Queries to find distance between two nodes of a Binary tree

Given a binary tree, the task is to find the distance between two keys in a binary tree, no parent pointers are given. The distance between two nodes is the minimum number of edges to be traversed to reach one node from other.

We have already discussed a method which uses segment tree to reduce the query time to O(logn), here the task is to reduce query time to O(1) by compromising with space complexity to O(nlogn). In this post, we will use Sparse table instead of segment tree for finding the minimum in given range, which uses dynamic programming and bit manipulation to achieve O(1) query time.

A sparse table will preprocess the minimum values of given range for L array in Nlogn space i.e. each node will contain chain of values of log(i) length where i is the index of the ith node in L array. Each entry in the sparse table says M[i][j] will represent the index of the minimum value in the subarray starting at i having length 2^j.



The distance between two nodes can be obtained in terms of lowest common ancestor.

Dist(n1, n2) = Level[n1] + Level[n2] - 2*Level[lca] 

This problem can be breakdown into:

  1. Finding levels of each node
  2. Finding the Euler tour of binary tree
  3. Building sparse table for LCA.

These steps are explained below :

  1. Find the levels of each node by applying level order traversal.
  2. Find the LCA of two nodes in binary tree in O(logn) by Storing Euler tour of Binary tree in array and computing two other arrays with the help of levels of each node and Euler tour.
    These steps are shown below:

    (I) First, find Euler Tour of binary tree.

    (II) Then, store levels of each node in Euler array.

    (III) Then, store First occurrences of all nodes of binary tree in Euler array. H stores the indices of nodes from Euler array, so that range of query for finding minimum can be minimized and their by further optimizing the query time.

  3. Then build sparse table on L array and find the minimum value say X in range (H[A] to H[B]). Then, we use the index of value X as an index to Euler array to get LCA, i.e. Euler[index(X)].

    Let, A=8 and B=5.
    (I) H[8]= 1 and H[5]=2
    (II) we get min value in L array between 1 and 2 as X=0, index=7
    (III) Then, LCA= Euler[7], i.e LCA=1.

  4. Finally, apply distance formula discussed above to get the distance between two nodes.
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#include <bits/stdc++.h>
#define MAX 100001
using namespace std;
  
/* A tree node structure */
struct Node {
    int data;
    struct Node* left;
    struct Node* right;
};
  
/* Utility function to create a new Binary Tree node */
struct Node* newNode(int data)
{
    struct Node* temp = new struct Node;
    temp->data = data;
    temp->left = temp->right = NULL;
    return temp;
}
  
// Array to store level of each node
int level[MAX];
  
// Utility Function to store level of all nodes
void FindLevels(struct Node* root)
{
    if (!root)
        return;
  
    // queue to hold tree node with level
    queue<pair<struct Node*, int> > q;
  
    // let root node be at level 0
    q.push({ root, 0 });
    pair<struct Node*, int> p;
  
    // Do level Order Traversal of tree
    while (!q.empty()) {
        p = q.front();
        q.pop();
  
        // Node p.first is on level p.second
        level[p.first->data] = p.second;
  
        // If left child exits, put it in queue
        // with current_level +1
        if (p.first->left)
            q.push({ p.first->left, p.second + 1 });
  
        // If right child exists, put it in queue
        // with current_level +1
        if (p.first->right)
            q.push({ p.first->right, p.second + 1 });
    }
}
  
// Stores Euler Tour
int Euler[MAX];
  
// index in Euler array
int idx = 0;
  
// Find Euler Tour
void eulerTree(struct Node* root)
{
  
    // store current node's data
    Euler[++idx] = root->data;
  
    // If left node exists
    if (root->left) {
  
        // traverse left subtree
        eulerTree(root->left);
  
        // store parent node's data
        Euler[++idx] = root->data;
    }
  
    // If right node exists
    if (root->right) {
  
        // traverse right subtree
        eulerTree(root->right);
  
        // store parent node's data
        Euler[++idx] = root->data;
    }
}
  
// checks for visited nodes
int vis[MAX];
  
// Stores level of Euler Tour
int L[MAX];
  
// Stores indices of the first occurrence 
// of nodes in Euler tour
int H[MAX];
  
// Preprocessing Euler Tour for finding LCA
void preprocessEuler(int size)
{
    for (int i = 1; i <= size; i++) {
        L[i] = level[Euler[i]];
  
        // If node is not visited before
        if (vis[Euler[i]] == 0) {
  
            // Add to first occurrence
            H[Euler[i]] = i;
  
