在前文中我们已经实现了二叉搜索树，但那个实现中二叉树的搜索性能可能因为树结构失去平衡而大幅退化。
AVL树通过树结构的等价变换实现平衡。两个等价的二叉树具有相同的中序遍历序列。AVL树的思路不复杂，但是需要考虑清楚四个不同的变换情形。
下面用Go来一步步实现AVL树，首先定义AVL树结构。在树的访问上，还加了一把读写锁支持并发访问。
/*definition of tree node*/
type AVLNode struct {
key string
val interface{}
left, right *AVLNode
height int
}
/*tree root node with read-write lock*/
type AVLTree struct {
lock sync.RWMutex
root *AVLNode
}
/*create new tree node*/
func newAVLNode(k string, v interface{}) (n *AVLNode) {
n = &AVLNode{key: k, val: v, height: 1}
return
}
//public function guarante concurrency-safty
func NewAVLTree() (t *AVLTree) {
t = &AVLTree{}
return t
}
AVL节点除了一个键值对，左右子节点外，还有一个节点高度(height)，nil节点的高度定义为0，叶子节点为1。树的平衡性判断通过比较左右子树的高度差完成。
/*get height of node*/
func (root *AVLNode) getHeight() int {
if root == nil {
return 0
}
return root.height
}
/*get balance factor of node*/
func (root *AVLNode) getBalance() int {
if root == nil {
return 0
}
return root.left.getHeight() - root.right.getHeight()
}
左旋转和右旋转是两种等价变换方法，它们的基本思路是通过改变根节点，同时保持中序顺序。
/*
y x
/ \ (R) / \
x T3 => T1 y
/ \ <= / \
T1 T2 (L) (T2) T3
do right/left roation and change root node,
for right roation, T2 will be moved from x.Right to y.Left, and root node will change from y to x
for left roation, T2 will be moved from y.Left to x.Right, root node
will be changed from x to y
*/
func rightRotate(y *AVLNode) *AVLNode {
if y == nil || y.left == nil {
return y
}
x := y.left
T2 := x.right
x.right = y
y.left = T2
y.height = Max(y.left.getHeight(), y.right.getHeight()) + 1
x.height = Max(x.left.getHeight(), x.right.getHeight()) + 1
return x
}
func leftRotate(x *AVLNode) *AVLNode {
if x == nil || x.right == nil {
return x
}
y := x.right
T2 := y.left
y.left = x
x.right = T2
x.height = Max(x.left.getHeight(), x.right.getHeight()) + 1
y.height = Max(y.left.getHeight(), y.right.getHeight()) + 1
return y
}
每次插入节点或删除节点后，树都可能失去平衡性。对插入节点的父节点进行调整。
/*rebalance the root node*/
func (root *AVLNode) reBalance(key string, balance int) *AVLNode {
if root.left != nil && balance > 1 {
if key < root.left.key { //left left case
/*
root 'z' is the first unbalanced parent after insert
*z (y)
/ \ / \
*y T4 x (z)
/ \ / \ / \
x *T3 => T1 T2 (T3) T4
/ \
T1 T2
during right rotation, T2 will change its parent node
*/
return rightRotate(root)
} else if key > root.left.key { //left right case
/*
z *z (x)
/ \ / \ / \
*y T3 *(x) T3 y (z)
/ \ / / \ \
T1 *x => (y) => T1 T2 T3
/ / \
*T2 T1 (T2)
firstly, left rotate the the left node (y) of root (z)
secondly, right rotate the root (z)
*/
root.left = leftRotate(root.left)
return rightRotate(root)
}
} else if root.right != nil && balance < -1 {
if key > root.right.key { //right right case
/*
*z (y)
/ \ / \
T1 *y (z) x
/ \ => / \ / \
*T2 x T1(T2) T3 T4
/ \
T3 T4
*/
return leftRotate(root)
} else if key < root.right.key { //right left case
/*
z *z (x)
/ \ / \ / \
T1 *y T1 *(x) (z) (y)
/ \ => / \ => / \ / \
*x T4 *T2 (y) T1 T2 T3 T4
/ \ / \
T2 *T3 (T3) T4
*/
root.right = rightRotate(root.right)
return leftRotate(root)
}
}
return root
}
以下是AVL平衡树的插入方法，在普通的二叉树插入的基础上，更新树节点高度后，对根节点进行平衡旋转。
/*insert key-value pair into tree*/
func (root *AVLNode) insertKv(k string, v interface{}) *AVLNode {
if root == nil {
return newAVLNode(k, v)
}
if k < root.key {
root.left = root.left.insertKv(k, v)
} else if k > root.key {
root.right = root.right.insertKv(k, v)
}
//update height of root node
root.height = Max(root.left.getHeight(), root.right.getHeight()) + 1
//balance the root and return new root node
balance := root.getBalance()
return root.reBalance(root.key, balance)
}
从AVL树上删除节点的方法也类似。
/*check links to node*/
func (root *AVLNode) hasLeft() bool {
return root.left != nil
}
func (root *AVLNode) hasRight() bool {
return root.right != nil
}
func (root *AVLNode) isLeaf() bool {
return !root.hasLeft() && !root.hasRight()
}
/*get the left-most node which preserve the min key*/
func (root *AVLNode) min() *AVLNode {
for ; root.hasLeft(); root = root.left {
}
return root
}
/*delete a node from AVL tree by key*/
func (root *AVLNode) delete(k string) (*AVLNode, error) {
var err error
if root == nil {
return nil, errors.New(NODE_NOT_FOUND)
