# Shortest distance between two nodes in an infinite binary tree

Consider you have an infinitely long binary tree having a pattern as below:

```
1
/      \
2        3
/  \      / \
4    5    6   7
/  \  / \  / \ / \
.  .  .  . .  .  .  .
```

Given two nodes with values x and y. The task is to find the length of the shortest path between the two nodes.

Examples:

```Input:  x = 2, y = 3
Output: 2

Input: x = 4, y = 6
Output: 4
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

A naive approach is to store all the ancestors of both nodes in 2 Data-structures(vectors, arrays, etc..) and do a binary search for the first element(let index i) in vector1, and check if it exists in the vector2 or not. If it does, return the index(let x) of the element in vector2. The answer will be thus

distance = v1.size() – 1 – i + v2.size() – 1 – x

Below is the implementation of the above approach.

## C++

 `// CPP program to find distance ` `// between two nodes ` `// in a infinite binary tree ` `#include ` `using` `namespace` `std; ` ` `  `// to stores ancestors of first given node ` `vector<``int``> v1; ` `// to stores ancestors of first given node ` `vector<``int``> v2; ` ` `  `// normal binary search to find the element ` `int` `BinarySearch(``int` `x) ` `{ ` `    ``int` `low = 0; ` `    ``int` `high = v2.size() - 1; ` ` `  `    ``while` `(low <= high) { ` `        ``int` `mid = (low + high) / 2; ` ` `  `        ``if` `(v2[mid] == x) ` `            ``return` `mid; ` `        ``else` `if` `(v2[mid] > x) ` `            ``high = mid - 1; ` `        ``else` `            ``low = mid + 1; ` `    ``} ` `    ``return` `-1; ` `} ` ` `  `// function to make ancestors of first node ` `void` `MakeAncestorNode1(``int` `x) ` `{ ` `    ``v1.clear(); ` `    ``while` `(x) { ` `        ``v1.push_back(x); ` `        ``x /= 2; ` `    ``} ` `    ``reverse(v1.begin(), v1.end()); ` `} ` ` `  `// function to make ancestors of second node ` `void` `MakeAncestorNode2(``int` `x) ` `{ ` `    ``v2.clear(); ` `    ``while` `(x) { ` `        ``v2.push_back(x); ` `        ``x /= 2; ` `    ``} ` `    ``reverse(v2.begin(), v2.end()); ` `} ` ` `  `// function to find distance bewteen two nodes ` `int` `Distance() ` `{ ` `    ``for` `(``int` `i = v1.size() - 1; i >= 0; i--) { ` `        ``int` `x = BinarySearch(v1[i]); ` `        ``if` `(x != -1) { ` `            ``return` `v1.size() - 1 - i + v2.size() - 1 - x; ` `        ``} ` `    ``} ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `node1 = 2, node2 = 3; ` ` `  `    ``// find ancestors ` `    ``MakeAncestorNode1(node1); ` `    ``MakeAncestorNode2(node2); ` ` `  `    ``cout << ``"Distance between "` `<< node1 << ` `    ``" and "` `<< node2 << ``" is : "` `<< Distance(); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to find distance ` `// between two nodes ` `// in a infinite binary tree ` `import` `java.util.