Algorithm 如何验证二进制搜索树?
我在这里读到了一个访谈练习,称为验证二叉搜索树Algorithm 如何验证二进制搜索树?,algorithm,data-structures,binary-search-tree,Algorithm,Data Structures,Binary Search Tree,我在这里读到了一个访谈练习,称为验证二叉搜索树 这到底是怎么回事?验证二叉搜索树需要什么?我写过一个基本的搜索树,但从未听说过这个概念。“验证”二叉搜索树意味着您要检查它确实在左侧有所有较小的项目,在右侧有较大的项目。本质上,这是一种检查,看看一棵二叉树是否是一棵二叉搜索树。实际上,这是每个人在面试中都会犯的错误 必须对照(minLimitof节点、node.value)检查Leftchild 必须对照(node.value,节点的MaxLimit)检查Rightchild isvalidbts
这到底是怎么回事?验证二叉搜索树需要什么?我写过一个基本的搜索树,但从未听说过这个概念。“验证”二叉搜索树意味着您要检查它确实在左侧有所有较小的项目,在右侧有较大的项目。本质上,这是一种检查,看看一棵二叉树是否是一棵二叉搜索树。实际上,这是每个人在面试中都会犯的错误 必须对照(minLimitof节点、node.value)检查Leftchild 必须对照(node.value,节点的MaxLimit)检查Rightchild
isvalidbtst(根,-无穷大,无穷大);
bool-isvalidbtst(二进制节点,int-MIN,int-MAX)
{
if(node==null)
返回true;
如果(node.element>MIN
&&node.element
另一种解决方案(如果空间不是约束):
按顺序遍历树并将节点值存储在数组中。如果数组是按排序的,则它是有效的BST,否则无效。以下是我在Clojure中的解决方案:
(defstruct BST :val :left :right)
(defn in-order [bst]
(when-let [{:keys [val, left, right]} bst]
(lazy-seq
(concat (in-order left) (list val) (in-order right)))))
(defn is-strictly-sorted? [col]
(every?
(fn [[a b]] (< a b))
(partition 2 1 col)))
(defn is-valid-BST [bst]
(is-strictly-sorted? (in-order bst)))
(defstruct BST:val:左:右)
(按顺序定义[bst]
(当let[{:keys[val,left,right]}bst]
(续)
(concat(按顺序左)(list val)(按顺序右(())))
(defn是严格分类的?[col]
(每个?
(fn[[ab]](
“最好先定义一个不变量。这里的不变量是——顺序遍历中BST的任意两个连续元素的出现顺序必须严格递增(不能相等,总是按顺序遍历递增)。因此,解决方案可以只是一个简单的按序遍历,记住上次访问的节点,并将当前节点与上次访问的节点进行比较(“)。”使用按序遍历的迭代解决方案
bool is_bst(Node *root) {
if (!root)
return true;
std::stack<Node*> stack;
bool started = false;
Node *node = root;
int prev_val;
while(true) {
if (node) {
stack.push(node);
node = node->left();
continue;
}
if (stack.empty())
break;
node = stack.top();
stack.pop();
/* beginning of bst check */
if(!started) {
prev_val = node->val();
started = true;
} else {
if (prev_val > node->val())
return false;
prev_val = node->val();
}
/* end of bst check */
node = node->right();
}
return true;
}
bool是(节点*根){
如果(!root)
返回true;
std::堆栈;
bool start=false;
节点*节点=根;
国际通行证;
while(true){
如果(节点){
栈推(节点);
node=node->left();
继续;
}
if(stack.empty())
打破
node=stack.top();
stack.pop();
/*开始bst检查*/
如果(!已启动){
prev_val=节点->val();
开始=真;
}否则{
如果(上一个值>节点->值())
返回false;
prev_val=节点->val();
}
/*bst检查结束*/
node=node->right();
}
返回true;
}
递归解决方案:
isBinary(root)
{
if root == null
return true
else if( root.left == NULL and root.right == NULL)
return true
else if(root.left == NULL)
if(root.right.element > root.element)
rerturn isBInary(root.right)
else if (root.left.element < root.element)
return isBinary(root.left)
else
return isBInary(root.left) and isBinary(root.right)
}
isBinary(根)
{
如果root==null
返回真值
else if(root.left==NULL和root.right==NULL)
返回真值
else if(root.left==NULL)
if(root.right.element>root.element)
重新返回isBInary(root.right)
else if(root.left.element
bool二进制搜索树::验证(){
int minVal=-1;
int maxVal=-1;
返回ValidateImpl(root、minVal、maxVal);
}
bool二进制搜索树::ValidateImpl(节点*currRoot、int&minVal、int&maxVal)
{
int leftMin=-1;
int leftMax=-1;
int rightMin=-1;
int rightMax=-1;
if(currRoot==NULL)返回true;
如果(curroot->left){
如果(curroot->left->valuevalue){
如果(!ValidateImpl(currRoot->left,leftMin,leftMax))返回false;
如果(leftMax!=currRoot->left->value&&currRoot->valuevalue;
}
如果(curroot->right){
如果(currRoot->right->value->currRoot->value){
如果(!ValidateImpl(currRoot->right,rightMin,rightMax))返回false;
如果(rightMin!=currRoot->right->value&&currRoot->value>rightMin)返回false;