            // Mark it visited
            vis[Euler[i]] = 1;
        }
    }
}
  
// Sparse table of size [MAX][LOGMAX]
// M[i][j] is the index of the minimum value in
// the sub array starting at i having length 2^j
int M[MAX][18];
  
// Utility function to preprocess Sparse table
void preprocessLCA(int N)
{
    for (int i = 0; i < N; i++)
        M[i][0] = i;
  
    for (int j = 1; 1 << j <= N; j++)
        for (int i = 0; i + (1 << j) - 1 < N; i++)
            if (L[M[i][j - 1]] < L[M[i + (1 << (j - 1))][j - 1]])
                M[i][j] = M[i][j - 1];
            else
                M[i][j] = M[i + (1 << (j - 1))][j - 1];
}
  
// Utility function to find the index of the minimum
// value in range a to b
int LCA(int a, int b)
{
    // Subarray of length 2^j
    int j = log2(b - a + 1);
    if (L[M[a][j]] <= L[M[b - (1 << j) + 1][j]])
        return M[a][j];
  
    else
        return M[b - (1 << j) + 1][j];
}
  
// Function to return distance between
// two nodes n1 and n2
int findDistance(int n1, int n2)
{
    // Maintain original Values
    int prevn1 = n1, prevn2 = n2;
  
    // Get First Occurrence of n1
    n1 = H[n1];
  
    // Get First Occurrence of n2
    n2 = H[n2];
  
    // Swap if low>high
    if (n2 < n1)
        swap(n1, n2);
  
    // Get position of minimum value
    int lca = LCA(n1, n2);
  
    // Extract value out of Euler tour
    lca = Euler[lca];
  
    // return calculated distance
    return level[prevn1] + level[prevn2] - 2 * level[lca];
}
  
void preProcessing(Node* root, int N)
{
    // Build Tree
    eulerTree(root);
  
    // Store Levels
    FindLevels(root);
  
    // Find L and H array
    preprocessEuler(2 * N - 1);
  
    // Build sparse table
    preprocessLCA(2 * N - 1);
}
  
/* Driver function to test above functions */
int main()
{
    // Number of nodes
    int N = 8;
  
    /* Constructing tree given in the above figure */
    Node* root = newNode(1);
    root->left = newNode(2);
    root->right = newNode(3);
    root->left->left = newNode(4);
    root->left->right = newNode(5);
    root->right->left = newNode(6);
    root->right->right = newNode(7);
    root->right->left->right = newNode(8);
  
    // Function to do all preprocessing
    preProcessing(root, N);
  
    cout << "Dist(4, 5) = " << findDistance(4, 5) << "\n";
    cout << "Dist(4, 6) = " << findDistance(4, 6) << "\n";
    cout << "Dist(3, 4) = " << findDistance(3, 4) << "\n";
    cout << "Dist(2, 4) = " << findDistance(2, 4) << "\n";
    cout << "Dist(8, 5) = " << findDistance(8, 5) << "\n";
  
    return 0;
}
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// Java implementation of the approach
import java.util.*;
  
class GFG {
  
    static class Pair<T, V> {
        T first;
        V second;
  
        Pair() {
        }
  
        Pair(T first, V second) {
            this.first = first;
            this.second = second;
        }
    }
  
    static class Node {
        int data;
        Node left, right;
  
        Node(int data) {
            this.data = data;
            this.left = this.right = null;
        }
    }
  
    static int MAX = 100001;
  
    // Array to store level of each node
    static int[] level = new int[MAX];
  
    // Utility Function to store level of all nodes
    static void FindLevels(Node root) {
        if (root == null)
            return;
  
        // queue to hold tree node with level
        Queue<Pair<Node, Integer>> q = new LinkedList<>();
  
        // let root node be at level 0
        q.add(new Pair<>(root, 0));
        Pair<Node, Integer> p = new Pair<>();
  
        // Do level Order Traversal of tree
        while (!q.isEmpty()) {
            p = q.poll();
  
            // Node p.first is on level p.second
            level[p.first.data] = p.second;
  
            // If left child exits, put it in queue
            // with current_level +1
            if (p.first.left != null)
                q.add(new Pair<>(p.first.left, p.second + 1));
  
            // If right child exists, put it in queue
            // with current_level +1
            if (p.first.right != null)
                q.add(new Pair<>(p.first.right, p.second + 1));
        }
    }
  
    // Stores Euler Tour
    static int[] Euler = new int[MAX];
  
    // index in Euler array
    static int idx = 0;
  
    // Find Euler Tour
    static void eulerTree(Node root) {
  
        // store current node's data
        Euler[++idx] = root.data;
  
        // If left node exists
        if (root.left != null) {
  
            // traverse left subtree
            eulerTree(root.left);
  