*; ` `class` `GFG  ` `{ ` ` `  `// to stores ancestors of first given node ` `static` `Vector v1 = ``new` `Vector(); ` ` `  `// to stores ancestors of first given node ` `static` `Vector v2 = ``new` `Vector(); ` ` `  `// normal binary search to find the element ` `static` `int` `BinarySearch(``int` `x) ` `{ ` `    ``int` `low = ``0``; ` `    ``int` `high = v2.size() - ``1``; ` ` `  `    ``while` `(low <= high) ` `    ``{ ` `        ``int` `mid = (low + high) / ``2``; ` ` `  `        ``if` `(v2.get(mid) == x) ` `            ``return` `mid; ` `        ``else` `if` `(v2.get(mid) > x) ` `            ``high = mid - ``1``; ` `        ``else` `            ``low = mid + ``1``; ` `    ``} ` `    ``return` `-``1``; ` `} ` ` `  `// function to make ancestors of first node ` `static` `void` `MakeAncestorNode1(``int` `x) ` `{ ` `    ``v1.clear(); ` `    ``while` `(x > ``0``) ` `    ``{ ` `        ``v1.add(x); ` `        ``x /= ``2``; ` `    ``} ` `    ``Collections.reverse(v1); ` `} ` ` `  `// function to make ancestors of second node ` `static` `void` `MakeAncestorNode2(``int` `x) ` `{ ` `    ``v2.clear(); ` `    ``while` `(x > ``0``)  ` `    ``{ ` `        ``v2.add(x); ` `        ``x /= ``2``; ` `    ``} ` `    ``Collections.reverse(v2); ` `} ` ` `  `// function to find distance bewteen two nodes ` `static` `int` `Distance() ` `{ ` `    ``for` `(``int` `i = v1.size() - ``1``; i >= ``0``; i--) ` `    ``{ ` `        ``int` `x = BinarySearch(v1.get(i)); ` `        ``if` `(x != -``1``) ` `        ``{ ` `            ``return` `v1.size() - ``1` `- i + ` `                   ``v2.size() - ``1` `- x; ` `        ``} ` `    ``} ` `    ``return` `Integer.MAX_VALUE; ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args)  ` `{ ` `    ``int` `node1 = ``2``, node2 = ``3``; ` ` `  `    ``// find ancestors ` `    ``MakeAncestorNode1(node1); ` `    ``MakeAncestorNode2(node2); ` ` `  `    ``System.out.print(``"Distance between "` `+ node1 + ` `                      ``" and "` `+ node2 + ``" is : "` `+  ` `                                      ``Distance()); ` `} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

## Python3

 `# Python3 program to find the distance between  ` `# two nodes in an infinite binary tree  ` ` `  `# normal binary search to find the element  ` `def` `BinarySearch(x):  ` ` `  `    ``low ``=` `0` `    ``high ``=` `len``(v2) ``-` `1` ` `  `    ``while` `low <``=` `high:  ` `        ``mid ``=` `(low ``+` `high) ``/``/` `2` ` `  `        ``if` `v2[mid] ``=``=` `x: ` `            ``return` `mid  ` `        ``elif` `v2[mid] > x:  ` `            ``high ``=` `mid ``-` `1` `        ``else``: ` `            ``low ``=` `mid ``+` `1` `     `  `    ``return` `-``1` ` `  `# Function to make ancestors of first node  ` `def` `MakeAncestorNode1(x):  ` ` `  `    ``v1.clear()  ` `    ``while` `x: ` `        ``v1.append(x)  ` `        ``x ``/``/``=` `2` `     `  `    ``v1.reverse()  ` ` `  `# Function to make ancestors of second node  ` `def` `MakeAncestorNode2(x): ` ` `  `    ``v2.clear()  ` `    ``while` `x: ` `        ``v2.append(x)  ` `        ``x ``/``/``=` `2` `     `  `    ``v2.reverse()  ` ` `  `# Function to find distance bewteen two nodes  ` `def` `Distance():  ` ` `  `    ``for` `i ``in` `range``(``len``(v1) ``-` `1``, ``-``1``, ``-``1``):  ` `        ``x ``=` `BinarySearch(v1[i])  ` `         `  `        ``if` `x !``=` `-``1``:  ` `            ``return` `(``len``(v1) ``-` `1` `-` `i ``+`  `                    ``len``(v2) ``-` `1` `-` `x)  ` `     `  `# Driver code  ` `if` `__name__ ``=``=` `"__main__"``:  ` ` `  `    ``node1, node2 ``=` `2``, ``3` `    ``v1, v2 ``=` `[], [] ` `     `  `    ``# Find ancestors  ` `    ``MakeAncestorNode1(node1)  ` `    ``MakeAncestorNode2(node2)  ` ` `  `    ``print``(``"Distance between"``, node1,  ` `          ``"and"``, node2, ``"is :"``, Distance())  ` ` `  `# This code is contributed by Rituraj Jain `

