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Longest path between any pair of vertices

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  • Difficulty Level : Hard
  • Last Updated : 30 Sep, 2021

We are given a map of cities connected with each other via cable lines such that there is no cycle between any two cities. We need to find the maximum length of cable between any two cities for given city map. 
 

Input : n = 6  
        1 2 3  // Cable length from 1 to 2 (or 2 to 1) is 3
        2 3 4
        2 6 2
        6 4 6
        6 5 5
Output: maximum length of cable = 12

Method 1 (Simple DFS) 
We create undirected graph for given city map and do DFS from every city to find maximum length of cable. While traversing, we look for total cable length to reach the current city and if it’s adjacent city is not visited then call DFS for it but if all adjacent cities are visited for current node, then update the value of max_length if previous value of max_length is less than current value of total cable length. 
 

C++




// C++ program to find the longest cable length
// between any two cities.
#include<bits/stdc++.h>
using namespace std;
 
// visited[] array to make nodes visited
// src is starting node for DFS traversal
// prev_len is sum of cable length till current node
// max_len is pointer which stores the maximum length
// of cable value after DFS traversal
void DFS(vector< pair<int,int> > graph[], int src,
         int prev_len, int *max_len,
         vector <bool> &visited)
{
    // Mark the src node visited
    visited[src] = 1;
 
    // curr_len is for length of cable from src
    // city to its adjacent city
    int curr_len = 0;
 
    // Adjacent is pair type which stores
    // destination city and cable length
    pair < int, int > adjacent;
 
    // Traverse all adjacent
    for (int i=0; i<graph[src].size(); i++)
    {
        // Adjacent element
        adjacent = graph[src][i];
 
        // If node or city is not visited
        if (!visited[adjacent.first])
        {
            // Total length of cable from src city
            // to its adjacent
            curr_len = prev_len + adjacent.second;
 
            // Call DFS for adjacent city
            DFS(graph, adjacent.first, curr_len,
                max_len,  visited);
        }
 
        // If total cable length till now greater than
        // previous length then update it
        if ((*max_len) < curr_len)
            *max_len = curr_len;
 
        // make curr_len = 0 for next adjacent
        curr_len = 0;
    }
}
 
// n is number of cities or nodes in graph
// cable_lines is total cable_lines among the cities
// or edges in graph
int longestCable(vector<pair<int,int> > graph[],
                                          int n)
{
    // maximum length of cable among the connected
    // cities
    int max_len = INT_MIN;
 
    // call DFS for each city to find maximum
    // length of cable
    for (int i=1; i<=n; i++)
    {
        // initialize visited array with 0
        vector< bool > visited(n+1, false);
 
        // Call DFS for src vertex i
        DFS(graph, i, 0, &max_len, visited);
    }
 
    return max_len;
}
 
// driver program to test the input
int main()
{
    // n is number of cities
    int n = 6;
 
    vector< pair<int,int> > graph[n+1];
 
    // create undirected graph
    // first edge
    graph[1].push_back(make_pair(2, 3));
    graph[2].push_back(make_pair(1, 3));
 
    // second edge
    graph[2].push_back(make_pair(3, 4));
    graph[3].push_back(make_pair(2, 4));
 
    // third edge
    graph[2].push_back(make_pair(6, 2));
    graph[6].push_back(make_pair(2, 2));
 
    // fourth edge
    graph[4].push_back(make_pair(6, 6));
    graph[6].push_back(make_pair(4, 6));
 
    // fifth edge
    graph[5].push_back(make_pair(6, 5));
    graph[6].push_back(make_pair(5, 5));
 
    cout << "Maximum length of cable = "
         << longestCable(graph, n);
 
    return 0;
}


Java




// Java program to find the longest cable length
// between any two cities.
import java.util.*;
public class GFG
{
   
    // visited[] array to make nodes visited
    // src is starting node for DFS traversal
    // prev_len is sum of cable length till current node
    // max_len is pointer which stores the maximum length
    // of cable value after DFS traversal
     
    // Class containing left and
    // right child of current
    // node and key value
    static class pair {
         
        public int x, y;
        public pair(int f, int s)
        {
            x = f;
            y = s;
        }
    }
       