}
//recursively find the parent node of to-delete node
if k < root.key {
root.left, err = root.left.delete(k)
return root, err
} else if k > root.key {
root.right, err = root.right.delete(k)
return root, err
}
if root.isLeaf() {
return nil, nil //if the to-delete node is leaf, return it as nil to remove link
} else if root.hasLeft() && !root.hasRight() {
return root.left, nil
} else if !root.hasLeft() && root.hasRight() {
return root.right, nil
} //if the to-delete node has only one child, replace the position of root
//with it child to remove the link
//exchange value of to-delete node with the left most node of its right
// node, which is the successor of to-delete node
min := root.right.min()
root.key, root.val = min.key, min.val
root.right, err = root.right.delete(min.key)
return root, err
}
遍历AVL树的方法是一样的，
/*iterative in-order traversal*/
func (root *AVLNode) inorder() (out []Item) {
if root == nil {
return
}
var stack []*AVLNode
curr := root
for true {
for curr != nil {
stack = append(stack, curr)
curr = curr.left
}
if len(stack) > 0 {
curr = stack[len(stack)-1]
stack = stack[:len(stack)-1]
out = append(out, Item{curr.key, curr.val})
}
curr = curr.right
if curr == nil && len(stack) == 0 {
break
}
}
return
}
/*iterative pre-order traversal*/
func (root *AVLNode) preorder() (out []Item) {
if root == nil {
return
}
queue := []*AVLNode{root}
for len(queue) > 0 {
curr := queue[0]
queue = queue[1:]
out = append(out, Item{curr.key, curr.val})
if curr.left != nil {
queue = append(queue, curr.left)
}
if curr.right != nil {
queue = append(queue, curr.right)
}
}
return
}
/*iterative post-order traversal with two stacks*/
func (root *AVLNode) postorder() (out []Item) {
if root == nil {
return
}
var stack1, stack2 []*AVLNode
stack1 = append(stack1, root)
for len(stack1) > 0 {
curr := stack1[len(stack1)-1]
stack1 = stack1[:len(stack1)-1]
stack2 = append(stack2, curr)
if curr.left != nil {
stack1 = append(stack1, curr.left)
}
if curr.right != nil {
stack1 = append(stack1, curr.right)
}
}
for len(stack2) > 0 {
curr := stack2[len(stack2)-1]
stack2 = stack2[:len(stack2)-1]
out = append(out, Item{curr.key, curr.val})
}
return
}
/*iterative level-order traversal with BFS*/
func (root *AVLNode) levelorder() (out [][]Item) {
if root == nil {
return
}
queue := []*AVLNode{root}
level := 0
for len(queue) > 0 {
size := len(queue)
if size > 0 {
out = append(out, []Item{})
} //allocate array to store next level
for ; size > 0; size-- {
curr := queue[0]
queue = queue[1:]
//travers node of current level
out[level] = append(out[level], Item{curr.key, curr.val})
//put next level nodes into queue
if curr.left != nil {
queue = append(queue, curr.left)
}
if curr.right != nil {
queue = append(queue, curr.right)
}
}
level++ //increase level
}
return
}
最后，为了支持并发访问，为树的访问加锁。
/*insert key-value pair to tree with tree gurannted to be balanced*/
func (t *AVLTree) BalancedInsert(key string, val interface{}) {
t.lock.Lock()
defer t.lock.Unlock()
t.root = t.root.insertKv(key, val)
}
/*get the height of tree*/
func (t *AVLTree) GetHeight() int {
t.lock.RLock()
defer t.lock.RUnlock()
return t.root.getHeight()
}
/*check if tree is balanced*/
func (t *AVLTree) IsBalanced() bool {
t.lock.RLock()
defer t.lock.RUnlock()
return Abs(t.root.getBalance()) <= 1
}
/*iterative in-order traversal*/
func (t *AVLTree) Inorder() (out []Item) {
t.lock.RLock()
defer t.lock.RUnlock()
return t.root.inorder()
}
最后添加一种迭代访问的方法
type Item struct {
Key string
Val interface{}
}
/*inorder traversal through channel*/
func (root *AVLNode) iter(ch chan<- Item) {
if root == nil {
return
}
root.left.iter(ch)
ch <- Item{Key: root.key, Val: root.val}
root.right.iter(ch)
}
/*inorder traversal through channel*/
func (t *AVLTree) Iter() <-chan Item {
ch := make(chan Item)
t.lock.RLock()
//delegate all tasks to a goroutine,
// keep program from blocking
go func() {
t.root.iter(ch)
t.lock.RUnlock() //unlock the mutex once
close(ch)
}()
return ch
}
代码清单：[https://github.com/KevinACoder/Learning/blob/master/ds_go/kit/avl_tree.go]
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