## C#

 `// C# program to find distance ` `// between two nodes ` `// in a infinite binary tree ` `using` `System; ` `using` `System.Collections.Generic; ` ` `  `class` `GFG  ` `{ ` ` `  `// to stores ancestors of first given node ` `static` `List<``int``> v1 = ``new` `List<``int``>(); ` ` `  `// to stores ancestors of first given node ` `static` `List<``int``> v2 = ``new` `List<``int``>(); ` ` `  `// normal binary search to find the element ` `static` `int` `BinarySearch(``int` `x) ` `{ ` `    ``int` `low = 0; ` `    ``int` `high = v2.Count - 1; ` ` `  `    ``while` `(low <= high) ` `    ``{ ` `        ``int` `mid = (low + high) / 2; ` ` `  `        ``if` `(v2[mid] == x) ` `            ``return` `mid; ` `        ``else` `if` `(v2[mid] > x) ` `            ``high = mid - 1; ` `        ``else` `            ``low = mid + 1; ` `    ``} ` `    ``return` `-1; ` `} ` ` `  `// function to make ancestors of first node ` `static` `void` `MakeAncestorNode1(``int` `x) ` `{ ` `    ``v1.Clear(); ` `    ``while` `(x > 0) ` `    ``{ ` `        ``v1.Add(x); ` `        ``x /= 2; ` `    ``} ` `    ``v1.Reverse(); ` `} ` ` `  `// function to make ancestors of second node ` `static` `void` `MakeAncestorNode2(``int` `x) ` `{ ` `    ``v2.Clear(); ` `    ``while` `(x > 0)  ` `    ``{ ` `        ``v2.Add(x); ` `        ``x /= 2; ` `    ``} ` `    ``v2.Reverse(); ` `} ` ` `  `// function to find distance bewteen two nodes ` `static` `int` `Distance() ` `{ ` `    ``for` `(``int` `i = v1.Count - 1; i >= 0; i--) ` `    ``{ ` `        ``int` `x = BinarySearch(v1[i]); ` `        ``if` `(x != -1) ` `        ``{ ` `            ``return` `v1.Count - 1 - i + ` `                ``v2.Count - 1 - x; ` `        ``} ` `    ``} ` `    ``return` `int``.MaxValue; ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main(String[] args)  ` `{ ` `    ``int` `node1 = 2, node2 = 3; ` ` `  `    ``// find ancestors ` `    ``MakeAncestorNode1(node1); ` `    ``MakeAncestorNode2(node2); ` ` `  `    ``Console.Write(``"Distance between "` `+ node1 + ` `                   ``" and "` `+ node2 + ``" is : "` `+  ` `                                   ``Distance()); ` `} ` `} ` ` `  `// This code is contributed by Princi Singh `

Output:

```Distance between 2 and 3 is : 2
```

Time Complexity: O(log(max(x, y)) * log(max(x, y)))
Auxiliary Space: O(log(max(x, y)))

An efficient approach is to use the property of 2*x and 2*x+1 given. Keep dividing the larger of the two nodes by 2. If the larger becomes the smaller one, then divide the other one. At a stage, both the values will be the same, keep a count on the number of divisions done which will be the answer.

Below is the implementation of the above approach.