            // store parent node's data
            Euler[++idx] = root.data;
        }
  
        // If right node exists
        if (root.right != null) {
  
            // traverse right subtree
            eulerTree(root.right);
  
            // store parent node's data
            Euler[++idx] = root.data;
        }
    }
  
    // checks for visited nodes
    static int[] vis = new int[MAX];
  
    // Stores level of Euler Tour
    static int[] L = new int[MAX];
  
    // Stores indices of the first occurrence
    // of nodes in Euler tour
    static int[] H = new int[MAX];
  
    // Preprocessing Euler Tour for finding LCA
    static void preprocessEuler(int size) {
        for (int i = 1; i <= size; i++) {
            L[i] = level[Euler[i]];
  
            // If node is not visited before
            if (vis[Euler[i]] == 0) {
  
                // Add to first occurrence
                H[Euler[i]] = i;
  
                // Mark it visited
                vis[Euler[i]] = 1;
            }
        }
    }
  
    // Sparse table of size [MAX][LOGMAX]
    // M[i][j] is the index of the minimum value in
    // the sub array starting at i having length 2^j
    static int[][] M = new int[MAX][18];
  
    // Utility function to preprocess Sparse table
    static void preprocessLCA(int N) {
        for (int i = 0; i < N; i++)
            M[i][0] = i;
  
        for (int j = 1; 1 << j <= N; j++)
            for (int i = 0; i + (1 << j) - 1 < N; i++)
                if (L[M[i][j - 1]] < L[M[i + (1 << (j - 1))][j - 1]])
                    M[i][j] = M[i][j - 1];
                else
                    M[i][j] = M[i + (1 << (j - 1))][j - 1];
    }
  
    // Utility function to find the index of the minimum
    // value in range a to b
    static int LCA(int a, int b) {
        // Subarray of length 2^j
        int j = (int) (Math.log(b - a + 1) / Math.log(2));
        if (L[M[a][j]] <= L[M[b - (1 << j) + 1][j]])
            return M[a][j];
  
        else
            return M[b - (1 << j) + 1][j];
    }
  
    // Function to return distance between
    // two nodes n1 and n2
    static int findDistance(int n1, int n2) {
        // Maintain original Values
        int prevn1 = n1, prevn2 = n2;
  
        // Get First Occurrence of n1
        n1 = H[n1];
  
        // Get First Occurrence of n2
        n2 = H[n2];
  
        // Swap if low>high
        if (n2 < n1) {
            int temp = n1;
            n1 = n2;
            n2 = temp;
        }
  
        // Get position of minimum value
        int lca = LCA(n1, n2);
  
        // Extract value out of Euler tour
        lca = Euler[lca];
  
        // return calculated distance
        return level[prevn1] + level[prevn2] - 2 * level[lca];
    }
  
    static void preProcessing(Node root, int N) {
        // Build Tree
        eulerTree(root);
  
        // Store Levels
        FindLevels(root);
  
        // Find L and H array
        preprocessEuler(2 * N - 1);
  
        // Build sparse table
        preprocessLCA(2 * N - 1);
    }
  
    // Driver Code
    public static void main(String[] args) {
        // Number of nodes
        int N = 8;
  
        /* Constructing tree given in the above figure */
        Node root = new Node(1);
        root.left = new Node(2);
        root.right = new Node(3);
        root.left.left = new Node(4);
        root.left.right = new Node(5);
        root.right.left = new Node(6);
        root.right.right = new Node(7);
        root.right.left.right = new Node(8);
  
        // Function to do all preprocessing
        preProcessing(root, N);
  
        System.out.println("Dist(4, 5) = " + findDistance(4, 5));
        System.out.println("Dist(4, 6) = " + findDistance(4, 6));
        System.out.println("Dist(3, 4) = " + findDistance(3, 4));
        System.out.println("Dist(2, 4) = " + findDistance(2, 4));
        System.out.println("Dist(8, 5) = " + findDistance(8, 5));
    }
}
  
// This code is contributed by
// sanjeev2552
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from collections import deque
from math import log2
  
MAX = 100001
  
# A tree node structure
class Node:
    def __init__(self, data):
        self.data = data
        self.left = None
        self.right = None
  
# Array to store level of each node
level = [0] * MAX
  
# Utility Function to store level of all nodes
def findLevels(root: Node):
    global level
  
    if root is None:
        return
  
    # queue to hold tree node with level
    q = deque()
  
    # let root node be at level 0
    q.append((root, 0))
  
    # Do level Order Traversal of tree
    while q:
        p = q[0]
        q.popleft()
  
        # Node p.first is on level p.second
        level[p[0].data] = p[1]
  
        # If left child exits, put it in queue
        # with current_level +1
        if p[0].left:
            q.append((p[0].left, p[1] + 1))
  
