Minimum value of distance of farthest node in a Graph

Last Updated : 09 Aug, 2021

Given an acyclic undirected graph having N nodes and N-1 edges in the form of a 2D array arr[][] in which every row consists of two numbers L and R which denotes the edge between L and R. For every node X in the tree, let dis(X) denotes the number of edges from X to the farthest node. The task is to find the minimum value of dis(x) for the given graph.

Examples:

Input: N = 6, arr[][] = { {1, 4}, {2, 3}, {3, 4}, {4, 5}, {5, 6} }
Output: 2
Explanation:
Below is the graph from the above information:

As we can see from the above graph the farthest node from vertex 0 is at distance 3. By repeating the DFS traversal for all the node in the graph, we have maximum distance[] from source node to farthest node as:
distance[] = {3, 4, 3, 2, 3, 4} and the minimum of the distances is the required result.

Input: N = 6, arr[][] = { {1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6} }
Output: 2
Explanation:
The distance[] from every node to farthest node for the above graph is:
distance[] = {3, 4, 3, 2, 3, 4} and the minimum of the distances is 1.

Approach:
The idea is to use DFS Traversal to solve this problem. Below are the steps:

For any Node(say a) traverse the graph using DFS Traversal with the distance of node with itself as 0.
For every recursive call for Node a, keep updating the distance of the recursive node with the node a in an array(say distance[]).
By taking the maximum of the distance with every recursive call for Node a give the number of edges between the nodes a and it's farthest node.
Repeat the above steps for all the node in the graph and keep updating the farthest node distance from every node in the distance array(distance[]).
The minimum value of the array distance[] is the desired result.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Distance vector to find the distance of
// a node to it's farthest node
vector<int> dist;

// To keep the track of visited array
// while DFS Traversal
vector<int> vis;

// Function for DFS traversal to update
// the distance vector
void dfs(int u, vector<int> Adj[], int s)
{
    // Mark the visited array for vertex u
    vis[u] = true;

    // Traverse the adjacency list for u
    for (auto& it : Adj[u]) {

        // If the any node is not visited,
        // then recursively call for next
        // vertex with distance increment
        // by 1
        if (vis[it] == false) {
            dfs(it, Adj, s + 1);
        }
    }

    // Update the maximum distance for the
    // farthest vertex from node u
    dist[u] = max(dist[u], s);
}

// Function to find the minimum of the
// farthest vertex for every vertex in
// the graph
void minFarthestDistance(int arr[][2], int n)
{

    // Resize distance vector
    dist.resize(n + 1, 0);

    // To create adjacency list for graph
    vector<int> Adj[n + 1];

    // Create Adjacency list for every
    // edge given in arr[][]
    for (int i = 0; i < n - 1; i++) {
        Adj[arr[i][0]].push_back(arr[i][1]);
        Adj[arr[i][1]].push_back(arr[i][0]);
    }

    // DFS Traversal for every node in the
    // graph to update the distance vector
    for (int i = 1; i <= n; i++) {

        // Clear and resize vis[] before
        // DFS traversal for every vertex
        vis.clear();
        vis.resize(n + 1, false);

        // DFS Traversal for vertex i
        dfs(i, Adj, 0);
    }

    cout << *min_element(dist.begin() + 1,
                         dist.end());
}

// Driver Code
int main()
{
    // Number of Nodes
    int N = 6;
    int arr[][2] = { { 1, 4 }, { 2, 3 }, { 3, 4 }, 
                     { 4, 5 }, { 5, 6 } };

    minFarthestDistance(arr, N);
    return 0;
}

Java

// Java program for the above approach 
import java.util.*;

class GFG{

// Distance vector to find the distance
// of a node to it's farthest node
static int[] dist;

// To keep the track of visited array
// while DFS Traversal
static boolean[] vis;

// Function for DFS traversal to update
// the distance vector
static void dfs(int u, Vector<Integer>[] Adj, int s)
{
    
    // Mark the visited array for vertex u
    vis[u] = true;

    // Traverse the adjacency list for u
    for (int it : Adj[u])
    {
        
        // If the any node is not visited,
        // then recursively call for next
        // vertex with distance increment
        // by 1
        if (vis[it] == false)
        {
            dfs(it, Adj, s + 1);
        }
    }