}
否则返回false;
}否则{
rightMin=rightMax=currRoot->value;
}
minVal=leftMinrightMax?leftMax:rightMax;
返回true;
}
//使用基于顺序遍历的Impl
bool二进制搜索树::validate(){
int val=-1;
返回ValidateImpl(root,val);
}
//基于顺序遍历的Impl
bool二进制搜索树::ValidateImpl(节点*currRoot,int&val){
if(currRoot==NULL)返回true;
如果(curroot->left){
如果(currRoot->left->value->currRoot->value)返回false;
如果(!ValidateImpl(currRoot->left,val))返回false;
}
如果(val>curroot->value)返回false;
val=currRoot->value;
如果(curroot->right){
如果(curroot->right->valuevalue)返回false;
如果(!ValidateImpl(currRoot->right,val))返回false;
}
返回true;
}
布尔isBST(节点根){
如果(root==null){return true;}
返回(isBST(root.left)和&(isBST(root.right)和&(root.left==null | | root.left.data root.data));
}
这里是不使用额外空间的迭代解决方案
Node{
int value;
Node right, left
}
public boolean ValidateBST(Node root){
Node currNode = root;
Node prevNode = null;
Stack<Node> stack = new Stack<Node>();
while(true){
if(currNode != null){
stack.push(currNode);
currNode = currNode.left;
continue;
}
if(stack.empty()){
return;
}
currNode = stack.pop();
if(prevNode != null){
if(currNode.value < prevNode.value){
return false;
}
}
prevNode = currNode;
currNode = currNode.right;
}
}
节点{
int值;
右节点,左节点
}
公共布尔ValidateBST(节点根){
节点=根节点;
节点prevNode=null;
堆栈=新堆栈();
while(true){
if(currNode!=null){
stack.push(currNode);
currNode=currNode.left;
继续;
}
if(stack.empty()){
返回;
}
currNode=stack.pop();
if(prevNode!=null){
if(currNode.valuebool BinarySearchTree::validate() {
int minVal = -1;
int maxVal = -1;
return ValidateImpl(root, minVal, maxVal);
}
bool BinarySearchTree::ValidateImpl(Node *currRoot, int &minVal, int &maxVal)
{
int leftMin = -1;
int leftMax = -1;
int rightMin = -1;
int rightMax = -1;
if (currRoot == NULL) return true;
if (currRoot->left) {
if (currRoot->left->value < currRoot->value) {
if (!ValidateImpl(currRoot->left, leftMin, leftMax)) return false;
if (leftMax != currRoot->left->value && currRoot->value < leftMax) return false;
}
else
return false;
} else {
leftMin = leftMax = currRoot->value;
}
if (currRoot->right) {
if (currRoot->right->value > currRoot->value) {
if(!ValidateImpl(currRoot->right, rightMin, rightMax)) return false;
if (rightMin != currRoot->right->value && currRoot->value > rightMin) return false;
}
else return false;
} else {
rightMin = rightMax = currRoot->value;
}
minVal = leftMin < rightMin ? leftMin : rightMin;
maxVal = leftMax > rightMax ? leftMax : rightMax;
return true;
}
// using inorder traverse based Impl
bool BinarySearchTree::validate() {
int val = -1;
return ValidateImpl(root, val);
}
// inorder traverse based Impl
bool BinarySearchTree::ValidateImpl(Node *currRoot, int &val) {
if (currRoot == NULL) return true;
if (currRoot->left) {
if (currRoot->left->value > currRoot->value) return false;
if(!ValidateImpl(currRoot->left, val)) return false;
}
if (val > currRoot->value) return false;
val = currRoot->value;
if (currRoot->right) {
if (currRoot->right->value < currRoot->value) return false;
if(!ValidateImpl(currRoot->right, val)) return false;
}
return true;
}
boolean isBST(Node root) {
if (root == null) { return true; }
return (isBST(root.left) && (isBST(root.right) && (root.left == null || root.left.data <= root.data) && (root.right == null || root.right.data > root.data));
}
Node{
int value;
Node right, left
}
public boolean ValidateBST(Node root){
Node currNode = root;
