Find the closest element in Binary Search Tree | Space Efficient Method

Given a binary search tree and a target node K. The task is to find the node with the minimum absolute difference with given target value K.

NOTE: The approach used should have constant extra space consumed O(1). No recursion or stack/queue like containers should be used.

Examples:

Input:  k = 4
Output:  4

Input:  k = 18
Output:  17

A simple solution mentioned in this post uses recursion to get the closest element to a key in Binary search tree. The method used in the above mentioned post consumes O(n) extra space due to recursion. 

Now we can easily modify the above mentioned approach using Morris traversal which is a space efficient approach to do inorder tree traversal without using recursion or stack/queue in constant space O(1).

Morris traversal is based on Threaded Binary trees which makes use of NULL pointers in a tree to make them point to some successor or predecessor nodes. As in a binary tree with n nodes, n+1 NULL pointers waste memory.

In the algorithm mentioned below we simply do inorder tree traversal and while doing inorder tree traversal using Morris Traversal we check for differences between the node’s data and the key and maintain two variables ‘diff’ and ‘closest’ which are updated when we find a closer node to the key. When we are done with the complete inorder tree traversal we have the closest node.

Algorithm : 

1) Initialize Current as root.

2) Initialize a variable diff as INT_MAX.

3)initialize a variable closest(pointer to node) which 
  will be returned.

4) While current is not NULL:

  4.1) If the current has no left child:
     a) If the absolute difference between current's data
        and the key is smaller than diff:
       1) Set diff as the absolute difference between the 
          current node and the key.
       2) Set closest as the current node. 

     b)Otherwise, Move to the right child of current.

  4.2) Else, here we have 2 cases:

   a) Find the inorder predecessor of the current node. 
      Inorder predecessor is the rightmost node 
      in the left subtree or left child itself.

   b) If the right child of the inorder predecessor is NULL:
      1) Set current as the right child of its inorder 
         predecessor(Making threads between nodes).
      2) Move current node to its left child.

   c) Else, if the threaded link between the current node 
      and it's inorder predecessor already exists :

      1) Set right pointer of the inorder predecessor node as NULL.

      2) If the absolute difference between current's data and 
         the key is smaller than diff:
        a) Set diff variable as the absolute difference between 
           the current node and the key.
        b) Set closest as the current node. 

      3) Move current to its right child.

5)By the time we have traversed the whole tree, we have the 
  closest node, so we simply return closest.

Below is the implementation of above approach: 

C++

// CPP program to find closest value in
// a Binary Search Tree.
#include <iostream>
#include <limits.h>
using namespace std;

// Tree Node
struct Node {
    int data;
    Node *left, *right;
};

// Utility function to create a new Node
Node* newNode(int data)
{
    Node* temp = new Node();
    temp->data = data;
    temp->left = temp->right = NULL;
    return temp;
}

// Function to find the Node closest to the
// given key in BST using Morris Traversal
Node* closestNodeUsingMorrisTraversal(Node* root,
                                         int key)
{
    int diff = INT_MAX;
    Node* curr = root;
    Node* closest;

    while (curr) {
        if (curr->left == NULL) {

            // updating diff if the current diff is
            // smaller than prev difference
            if (diff > abs(curr->data - key)) {
                diff = abs(curr->data - key);
                closest = curr;
            }

            curr = curr->right;
        }

        else {

            // finding the inorder predecessor
            Node* pre = curr->left;
            while (pre->right != NULL &&
                   pre->right != curr)
                pre = pre->right;

            if (pre->right == NULL) {
                pre->right = curr;
                curr = curr->left;
            }

            // threaded link between curr and
            // its predecessor already exists
            else {
                pre->right = NULL;

                // if a closer Node found, then update
                // the diff and set closest to current
                if (diff > abs(curr->data - key)) {
                    diff = abs(curr->data - key);
                    closest = curr;
                }