    // Update the maximum distance for 
    // the farthest vertex from node u
    dist[u] = Math.max(dist[u], s);
}

// Function to find the minimum of the
// farthest vertex for every vertex in
// the graph
static void minFarthestDistance(int[][] arr, int n) 
{
    
    // Resize distance vector
    dist = new int[n + 1];
    Arrays.fill(dist, 0);

    // To create adjacency list for graph
    @SuppressWarnings("unchecked")
    Vector<Integer>[] Adj = new Vector[n + 1];

    for(int i = 0; i < n + 1; i++)
    {
        Adj[i] = new Vector<>();
    }

    // Create Adjacency list for every
    // edge given in arr[][]
    for(int i = 0; i < n - 1; i++) 
    {
        Adj[arr[i][0]].add(arr[i][1]);
        Adj[arr[i][1]].add(arr[i][0]);
    }

    // DFS Traversal for every node in the
    // graph to update the distance vector
    for(int i = 1; i <= n; i++) 
    {
        
        // Clear and resize vis[] before
        // DFS traversal for every vertex
        vis = new boolean[n + 1];
        Arrays.fill(vis, false);

        // DFS Traversal for vertex i
        dfs(i, Adj, 0);
    }

    int min = Integer.MAX_VALUE;
    for(int i = 1; i < dist.length; i++) 
    {
        if (dist[i] < min)
            min = dist[i];
    }
    System.out.println(min);
}

// Driver Code
public static void main(String[] args) 
{
    
    // Number of Nodes
    int N = 6;
    int[][] arr = { { 1, 4 }, { 2, 3 },
                    { 3, 4 }, { 4, 5 }, 
                    { 5, 6 } };

    minFarthestDistance(arr, N);
}
}

// This code is contributed by sanjeev2552

Python3

# Python3 program for the above approach

# Function for DFS traversal to update
# the distance vector
def dfs(u, s):
    
    global vis, Adj, dist

    # Mark the visited array for vertex u
    vis[u] = True

    # Traverse the adjacency list for u
    for it in Adj[u]:
        
        # If the any node is not visited,
        # then recursively call for next
        # vertex with distance increment
        # by 1
        if (vis[it] == False):
            dfs(it, s + 1)

    # Update the maximum distance for the
    # farthest vertex from node u
    dist[u] = max(dist[u], s)

# Function to find the minimum of the
# farthest vertex for every vertex in
# the graph
def minFarthestDistance(arr, n):
    
    global dist, vis, Adj
    
    # Create Adjacency list for every
    # edge given in arr[][]
    for i in range(n - 1):
        Adj[arr[i][0]].append(arr[i][1])
        Adj[arr[i][1]].append(arr[i][0])

    # DFS Traversal for every node in the
    # graph to update the distance vector
    for i in range(1, n + 1):
        
        # Clear and resize vis[] before
        # DFS traversal for every vertex
        # vis.clear()
        for j in range(n + 1):
            vis[j] = False
            
        # vis.resize(n + 1, false)

        # DFS Traversal for vertex i
        dfs(i, 0)

    print(min(dist[i] for i in range(1, n + 1)))

# Driver Code
if __name__ == '__main__':
    
    dist = [0 for i in range(1001)]
    vis = [False for i in range(1001)]
    Adj = [[] for i in range(1001)]

    # Number of Nodes
    N = 6
    arr = [ [ 1, 4 ], [ 2, 3 ], 
            [ 3, 4 ], [ 4, 5 ], [ 5, 6 ] ]

    minFarthestDistance(arr, N)

# This code is contributed by mohit kumar 29

// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG
{

  // Distance vector to find the distance
  // of a node to it's farthest node
  static int[] dist;

  // To keep the track of visited array
  // while DFS Traversal
  static bool[] vis;

  // Function for DFS traversal to update
  // the distance vector
  static void dfs(int u, List<List<int>> Adj, int s)
  {

    // Mark the visited array for vertex u
    vis[u] = true;

    // Traverse the adjacency list for u
    foreach(int it in Adj[u])
    {

      // If the any node is not visited,
      // then recursively call for next
      // vertex with distance increment
      // by 1
      if (vis[it] == false)
      {
        dfs(it, Adj, s + 1);
      }
    }

    // Update the maximum distance for 
    // the farthest vertex from node u
    dist[u] = Math.Max(dist[u], s);        
  }