Node prevNode = null;
Stack<Node> stack = new Stack<Node>();
while(true){
if(currNode != null){
stack.push(currNode);
currNode = currNode.left;
continue;
}
if(stack.empty()){
return;
}
currNode = stack.pop();
if(prevNode != null){
if(currNode.value < prevNode.value){
return false;
}
}
prevNode = currNode;
currNode = currNode.right;
}
}
bool ValidateBST(Node *pCurrentNode, int nMin = INT_MIN, int nMax = INT_MAX)
{
return
(
pCurrentNode == NULL
)
||
(
(
!pCurrentNode->pLeftNode ||
(
pCurrentNode->pLeftNode->value < pCurrentNode->value &&
pCurrentNode->pLeftNode->value < nMax &&
ValidateBST(pCurrentNode->pLeftNode, nMin, pCurrentNode->value)
)
)
&&
(
!pCurrentNode->pRightNode ||
(
pCurrentNode->pRightNode->value > pCurrentNode->value &&
pCurrentNode->pRightNode->value > nMin &&
ValidateBST(pCurrentNode->pRightNode, pCurrentNode->value, nMax)
)
)
);
}
static struct node *prev = NULL;
bool isBST(struct node* root)
{
// traverse the tree in inorder fashion and keep track of prev node
if (root)
{
if (!isBST(root->left))
return false;
// Allows only distinct valued nodes
if (prev != NULL && root->data <= prev->data)
return false;
prev = root;
return isBST(root->right);
}
return true;
}
#define MIN (FirstVal, SecondVal) ((FirstVal) < (SecondVal)) ? (FirstVal):(SecondVal)
#define MAX (FirstVal, SecondVal) ((FirstVal) > (SecondVal)) ? (FirstVal):(SecondVal)
template <class T>
bool IsValidBST (treeNode &root)
{
T min, max;
return IsValidBST (root, &min, &max);
}
template <class T>
bool IsValidBST (treeNode *root, T *MIN , T *MAX)
{
T leftMin, leftMax, rightMin, rightMax;
bool isValidBST;
if (root->leftNode == NULL && root->rightNode == NULL)
{
*MIN = root->element;
*MAX = root->element;
return true;
}
isValidBST = IsValidBST (root->leftNode, &leftMin, &leftMax);
if (isValidBST)
isValidBST = IsValidBST (root->rightNode, &rightMin, &rightMax);
if (isValidBST)
{
*MIN = MIN (leftMIN, rightMIN);
*Max = MAX (rightMax, leftMax);
}
return isValidBST;
}
bool isBST(struct node* root)
{
static struct node *prev = NULL;
// traverse the tree in inorder fashion and keep track of prev node
if (root)
{
if (!isBST(root->left))
return false;
// Allows only distinct valued nodes
if (prev != NULL && root->data <= prev->data)
return false;
prev = root;
return isBST(root->right);
}
return true;
}
struct TreeNode
{
int value;
TreeNode* left;
TreeNode* right;
};
bool isBST(TreeNode* root) {
vector<TreeNode*> stack;
TreeNode* prev = nullptr;
while (root || stack.size()) {
if (root) {
stack.push_back(root);
root = root->left;
} else {
if (prev && stack.back()->value <= prev->value)
return false;
prev = stack.back();
root = prev->right;
stack.pop_back();
}
}
return true;
}
public boolean isValidBST(TreeNode root, TreeNode predecessor) {
boolean left = true, right = true;
if (root.left != null) {
left = isValidBST(root.left, predecessor);
}
if (!left)
return false;
if (predecessor.val > root.val)
return false;
predecessor.val = root.val;
if (root.right != null) {
right = isValidBST(root.right, predecessor);
}
if (!right)
return false;
return true;
}
static class BSTValidator {
private boolean lastNumberInitialized = false;
private int lastNumber = -1;
boolean isValidBST(TreeNode node) {
if (node.left != null && !isValidBST(node.left)) return false;
// In-order visiting should never see number less than previous
// in valid BST.