## C++

 `// C++ program to find the distance ` `// between two nodes in an infinite ` `// binary tree ` `#include ` `using` `namespace` `std; ` ` `  `// function to find the distance ` `// between two nodes in an infinite ` `// binary tree ` `int` `Distance(``int` `x, ``int` `y) ` `{ ` `    ``// swap the smaller ` `    ``if` `(x < y) { ` `        ``swap(x, y); ` `    ``} ` `    ``int` `c = 0; ` ` `  `    ``// divide till x!=y ` `    ``while` `(x != y) { ` ` `  `        ``// keep a count ` `        ``++c; ` ` `  `        ``// perform divison ` `        ``if` `(x > y) ` `            ``x = x >> 1; ` ` `  `        ``// when the smaller ` `        ``// becomes the greater ` `        ``if` `(y > x) { ` `            ``y = y >> 1; ` `            ``++c; ` `        ``} ` `    ``} ` `    ``return` `c; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `x = 4, y = 6; ` `    ``cout << Distance(x, y); ` ` `  ` ``return` `0; ` `} `

## Java

 `// Java program to find the distance ` `// between two nodes in an infinite ` `// binary tree ` `class` `GFG ` `{ ` ` `  `// function to find the distance ` `// between two nodes in an infinite ` `// binary tree ` `static` `int` `Distance(``int` `x, ``int` `y) ` `{ ` `    ``// swap the smaller ` `    ``if` `(x < y) ` `    ``{ ` `        ``int` `temp = x; ` `        ``x = y; ` `        ``y = temp; ` `    ``} ` `    ``int` `c = ``0``; ` ` `  `    ``// divide till x!=y ` `    ``while` `(x != y)  ` `    ``{ ` ` `  `        ``// keep a count ` `        ``++c; ` ` `  `        ``// perform divison ` `        ``if` `(x > y) ` `            ``x = x >> ``1``; ` ` `  `        ``// when the smaller ` `        ``// becomes the greater ` `        ``if` `(y > x)  ` `        ``{ ` `            ``y = y >> ``1``; ` `            ``++c; ` `        ``} ` `    ``} ` `    ``return` `c; ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `x = ``4``, y = ``6``; ` `    ``System.out.println(Distance(x, y)); ` `} ` `} ` ` `  `// This code is contributed by PrinciRaj1992  `

## Python3

 `# Python3 program to find the distance between ` `# two nodes in an infinite binary tree  ` ` `  `# Function to find the distance between ` `# two nodes in an infinite binary tree  ` `def` `Distance(x, y): ` ` `  `    ``# Swap the smaller  ` `    ``if` `x < y: ` `        ``x, y ``=` `y, x  ` `     `  `    ``c ``=` `0` `     `  `    ``# divide till x != y  ` `    ``while` `x !``=` `y:  ` ` `  `        ``# keep a count  ` `        ``c ``+``=` `1` ` `  `        ``# perform divison  ` `        ``if` `x > y:  ` `            ``x ``=` `x >> ``1` ` `  `        ``# when the smaller becomes ` `        ``# the greater  ` `        ``if` `y > x:  ` `            ``y ``=` `y >> ``1` `            ``c ``+``=` `1` `     `  `    ``return` `c  ` ` `  `# Driver code  ` `if` `__name__ ``=``=` `"__main__"``: ` ` `  `    ``x, y ``=` `4``, ``6` `    ``print``(Distance(x, y))  ` ` `  `# This code is contributed by ` `# Rituraj Jain `

## C#

 `// C# program to find the distance ` `// between two nodes in an infinite ` `// binary tree ` `using` `System; ` ` `  `class` `GFG ` `{ ` ` `  `// function to find the distance ` `// between two nodes in an infinite ` `// binary tree ` `static` `int` `Distance(``int` `x, ``int` `y) ` `{ ` `    ``// swap the smaller ` `    ``if` `(x < y) ` `    ``{ ` `        ``int` `temp = x; ` `        ``x = y; ` `        ``y = temp; ` `    ``} ` `    ``int` `c = 0; ` ` `  `    ``// divide till x!=y ` `    ``while` `(x != y)  ` `    ``{ ` ` `  `        ``// keep a count ` `        ``++c; ` ` `  `        ``// perform divison ` `        ``if` `(x > y) ` `            ``x = x >> 1; ` ` `  `        ``// when the smaller ` `        ``// becomes the greater ` `        ``if` `(y > x)  ` `        ``{ ` `            ``y = y >> 1; ` `            ``++c; ` `        ``} ` `    ``} ` `    ``return` `c; ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main(String[] args) ` `{ ` `    ``int` `x = 4, y = 6; ` `    ``Console.WriteLine(Distance(x, y)); ` `} ` `} ` ` `  `// This code contributed by Rajput-Ji `

Output:

```4
```

Time Complexity: O(log(max(x, y)))
Auxiliary Space: O(1)

The efficient approach has been suggested by Striver.

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