  // Function to find the minimum of the
  // farthest vertex for every vertex in
  // the graph
  static void minFarthestDistance(int[,] arr, int n) 
  {

    // Resize distance vector
    dist = new int[n + 1];
    Array.Fill(dist, 0);

    // To create adjacency list for graph
    List<List<int>> Adj = new List<List<int>>();
    for(int i = 0; i < n + 1; i++)
    {
      Adj.Add(new List<int>());
    }

    // Create Adjacency list for every
    // edge given in arr[][]
    for(int i = 0; i < n - 1; i++) 
    {
      Adj[arr[i, 0]].Add(arr[i, 1]);
      Adj[arr[i, 1]].Add(arr[i, 0]);
    }

    // DFS Traversal for every node in the
    // graph to update the distance vector
    for(int i = 1; i <= n; i++) 
    {

      // Clear and resize vis[] before
      // DFS traversal for every vertex
      vis = new bool[n + 1];
      Array.Fill(vis, false);

      // DFS Traversal for vertex i
      dfs(i, Adj, 0);
    }
    int min = Int32.MaxValue;
    for(int i = 1; i < dist.Length; i++) 
    {
      if (dist[i] < min)
      {
        min = dist[i];
      }

    }
    Console.WriteLine(min);   
  }

  // Driver Code
  static public void Main ()
  {

    // Number of Nodes
    int N = 6;
    int[,] arr = { { 1, 4 }, { 2, 3 },{ 3, 4 }, 
                  { 4, 5 }, { 5, 6 } };
    minFarthestDistance(arr, N);
  }
}

// This code is contributed by rag2127

JavaScript

<script>
// Javascript program for the above approach

// Distance vector to find the distance
// of a node to it's farthest node
let dist=[];

// To keep the track of visited array
// while DFS Traversal
let vis=[];

// Function for DFS traversal to update
// the distance vector
function dfs(u,Adj,s)
{
    // Mark the visited array for vertex u
    vis[u] = true;
 
    // Traverse the adjacency list for u
    for (let it=0;it<Adj[u].length;it++)
    {
         
        // If the any node is not visited,
        // then recursively call for next
        // vertex with distance increment
        // by 1
        if (vis[Adj[u][it]] == false)
        {
            dfs(Adj[u][it], Adj, s + 1);
        }
    }
 
    // Update the maximum distance for
    // the farthest vertex from node u
    dist[u] = Math.max(dist[u], s);
}

// Function to find the minimum of the
// farthest vertex for every vertex in
// the graph
function minFarthestDistance(arr,n)
{
    // Resize distance vector
    dist = new Array(n + 1);
    for(let i=0;i<(n+1);i++)
    {
        dist[i]=0;
    }
 
    // To create adjacency list for graph
    
    let Adj = new Array(n + 1);
 
    for(let i = 0; i < n + 1; i++)
    {
        Adj[i] = [];
    }
 
    // Create Adjacency list for every
    // edge given in arr[][]
    for(let i = 0; i < n - 1; i++)
    {
        Adj[arr[i][0]].push(arr[i][1]);
        Adj[arr[i][1]].push(arr[i][0]);
    }
 
    // DFS Traversal for every node in the
    // graph to update the distance vector
    for(let i = 1; i <= n; i++)
    {
         
        // Clear and resize vis[] before
        // DFS traversal for every vertex
        vis = new Array(n + 1);
        for(let i=0;i<(n+1);i++)
        {
            vis[i]=false;
        }
 
        // DFS Traversal for vertex i
        dfs(i, Adj, 0);
    }
 
    let min = Number.MAX_VALUE;
    for(let i = 1; i < dist.length; i++)
    {
        if (dist[i] < min)
            min = dist[i];
    }
    document.write(min);
}

// Driver Code
// Number of Nodes
let N = 6;
let arr=[[ 1, 4 ], [ 2, 3 ],
                    [ 3, 4 ], [ 4, 5 ],
                    [ 5, 6 ]];
minFarthestDistance(arr, N);



// This code is contributed by patel2127
</script>

Output:

Time Complexity: O(V*(V+E)), where V is the number of vertices and E is the number of edges.
Auxiliary Space: O(V + E)

Minimum value of distance of farthest node in a Graph

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