        # If right child exists, put it in queue
        # with current_level +1
        if p[0].right:
            q.append((p[0].right, p[1] + 1))
  
# Stores Euler Tour
Euler = [0] * MAX
  
# index in Euler array
idx = 0
  
# Find Euler Tour
def eulerTree(root: Node):
    global Euler, idx
    idx += 1
  
    # store current node's data
    Euler[idx] = root.data
  
    # If left node exists
    if root.left:
  
        # traverse left subtree
        eulerTree(root.left)
        idx += 1
  
        # store parent node's data
        Euler[idx] = root.data
  
    # If right node exists
    if root.right:
  
        # traverse right subtree
        eulerTree(root.right)
        idx += 1
  
        # store parent node's data
        Euler[idx] = root.data
  
# checks for visited nodes
vis = [0] * MAX
  
# Stores level of Euler Tour
L = [0] * MAX
  
# Stores indices of the first occurrence
# of nodes in Euler tour
H = [0] * MAX
  
# Preprocessing Euler Tour for finding LCA
def preprocessEuler(size: int):
    global L, H, vis
    for i in range(1, size + 1):
        L[i] = level[Euler[i]]
  
        # If node is not visited before
        if vis[Euler[i]] == 0:
  
            # Add to first occurrence
            H[Euler[i]] = i
  
            # Mark it visited
            vis[Euler[i]] = 1
  
# Sparse table of size [MAX][LOGMAX]
# M[i][j] is the index of the minimum value in
# the sub array starting at i having length 2^j
M = [[0 for i in range(18)] for j in range(MAX)]
  
# Utility function to preprocess Sparse table
def preprocessLCA(N: int):
    global M
    for i in range(N):
        M[i][0] = i
  
    j = 1
    while 1 << j <= N:
        i = 0
        while i + (1 << j) - 1 < N:
            if L[M[i][j - 1]] < L[M[i + 
                (1 << (j - 1))][j - 1]]:
                M[i][j] = M[i][j - 1]
            else:
                M[i][j] = M[i + (1 << (j - 1))][j - 1]
            i += 1
        j += 1
  
# Utility function to find the index of the minimum
# value in range a to b
def LCA(a: int, b: int) -> int:
  
    # Subarray of length 2^j
    j = int(log2(b - a + 1))
    if L[M[a][j]] <= L[M[b - (1 << j) + 1][j]]:
        return M[a][j]
    else:
        return M[b - (1 << j) + 1][j]
  
# Function to return distance between
# two nodes n1 and n2
def findDistance(n1: int, n2: int) -> int:
  
    # Maintain original Values
    prevn1 = n1
    prevn2 = n2
  
    # Get First Occurrence of n1
    n1 = H[n1]
  
    # Get First Occurrence of n2
    n2 = H[n2]
  
    # Swap if low>high
    if n2 < n1:
        n1, n2 = n2, n1
  
    # Get position of minimum value
    lca = LCA(n1, n2)
  
    # Extract value out of Euler tour
    lca = Euler[lca]
  
    # return calculated distance
    return level[prevn1] + level[prevn2] - 2 * level[lca]
  
def preProcessing(root: Node, N: int):
  
    # Build Tree
    eulerTree(root)
  
    # Store Levels
    findLevels(root)
  
    # Find L and H array
    preprocessEuler(2 * N - 1)
  
    # Build sparse table
    preprocessLCA(2 * N - 1)
  
# Driver Code
if __name__ == "__main__":
  
    # Number of nodes
    N = 8
  
    # Constructing tree given in the above figure
    root = Node(1)
    root.left = Node(2)
    root.right = Node(3)
    root.left.left = Node(4)
    root.left.right = Node(5)
    root.right.left = Node(6)
    root.right.right = Node(7)
    root.right.left.right = Node(8)
  
    # Function to do all preprocessing
    preProcessing(root, N)
  
    print("Dist(4, 5) =", findDistance(4, 5))
    print("Dist(4, 6) =", findDistance(4, 6))
    print("Dist(3, 4) =", findDistance(3, 4))
    print("Dist(2, 4) =", findDistance(2, 4))
    print("Dist(8, 5) =", findDistance(8, 5))
  
# This code is contributed by
# sanjeev2552
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Output:

Dist(4, 5) = 2
Dist(4, 6) = 4
Dist(3, 4) = 3
Dist(2, 4) = 1
Dist(8, 5) = 5

Time Complexity: O(1)
Space Complexity: O(N log N)




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