Diameter of a Binary Indexed Tree with N nodes

Given a Binary Indexed Tree with N nodes except root node 0 (Numbered from 1 to N), Find its diameter.

Binary Indexed Tree is a tree where parent of a node number X = X – (X & (X – 1)) i.e. last bit is unset in X. The diameter of a tree is the longest simple path between any two leaves.

Examples:



Input: N = 12
Output: 6
Explanation: Path from node 7 to node 11.

BIT with n  = 11

Input : n = 15
Output : 7

Approach:

  • In a BIT, root is always node 0. In first level, all nodes are of power of 2 . (1, 2, 4, 8, ….)
  • Consider any node in the first level (1, 2, 4, 8, ) its sub-tree will include all the nodes which has same number of bits as that of the root.
    1. Sub-Tree with root 1 will have no child.
    2. Sub-Tree with root 2 will have 3 as a child.
    3. Sub-Tree with root 4 will have 5, 6, 7 as a child.
    4. Sub-Tree with root 8 will have 9, 10, 11, 12, 13, 14, 15 as a child. (Double the size of the previous subtree)
    5. So subtree with root K will have K nodes including root. And the height of each subtree would be equal:
      • for subtree with root 1
      • for subtree with root 2
      • for subtree with root 4
  • Now, we need to find the subtree in which N lies. Say, the height of subtree just before the subtree in which N lies is H and size is L. So, the following cases are possible :
    • Case 1 : When N >= L*2 – 1, in such a scenario N is in last level of its subtree. Thus, the diameter will be 2*H + 1. (Path from the lowest level leaf of the previous subtree to the N ).
    • Case 2 : When N >= L + L/2 – 1, in such a scenario N is at level H in its subtree. Thus, diameter will be 2*H.
    • Case 3 : Otherwise, it is optimal to consider the maximum path length between leaf nodes of two subtree just before the subtree in which N lies i.e diameter is 2*H – 1.

Below are the implementation of the above approach:

C++

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#include <bits/stdc++.h>
using namespace std;
  
// Function to find diameter
// of BIT with N + 1 nodes
int diameter(int n)
{
    // L is size of subtree just before subtree
    // in which N lies
    int L, H, templen;
    L = 1;
  
    // H is the height of subtree just before
    // subtree in which N lies
    H = 0;
  
    // Base Cases
    if (n == 1) {
        return 1;
    }
    if (n == 2) {
        return 2;
    }
    if (n == 3) {
        return 3;
    }
  
    // Size of subtree are power of 2
    while (L * 2 <= n) {
        L *= 2;
        H++;
    }
  
    // 3 Cases as explained in Approach
    if (n >= L * 2 - 1)
        return 2 * H + 1;
    else if (n >= L + (L / 2) - 1)
        return 2 * H;
    return 2 * H - 1;
}
  
// Driver Code
int main()
{
    int n = 15;
    cout << diameter(n) << endl;
}

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Java

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// Java implementation of the approach
class GFG 
{
  
// Function to find diameter
// of BIT with N + 1 nodes
static int diameter(int n)
{
    // L is size of subtree just before subtree
    // in which N lies
    int L, H, templen;
    L = 1;
   
    // H is the height of subtree just before
    // subtree in which N lies
    H = 0;
   
    // Base Cases
    if (n == 1) {
        return 1;
    }
    if (n == 2) {
        return 2;
    }
    if (n == 3) {
        return 3;
    }
   
    // Size of subtree are power of 2
    while (L * 2 <= n) {
        L *= 2;
        H++;
    }
   
    // 3 Cases as explained in Approach
    if (n >= L * 2 - 1)
        return 2 * H + 1;
    else if (n >= L + (L / 2) - 1)
        return 2 * H;
    return 2 * H - 1;
}
   
// Driver Code
public static void main(String []args) 
{
    int n = 15;
  
    System.out.println(diameter(n));
}
}
  
// This code contributed by PrinciRaj1992

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C#

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// C# implementation of the approach
using System;
  
class GFG 
{
  
// Function to find diameter
// of BIT with N + 1 nodes
static int diameter(int n)
{
    // L is size of subtree just before subtree
    // in which N lies
    int L, H;
    L = 1;
  
    // H is the height of subtree just before
    // subtree in which N lies
    H = 0;
  
    // Base Cases
    if (n == 1)
    {
        return 1;
    }
    if (n == 2) 
    {
        return 2;
    }
    if (n == 3) 
    {
        return 3;
    }
  
    // Size of subtree are power of 2
    while (L * 2 <= n) 
    {
        L *= 2;
        H++;
    }
  
    // 3 Cases as explained in Approach
    if (n >= L * 2 - 1)
        return 2 * H + 1;
    else if (n >= L + (L / 2) - 1)
        return 2 * H;
    return 2 * H - 1;
}
  
// Driver Code
public static void Main(String []args) 
{
    int n = 15;
  
    Console.WriteLine(diameter(n));
}
}
  
// This code is contributed by 29AjayKumar

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Output:

7


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