                // moving to the right child
                curr = curr->right;
            }
        }
    }

    return closest;
}

// Driver Code
int main()
{
    /* Constructed binary tree is
          5
        /   \
       3     9
     /  \   /  \
    1    2  8    12 */
    Node* root = newNode(5);
    root->left = newNode(3);
    root->right = newNode(9);
    root->left->left = newNode(1);
    root->left->right = newNode(2);
    root->right->left = newNode(8);
    root->right->right = newNode(12);

    cout << closestNodeUsingMorrisTraversal(root, 10)->data;

    return 0;
}

Java

// Java program to find closest value in
// a Binary Search Tree.
class GFG
{


// Tree Node
static class Node
{
    int data;
    Node left, right;
};

// Utility function to create a new Node
static Node newNode(int data)
{
    Node temp = new Node();
    temp.data = data;
    temp.left = temp.right = null;
    return temp;
}

// Function to find the Node closest to the
// given key in BST using Morris Traversal
static Node closestNodeUsingMorrisTraversal(Node root,
                                        int key)
{
    int diff = Integer.MAX_VALUE;
    Node curr = root;
    Node closest = null;

    while (curr != null)
    {
        if (curr.left == null)
        {

            // updating diff if the current diff is
            // smaller than prev difference
            if (diff > Math.abs(curr.data - key))
            {
                diff = Math.abs(curr.data - key);
                closest = curr;
            }

            curr = curr.right;
        }

        else
        {

            // finding the inorder predecessor
            Node pre = curr.left;
            while (pre.right != null &&
                pre.right != curr)
                pre = pre.right;

            if (pre.right == null)
            {
                pre.right = curr;
                curr = curr.left;
            }

            // threaded link between curr and
            // its predecessor already exists
            else
            {
                pre.right = null;

                // if a closer Node found, then update
                // the diff and set closest to current
                if (diff > Math.abs(curr.data - key))
                {
                    diff = Math.abs(curr.data - key);
                    closest = curr;
                }

                // moving to the right child
                curr = curr.right;
            }
        }
    }

    return closest;
}

// Driver Code
public static void main(String[] args)
{
    /* Constructed binary tree is
        5
        / \
    3     9
    / \ / \
    1 2 8 12 */
    Node root = newNode(5);
    root.left = newNode(3);
    root.right = newNode(9);
    root.left.left = newNode(1);
    root.left.right = newNode(2);
    root.right.left = newNode(8);
    root.right.right = newNode(12);

    System.out.println(closestNodeUsingMorrisTraversal(root, 10).data);
}
}

// This code is contributed by Rajput-Ji

Python3

# Python program to find closest value in
# Binary search Tree

_MIN = -2147483648
_MAX = 2147483648

# Helper function that allocates a new
# node with the given data and None left
# and right pointers.
class newNode:

    # Constructor to create a new node
    def __init__(self, data):
        self.data = data
        self.left = None
        self.right = None

# Function to find the Node closest to the
# given key in BST using Morris Traversal
def closestNodeUsingMorrisTraversal(root,key):
    diff = _MAX
    curr = root
    closest=0

    while (curr) :
        if (curr.left == None) :

            # updating diff if the current diff is
            # smaller than prev difference
            if (diff > abs(curr.data - key)) :
                diff = abs(curr.data - key)
                closest = curr

            curr = curr.right


        else :

            # finding the inorder predecessor
            pre = curr.left
            while (pre.right != None and
                    pre.right != curr):
                pre = pre.right

            if (pre.right == None):
                pre.right = curr
                curr = curr.left


            # threaded link between curr and
            # its predecessor already exists
            else :
                pre.right = None

                # if a closer Node found, then update
                # the diff and set closest to current
                if (diff > abs(curr.data - key)) :
                    diff = abs(curr.data - key)
                    closest = curr

                # moving to the right child
                curr = curr.right

    return closest


# Driver Code
if __name__ == '__main__':
    """ /* Constructed binary tree is
        5
        / \
    3 9
    / \ / \
    1 2 8 12 */ """

    root = newNode(5)
    root.left = newNode(3)
    root.right = newNode(9)
    root.left.right = newNode(2)
    root.left.left = newNode(1)
    root.right.right = newNode(12)
    root.right.left = newNode(8)
    print(closestNodeUsingMorrisTraversal(root, 10).data)

