) > 1)
System.out.println("unbalanced: " + banlanceFactor);
//平衡維護(hù)
// LL
if (banlanceFactor > 1 && getBalanceFactor(node.left) >= 0)
return rightRotate(node);
//RR
if (banlanceFactor < -1 && getBalanceFactor(node.right) <= 0)
return leftRotate(node);
//LR
if (banlanceFactor > 1 && getBalanceFactor(node.left) < 0){
node.left = leftRotate(node.left);
return rightRotate(node);
}
//RL
if (banlanceFactor < -1 && getBalanceFactor(node.right) > 0){
node.right = rightRotate(node.right);
return leftRotate(node);
}
return node;
}
刪除一個(gè)元素
public V remove(K key) {
Node node = getNode(root, key);
if (node != null){
root = remove(root, key);
return node.value;
}
return null;
}
//刪除以node為根的二分搜索樹(shù)中值鍵為Key的節(jié)點(diǎn) 遞歸算法
//返回刪除節(jié)點(diǎn)后新的二分搜索樹(shù)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 {
if (node.left == null) {
Node right = node.right;
node.right = null;
size --;
retNode = right;
} else if (node.right == null) {
Node left = node.left;
node.left = null;
size --;
retNode = left;
} else {
//待刪除節(jié)點(diǎn)左右子樹(shù)均不為空的情況
//找到比待刪除節(jié)點(diǎn)大的最小元素，即待刪除節(jié)點(diǎn)右子樹(shù)的最小節(jié)點(diǎn)
//用這個(gè)節(jié)點(diǎn)頂替待刪除節(jié)點(diǎn)的位置
Node successor = minimum(node.right);
//removeMin 沒(méi)有維護(hù)平衡 所以會(huì)影響平衡因子
//successor.right = removeMin(node.right);
//調(diào)用自己也會(huì)刪除 且維護(hù)平衡
successor.right = remove(node.right, successor.key);
successor.left = node.left;
node.left = node.right = null;
retNode = successor;
}
}
if (retNode == null)
return null;
//更新height
retNode.height = 1 + Math.max(getHeight(retNode.left), getHeight(retNode.right));
//計(jì)算平衡因子
int banlanceFactor = getBalanceFactor(retNode);
if (Math.abs(banlanceFactor) > 1)
System.out.println("unbalanced: " + banlanceFactor);
//平衡維護(hù)
// LL
if (banlanceFactor > 1 && getBalanceFactor(retNode.left) >= 0)
return rightRotate(retNode);
//RR
if (banlanceFactor < -1 && getBalanceFactor(retNode.right) <= 0)
return leftRotate(retNode);
//LR
if (banlanceFactor > 1 && getBalanceFactor(retNode.left) < 0){
retNode.left = leftRotate(retNode.left);
return rightRotate(retNode);
}
//RL
if (banlanceFactor < -1 && getBalanceFactor(retNode.right) > 0){
retNode.right = rightRotate(retNode.right);
return leftRotate(retNode);
}
return retNode;
}
時(shí)間復(fù)雜度：O(logn)