    // maximum length of cable among the connected
    // cities
    static int max_len = Integer.MIN_VALUE;
      
    static void DFS(Vector<Vector<pair>> graph, int src,
                    int prev_len, boolean[] visited)
    {
       
        // Mark the src node visited
        visited[src] = true;
   
        // curr_len is for length of cable
        // from src city to its adjacent city
        int curr_len = 0;
   
        // Adjacent is pair type which stores
        // destination city and cable length
        pair adjacent;
   
        // Traverse all adjacent
        for(int i = 0; i < graph.get(src).size(); i++)
        {
            // Adjacent element
            adjacent = graph.get(src).get(i);
   
            // If node or city is not visited
            if (!visited[adjacent.x])
            {
                // Total length of cable from
                // src city to its adjacent
                curr_len = prev_len + adjacent.y;
   
                // Call DFS for adjacent city
                DFS(graph, adjacent.x, curr_len, visited);
            }
   
            // If total cable length till
            // now greater than previous
            // length then update it
            if (max_len < curr_len)
            {
                max_len = curr_len;
            }
   
            // make curr_len = 0 for next adjacent
            curr_len = 0;
        }
    }
       
    // n is number of cities or nodes in graph
    // cable_lines is total cable_lines among the cities
    // or edges in graph
    static int longestCable(Vector<Vector<pair>> graph, int n)
    {
        // call DFS for each city to find maximum
        // length of cable
        for (int i=1; i<=n; i++)
        {
            // initialize visited array with 0
            boolean[] visited = new boolean[n+1];
   
            // Call DFS for src vertex i
            DFS(graph, i, 0, visited);
        }
   
        return max_len;
    }
     
    public static void main(String[] args) {
        // n is number of cities
        int n = 6;
       
        Vector<Vector<pair>> graph = new Vector<Vector<pair>>();
        for(int i = 0; i < n + 1; i++)
        {
            graph.add(new Vector<pair>());
        }
       
        // create undirected graph
        // first edge
        graph.get(1).add(new pair(2, 3));
        graph.get(2).add(new pair(1, 3));
       
        // second edge
        graph.get(2).add(new pair(3, 4));
        graph.get(3).add(new pair(2, 4));
       
        // third edge
        graph.get(2).add(new pair(6, 2));
        graph.get(6).add(new pair(2, 2));
       
        // fourth edge
        graph.get(4).add(new pair(6, 6));
        graph.get(6).add(new pair(4, 6));
       
        // fifth edge
        graph.get(5).add(new pair(6, 5));
        graph.get(6).add(new pair(5, 5));
       
        System.out.print("Maximum length of cable = "
             + longestCable(graph, n));
    }
}
 
// This code is contributed by suresh07.


Python3




# Python3 program to find the longest
# cable length between any two cities.
 
# visited[] array to make nodes visited
# src is starting node for DFS traversal
# prev_len is sum of cable length till
# current node max_len is pointer which
# stores the maximum length of cable
# value after DFS traversal
def DFS(graph, src, prev_len,
        max_len, visited):
     
    # Mark the src node visited
    visited[src] = 1
 
    # curr_len is for length of cable
    # from src city to its adjacent city
    curr_len = 0
 
    # Adjacent is pair type which stores
    # destination city and cable length
    adjacent = None
 
    # Traverse all adjacent
    for i in range(len(graph[src])):
         
        # Adjacent element
        adjacent = graph[src][i]
 
        # If node or city is not visited
        if (not visited[adjacent[0]]):
             
            # Total length of cable from
            # src city to its adjacent
            curr_len = prev_len + adjacent[1]
 
            # Call DFS for adjacent city
            DFS(graph, adjacent[0], curr_len,
                            max_len, visited)
 
        # If total cable length till
        # now greater than previous
        # length then update it
        if (max_len[0] < curr_len):
            max_len[0] = curr_len
 
        # make curr_len = 0 for next adjacent
        curr_len = 0
 
# n is number of cities or nodes in
# graph cable_lines is total cable_lines 
# among the cities or edges in graph
def longestCable(graph, n):
     