# This code is contributed
# Shubham Singh(SHUBHAMSINGH10)

C#

// C# program to find closest value in
// a Binary Search Tree.
using System;

class GFG
{


// Tree Node
public class Node
{
    public int data;
    public Node left, right;
};

// Utility function to create a new Node
static Node newNode(int data)
{
    Node temp = new Node();
    temp.data = data;
    temp.left = temp.right = null;
    return temp;
}

// Function to find the Node closest to the
// given key in BST using Morris Traversal
static Node closestNodeUsingMorrisTraversal(Node root,
                                        int key)
{
    int diff = int.MaxValue;
    Node curr = root;
    Node closest = null;

    while (curr != null)
    {
        if (curr.left == null)
        {

            // updating diff if the current diff is
            // smaller than prev difference
            if (diff > Math.Abs(curr.data - key))
            {
                diff = Math.Abs(curr.data - key);
                closest = curr;
            }

            curr = curr.right;
        }

        else
        {

            // finding the inorder predecessor
            Node pre = curr.left;
            while (pre.right != null &&
                pre.right != curr)
                pre = pre.right;

            if (pre.right == null)
            {
                pre.right = curr;
                curr = curr.left;
            }

            // threaded link between curr and
            // its predecessor already exists
            else
            {
                pre.right = null;

                // if a closer Node found, then update
                // the diff and set closest to current
                if (diff > Math.Abs(curr.data - key))
                {
                    diff = Math.Abs(curr.data - key);
                    closest = curr;
                }

                // moving to the right child
                curr = curr.right;
            }
        }
    }

    return closest;
}

// Driver Code
public static void Main(String[] args)
{
    /* Constructed binary tree is
        5
        / \
    3     9
    / \ / \
    1 2 8 12 */
    Node root = newNode(5);
    root.left = newNode(3);
    root.right = newNode(9);
    root.left.left = newNode(1);
    root.left.right = newNode(2);
    root.right.left = newNode(8);
    root.right.right = newNode(12);

    Console.WriteLine(closestNodeUsingMorrisTraversal(root, 10).data);
}
}

/* This code is contributed by PrinciRaj1992 */

Javascript

<script>

// Javascript program to find closest value in
// a Binary Search Tree.

// Tree Node
class Node
{
    constructor()
    {
        this.data = 0;
        this.left = null;
        this.right = null;
    }
};

// Utility function to create a new Node
function newNode(data)
{
    var temp = new Node();
    temp.data = data;
    temp.left = temp.right = null;
    return temp;
}

// Function to find the Node closest to the
// given key in BST using Morris Traversal
function closestNodeUsingMorrisTraversal(root, key)
{
    var diff = 1000000000;
    var curr = root;
    var closest = null;

    while (curr != null)
    {
        if (curr.left == null)
        {

            // updating diff if the current diff is
            // smaller than prev difference
            if (diff > Math.abs(curr.data - key))
            {
                diff = Math.abs(curr.data - key);
                closest = curr;
            }

            curr = curr.right;
        }

        else
        {

            // finding the inorder predecessor
            var pre = curr.left;
            while (pre.right != null &&
                pre.right != curr)
                pre = pre.right;

            if (pre.right == null)
            {
                pre.right = curr;
                curr = curr.left;
            }

            // threaded link between curr and
            // its predecessor already exists
            else
            {
                pre.right = null;

                // if a closer Node found, then update
                // the diff and set closest to current
                if (diff > Math.abs(curr.data - key))
                {
                    diff = Math.abs(curr.data - key);
                    closest = curr;
                }

                // moving to the right child
                curr = curr.right;
            }
        }
    }

    return closest;
}

// Driver Code
/* Constructed binary tree is
    5
    / \
3     9
/ \ / \
1 2 8 12 */
var root = newNode(5);
root.left = newNode(3);
root.right = newNode(9);
root.left.left = newNode(1);
root.left.right = newNode(2);
root.right.left = newNode(8);
root.right.right = newNode(12);
document.write(closestNodeUsingMorrisTraversal(root, 10).data);

// This code is contributed by itsok.
</script>

Output

9

Complexity Analysis:

• Time Complexity: O(n) 
• Auxiliary Space : O(1)