if (lastNumberInitialized && (lastNumber > node.getData())) return false;
if (!lastNumberInitialized) lastNumberInitialized = true;
lastNumber = node.getData();
if (node.right != null && !isValidBST(node.right)) return false;
return true;
}
}
public boolean isValidBST(TreeNode root) {
TreeNode pre = null, cur = root, tmp;
while(cur != null) {
if(cur.left == null) {
if(pre != null && pre.val >= cur.val)
return false;
pre = cur;
cur = cur.right;
}
else {
tmp = cur.left;
while(tmp.right != null && tmp.right != cur)
tmp = tmp.right;
if(tmp.right == null) { // left child has not been visited
tmp.right = cur;
cur = cur.left;
}
else { // left child has been visited already
tmp.right = null;
if(pre != null && pre.val >= cur.val)
return false;
pre = cur;
cur = cur.right;
}
}
}
return true;
}
public <T extends Comparable<T>> boolean isBst(TreeNode<T> root) {
return isBst(root, null);
}
private <T extends Comparable<T>> boolean isBst(TreeNode<T> node, TreeNode<T> prev) {
if (node == null)
return true;
if (isBst(node.left, prev) && (prev == null || prev.compareTo(node) < 0 ))
return isBst(node.right, node);
return false;
}
private static boolean checkBst(bst node) {
Stack<bst> s = new Stack<bst>();
bst temp;
while(node!=null){
s.push(node);
node=node.left;
}
while (!s.isEmpty()){
node = s.pop();
System.out.println(node.val);
temp = node;
if(node.right!=null){
node = node.right;
while(node!=null)
{
//Checking if the current value is lesser than the previous value and ancestor.
if(node.val < temp.val)
return false;
if(!s.isEmpty())
if(node.val>s.peek().val)
return false;
s.push(node);
if(node!=null)
node=node.left;
}
}
}
return true;
}
// time O(n), space O(logn)
// pseudocode
is-bst(node, min = int.min, max = int.max):
if node == null:
return true
if node.value <= min || max < node.value:
return false
return is-bst(node.left, min, node.value)
&& is-bst(node.right, node.value, max)
// time O(n), space O(logn)
// pseudocode
is-bst(node, min = null, max = null):
if node == null:
return true
if min != null && node.value <= min
return false
if max != null && max < node.value:
return false
return is-bst(node.left, min, node.value)
&& is-bst(node.right, node.value, max)
public class validateBinarySearchTree {
public boolean isValidBST(TreeNode root) {
return isBSTTraversal(root) && isBSTDivideAndConquer(root);
}
// Solution 1: Traversal
// The inorder sequence of a BST is a sorted ascending list
private int lastValue = 0; // the init value of it doesn't matter.
private boolean firstNode = true;
public boolean isBSTTraversal(TreeNode root) {
if (root == null) {
return true;
}
if (!isValidBST(root.left)) {
return false;
}
// firstNode is needed because of if firstNode is Integer.MIN_VALUE,
// even if we set lastValue to Integer.MIN_VALUE, it will still return false
if (!firstNode && lastValue >= root.val) {
return false;
}
firstNode = false;
lastValue = root.val;
if (!isValidBST(root.right)) {
return false;
}
return true;
}
// Solution 2: divide && conquer
private class Result {
int min;
int max;
boolean isBST;
Result(int min, int max, boolean isBST) {
this.min = min;
this.max = max;
this.isBST = isBST;
}
}
public boolean isBSTDivideAndConquer(TreeNode root) {
return isBSTHelper(root).isBST;
}
public Result isBSTHelper(TreeNode root) {
// For leaf node's left or right
if (root == null) {
// we set min to Integer.MAX_VALUE and max to Integer.MIN_VALUE
// because of in the previous level which is the leaf level,
// we want to set the min or max to that leaf node's val (in the last return line)
return new Result(Integer.MAX_VALUE, Integer.MIN_VALUE, true);
}
Result left = isBSTHelper(root.left);
Result right = isBSTHelper(root.right);
if (!left.isBST || !right.isBST) {
return new Result(0,0, false);
}
// For non-leaf node
if (root.left != null && left.max >= root.val
&& root.right != null && right.min <= root.val) {
return new Result(0, 0, false);
}
return new Result(Math.min(left.min, root.val),
Math.max(right.max, root.val), true);
}
}
public boolean isValid(Node node) {
return isValid(node, Integer.MIN_VALUE, Integer.MAX_VALUE);
}
private boolean isValid(Node node, int minLimit, int maxLimit) {
if (node == null)
return true;
return minLimit <= node.value && node.value <= maxLimit
&& isValid(node.left, minLimit, node.value)