    # maximum length of cable among
    # the connected cities
    max_len = [-999999999999]
 
    # call DFS for each city to find
    # maximum length of cable
    for i in range(1, n + 1):
         
        # initialize visited array with 0
        visited = [False] * (n + 1)
 
        # Call DFS for src vertex i
        DFS(graph, i, 0, max_len, visited)
 
    return max_len[0]
 
# Driver Code
if __name__ == '__main__':
 
    # n is number of cities
    n = 6
 
    graph = [[] for i in range(n + 1)]
 
    # create undirected graph
    # first edge
    graph[1].append([2, 3])
    graph[2].append([1, 3])
 
    # second edge
    graph[2].append([3, 4])
    graph[3].append([2, 4])
 
    # third edge
    graph[2].append([6, 2])
    graph[6].append([2, 2])
 
    # fourth edge
    graph[4].append([6, 6])
    graph[6].append([4, 6])
 
    # fifth edge
    graph[5].append([6, 5])
    graph[6].append([5, 5])
 
    print("Maximum length of cable =",
               longestCable(graph, n))
                 
# This code is contributed by PranchalK


C#




// C# program to find the longest cable length
// between any two cities.
using System;
using System.Collections.Generic;
class GFG {
     
    // visited[] array to make nodes visited
    // src is starting node for DFS traversal
    // prev_len is sum of cable length till current node
    // max_len is pointer which stores the maximum length
    // of cable value after DFS traversal
      
    // maximum length of cable among the connected
    // cities
    static int max_len = Int32.MinValue;
     
    static void DFS(List<List<Tuple<int,int>>> graph, int src, int prev_len, bool[] visited)
    {
        // Mark the src node visited
        visited[src] = true;
  
        // curr_len is for length of cable
        // from src city to its adjacent city
        int curr_len = 0;
  
        // Adjacent is pair type which stores
        // destination city and cable length
        Tuple<int,int> adjacent;
  
        // Traverse all adjacent
        for(int i = 0; i < graph[src].Count; i++)
        {
            // Adjacent element
            adjacent = graph[src][i];
  
            // If node or city is not visited
            if (!visited[adjacent.Item1])
            {
                // Total length of cable from
                // src city to its adjacent
                curr_len = prev_len + adjacent.Item2;
  
                // Call DFS for adjacent city
                DFS(graph, adjacent.Item1, curr_len, visited);
            }
  
            // If total cable length till
            // now greater than previous
            // length then update it
            if (max_len < curr_len)
            {
                max_len = curr_len;
            }
  
            // make curr_len = 0 for next adjacent
            curr_len = 0;
        }
    }
      
    // n is number of cities or nodes in graph
    // cable_lines is total cable_lines among the cities
    // or edges in graph
    static int longestCable(List<List<Tuple<int,int>>> graph, int n)
    {
        // call DFS for each city to find maximum
        // length of cable
        for (int i=1; i<=n; i++)
        {
            // initialize visited array with 0
            bool[] visited = new bool[n+1];
  
            // Call DFS for src vertex i
            DFS(graph, i, 0, visited);
        }
  
        return max_len;
    }
     
  static void Main() {
    // n is number of cities
    int n = 6;
  
    List<List<Tuple<int,int>>> graph = new List<List<Tuple<int,int>>>();
    for(int i = 0; i < n + 1; i++)
    {
        graph.Add(new List<Tuple<int,int>>());
    }
  
    // create undirected graph
    // first edge
    graph[1].Add(new Tuple<int,int>(2, 3));
    graph[2].Add(new Tuple<int,int>(1, 3));
  
    // second edge
    graph[2].Add(new Tuple<int,int>(3, 4));
    graph[3].Add(new Tuple<int,int>(2, 4));
  
    // third edge
    graph[2].Add(new Tuple<int,int>(6, 2));
    graph[6].Add(new Tuple<int,int>(2, 2));
  
    // fourth edge
    graph[4].Add(new Tuple<int,int>(6, 6));
    graph[6].Add(new Tuple<int,int>(4, 6));
  
    // fifth edge
    graph[5].Add(new Tuple<int,int>(6, 5));
    graph[6].Add(new Tuple<int,int>(5, 5));
  
    Console.Write("Maximum length of cable = "
         + longestCable(graph, n));
  }
}
 
// This code is contributed by decode2207.


Javascript




<script>
    // Javascript program to find the longest cable length
    // between any two cities.
     
    // visited[] array to make nodes visited
    // src is starting node for DFS traversal
    // prev_len is sum of cable length till current node
    // max_len is pointer which stores the maximum length
    // of cable value after DFS traversal
     
    // maximum length of cable among the connected
    // cities
    let max_len = Number.MIN_VALUE;
         
    function DFS(graph, src, prev_len, visited)
    {
        // Mark the src node visited
        visited[src] = true;
 
        // curr_len is for length of cable
        // from src city to its adjacent city
        let curr_len = 0;
 
        // Adjacent is pair type which stores
        // destination city and cable length
        let adjacent = [];
 
        // Traverse all adjacent
        for(let i = 0; i < graph[src].length; i++)
        {
            // Adjacent element
            adjacent = graph[src][i];
 
            // If node or city is not visited
            if (!visited[adjacent[0]])
            {
                // Total length of cable from
                // src city to its adjacent
                curr_len = prev_len + adjacent[1];
 
                // Call DFS for adjacent city
                DFS(graph, adjacent[0], curr_len, visited);
            }
 
            // If total cable length till
            // now greater than previous
            // length then update it
            if (max_len < curr_len)
            {
                max_len = curr_len;
            }
 
            // make curr_len = 0 for next adjacent
            curr_len = 0;
        }
    }
     
    // n is number of cities or nodes in graph
    // cable_lines is total cable_lines among the cities
    // or edges in graph
    function longestCable(graph, n)
    {
        // call DFS for each city to find maximum
        // length of cable
        for (let i=1; i<=n; i++)
        {
            // initialize visited array with 0
            let visited = new Array(n+1);
            visited.fill(false);
 
            // Call DFS for src vertex i
            DFS(graph, i, 0, visited);
        }
 
        return max_len;
    }
     
    // n is number of cities
    let n = 6;
  
    let graph = [];
    for(let i = 0; i < n + 1; i++)
    {
        graph.push([]);
    }
  
    // create undirected graph
    // first edge
    graph[1].push([2, 3]);
    graph[2].push([1, 3]);
  
    // second edge
    graph[2].push([3, 4]);
    graph[3].push([2, 4]);
  
    // third edge
    graph[2].push([6, 2]);
    graph[6].push([2, 2]);
  
    // fourth edge
    graph[4].push([6, 6]);
    graph[6].push([4, 6]);
  
    // fifth edge
    graph[5].push([6, 5]);
    graph[6].push([5, 5]);
  
    document.write("Maximum length of cable = " +
               longestCable(graph, n));
     
    // This code is contributed by divyesh072019.
</script>


Output: 
 

Maximum length of cable = 12

Time Complexity : O(V * (V + E))
  
Method 2 (Efficient : Works only if Graph is Directed) 
We can solve above problem in O(V+E) time if the given graph is directed instead of undirected graph. Below are steps.
1) Create a distance array dist[] and initialize all entries of it as minus infinite 
2) Order all vertices in topological order
3) Do following for every vertex u in topological order. 
…..Do following for every adjacent vertex v of u 
………..if (dist[v] < dist[u] + weight(u, v)) 
……………dist[v] = dist[u] + weight(u, v) 
4) Return maximum value from dist[] 
 
Since there is no negative weight, processing vertices in topological order would always produce an array of longest paths dist[] such that dist[u] indicates longest path ending at vertex ‘u’.
The implementation of above approach can be easily adopted from here. The differences here are, there are no negative weight edges and we need overall longest path (not longest paths from a source vertex). Finally we return maximum of all values in dist[].
Time Complexity : O(V + E)
This article is contributed by Shashank Mishra ( Gullu ). This article is reviewed by team GeeksForGeeks. If you have any better approach for this problem then please share.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 


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