&& isValid(node.right, node.value, maxLimit);
}
private void validateBinarySearchTree(Node node) {
if (node == null) return;
Node left = node.getLeft();
if (left != null) {
if (left.getData() < node.getData()) {
validateBinarySearchTree(left);
} else {
throw new IllegalStateException("Not a valid Binary Search tree");
}
}
Node right = node.getRight();
if (right != null) {
if (right.getData() > node.getData()) {
validateBinarySearchTree(right);
} else {
throw new IllegalStateException("Not a valid Binary Search tree");
}
}
}
""" Node is defined as
class node:
def __init__(self, data):
self.data = data
self.left = None
self.right = None
"""
def checkBST(root):
return checkLeftSubTree(root, root.left) and checkRightSubTree(root, root.right)
def checkLeftSubTree(root, subTree):
if not subTree:
return True
else:
return root.data > subTree.data \
and checkLeftSubTree(root, subTree.left) \
and checkLeftSubTree(root, subTree.right) \
and checkLeftSubTree(subTree, subTree.left) \
and checkRightSubTree(subTree, subTree.right)
def checkRightSubTree(root, subTree):
if not subTree:
return True
else:
return root.data < subTree.data \
and checkRightSubTree(root, subTree.left) \
and checkRightSubTree(root, subTree.right) \
and checkRightSubTree(subTree, subTree.right) \
and checkLeftSubTree(subTree, subTree.left)
bool is_bst(Node *root, int from, int to) {
return (root == NULL) ? true :
root->val >= from && root->val <= to &&
is_bst(root->left, from, root->val) &&
is_bst(root->right, root->val, to);
}
private boolean isBST() {
return isBST(root, null, null);
}
private boolean isBST(Node x, Key min, Key max) {
if (x == null) return true;
// when checking right subtree min is key of x's parent
if (min != null && x.key.compareTo(min) <= 0) return false;
// when checking left subtree, max is key of x's parent
if (max != null && x.key.compareTo(max) >= 0) return false;
// check left subtree and right subtree
return isBST(x.left, min, x.key) && isBST(x.right, x.key, max);
}
class Solution:
def isValidBST(self, root):
"""
:type root: TreeNode
:rtype: bool
"""
return self.iterative(root)
# return self.recurse(root, float("inf"), float("-inf"))
def iterative(self, root):
if not root:
return True
level = [[root, -float("inf"), float("inf")]]
while level:
next_level = []
for element in level:
node, min_val, max_val = element
if min_val<node.val<max_val:
if node.left:
next_level.append([node.left, min_val, node.val])
if node.right:
next_level.append([node.right, node.val, max_val])
else:
return False
level = next_level
return True
def recurse(self, root, maxi, mini):
if root is None:
return True
if root.val < mini or root.val > maxi:
return False
return self.recurse(root.left, root.val-1, mini) and self.recurse(root.right, maxi, root.val+1)
from __future__ import annotations
"""Checks if Binary Search Tree (BST) is balanced"""
from __future__ import annotations
import sys
from dataclasses import dataclass
MAX_KEY = sys.maxsize
MIN_KEY = -sys.maxsize - 1
@dataclass
class Node:
value: int
left: Node
right: Node
@property
def is_leaf(self) -> bool:
"""Check if node is a leaf"""
return not self.left and not self.right
def is_bst(node: Node, min_value: int, max_value: int) -> bool:
if node.value < min_value or max_value < node.value:
return False
elif node.is_leaf:
return True
return is_bst(node.left, min_value, node.value) and is_bst(
node.right, node.value, max_value
)
if __name__ == "__main__":
node5 = Node(5, None, None)
node25 = Node(25, None, None)
node40 = Node(40, None, None)
node10 = Node(10, None, None)
# balanced tree
node30 = Node(30, node25, node40)
root = Node(20, node10, node30)
print(is_bst(root, MIN_KEY, MAX_KEY))
# unbalanced tree
node30 = Node(30, node5, node40)
root = Node(20, node10, node30)
print(is_bst(root, MIN_KEY, MAX_KEY))
public bool IsBinarySearchTree(TreeNode root)
{
return IsValid(root, long.MinValue, long.MaxValue);
}
private static bool IsValid(TreeNode node, long min, long max)
{
if (node == null)
{
return true;
}
if (node.Value >= max || node.Value <= min)
{
return false;
}
return IsValid(node.Left, min, node.Value) && IsValid(node.Right, node.Value, max);
}
public class TreeNode
{
public int Value;
public TreeNode Left;
public TreeNode Right;
public TreeNode(int value)
{
Value = value;
}
}