Longest path in an undirected tree

Given an undirected tree, we need to find the longest path of this tree where a path is defined as a sequence of nodes. 

Example:

Input : Below shown Tree using adjacency list 
        representation:
Output : 5
In below tree longest path is of length 5
from node 5 to node 7

This problem is the same as the diameter of the n-ary tree. We have discussed a simple solution here. 

In this post, an efficient solution is discussed. We can find the longest path using two BFSs. The idea is based on the following fact: If we start BFS from any node x and find a node with the longest distance from x, it must be an endpoint of the longest path. It can be proved using contradiction. So our algorithm reduces to simple two BFSs. First BFS to find an endpoint of the longest path and second BFS from this endpoint to find the actual longest path. 

For the proof of why does this algorithm works, there is a nice explanation here Proof of correctness: Algorithm for the diameter of a tree in graph theory

As we can see in the above diagram, if we start our BFS from node-0, the node at the farthest distance from it will be node-5, now if we start our BFS from node-5 the node at the farthest distance will be node-7, finally, the path from node-5 to node-7 will constitute our longest path.

Implementation:

C++

// C++ program to find longest path of the tree
#include <bits/stdc++.h>
using namespace std;

// This class represents a undirected graph using adjacency list
class Graph
{
    int V;              // No. of vertices
    list<int> *adj;     // Pointer to an array containing
                        // adjacency lists
public:
    Graph(int V);              // Constructor
    void addEdge(int v, int w);// function to add an edge to graph
    void longestPathLength();  // prints longest path of the tree
    pair<int, int> bfs(int u); // function returns maximum distant
                               // node from u with its distance
};

Graph::Graph(int V)
{
    this->V = V;
    adj = new list<int>[V];
}

void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w);    // Add w to v’s list.
    adj[w].push_back(v);    // Since the graph is undirected
}

//  method returns farthest node and its distance from node u
pair<int, int> Graph::bfs(int u)
{
    //  mark all distance with -1
    int dis[V];
    memset(dis, -1, sizeof(dis));

    queue<int> q;
    q.push(u);

    //  distance of u from u will be 0
    dis[u] = 0;

    while (!q.empty())
    {
        int t = q.front();       q.pop();

        //  loop for all adjacent nodes of node-t
        for (auto it = adj[t].begin(); it != adj[t].end(); it++)
        {
            int v = *it;

            // push node into queue only if
            // it is not visited already
            if (dis[v] == -1)
            {
                q.push(v);

                // make distance of v, one more
                // than distance of t
                dis[v] = dis[t] + 1;
            }
        }
    }

    int maxDis = 0;
    int nodeIdx;

    //  get farthest node distance and its index
    for (int i = 0; i < V; i++)
    {
        if (dis[i] > maxDis)
        {
            maxDis = dis[i];
            nodeIdx = i;
        }
    }
    return make_pair(nodeIdx, maxDis);
}

//  method prints longest path of given tree
void Graph::longestPathLength()
{
    pair<int, int> t1, t2;

    // first bfs to find one end point of
    // longest path
    t1 = bfs(0);

    //  second bfs to find actual longest path
    t2 = bfs(t1.first);

    cout << "Longest path is from " << t1.first << " to "
         << t2.first << " of length " << t2.second;
}

// Driver code to test above methods
int main()
{
    // Create a graph given in the example
    Graph g(10);
    g.addEdge(0, 1);
    g.addEdge(1, 2);
    g.addEdge(2, 3);
    g.addEdge(2, 9);
    g.addEdge(2, 4);
    g.addEdge(4, 5);
    g.addEdge(1, 6);
    g.addEdge(6, 7);
    g.addEdge(6, 8);

    g.longestPathLength();
    return 0;
}

Java

// Java program to find longest path of the tree

import java.util.Arrays;
import java.util.LinkedList;
import java.util.Queue;


class LongestPathUndirectedTree {

    // Utility Pair class for storing maximum distance
    // Node with its distance
    static class Pair<T,V> {
        T first; // maximum distance Node
        V second; // distance of maximum distance node

        //Constructor
        Pair(T first, V second) {
            this.first = first;
            this.second = second;
        }
    }

    // This class represents a undirected graph using adjacency list
    static class Graph {
        int V; // No. of vertices
        LinkedList<Integer>[] adj; //Adjacency List

        // Constructor
        Graph(int V) {
            this.V = V;
            // Initializing Adjacency List
            adj = new LinkedList[V];
            for(int i = 0; i < V; ++i) {
                adj[i] = new LinkedList<Integer>();
            }
        }

        // function to add an edge to graph
        void addEdge(int s, int d) {
            adj[s].add(d); // Add d to s's list.
            adj[d].add(s); // Since the graph is undirected
        }


        // method returns farthest node and its distance from node u
        Pair<Integer, Integer> bfs(int u) {
            int[] dis = new int[V];

            // mark all distance with -1
            Arrays.fill(dis, -1);

            Queue<Integer> q = new LinkedList<>();

            q.add(u);

            // distance of u from u will be 0
            dis[u] = 0;
            while (!q.isEmpty()) {
                int t = q.poll();

                // loop for all adjacent nodes of node-t
                for(int i = 0; i < adj[t].size(); ++i) {
                    int v = adj[t].get(i);

                    // push node into queue only if
                    // it is not visited already
                    if(dis[v] == -1) {
                        q.add(v);
                        // make distance of v, one more
                        // than distance of t
                        dis[v] = dis[t] + 1;
                    }
                }
            }

            int maxDis = 0;
            int nodeIdx = 0;

            // get farthest node distance and its index
            for(int i = 0; i < V; ++i) {
                if(dis[i] > maxDis) {
                    maxDis = dis[i];
                    nodeIdx = i;
                }
            }

            return new Pair<Integer, Integer>(nodeIdx, maxDis);
        }

        // method prints longest path of given tree
        void longestPathLength() {
            Pair<Integer, Integer> t1, t2;

            // first bfs to find one end point of
            // longest path
            t1 = bfs(0);

            // second bfs to find actual longest path
            t2 = bfs(t1.first);

            System.out.println("Longest path is from "+ t1.first
            + " to "+ t2.first +" of length "+t2.second);
        }
    }

    // Driver code to test above methods
    public static void main(String[] args){
        // Create a graph given in the example

        Graph graph = new Graph(10);
        graph.addEdge(0, 1);
        graph.addEdge(1, 2);
        graph.addEdge(2, 3);
        graph.addEdge(2, 9);
        graph.addEdge(2, 4);
        graph.addEdge(4, 5);
        graph.addEdge(1, 6);
        graph.addEdge(6, 7);
        graph.addEdge(6, 8);

        graph.longestPathLength();
    }

}
// Added By Brij Raj Kishore

C#

// C# program to find longest path of the tree
using System;
using System.Collections.Generic;

class GFG
{

// Utility Pair class for storing
// maximum distance Node with its distance
public class Pair<T, V>
{
    // maximum distance Node
    public T first;

    // distance of maximum distance node
    public V second;

    // Constructor
    public Pair(T first, V second)
    {
        this.first = first;
        this.second = second;
    }
}

// This class represents a undirected graph
// using adjacency list
class Graph
{
    int V; // No. of vertices
    List<int>[] adj; //Adjacency List

    // Constructor
    public Graph(int V)
    {
        this.V = V;

        // Initializing Adjacency List
        adj = new List<int>[V];
        for(int i = 0; i < V; ++i)
        {
            adj[i] = new List<int>();
        }
    }

    // function to add an edge to graph
    public void addEdge(int s, int d)
    {
        adj[s].Add(d); // Add d to s's list.
        adj[d].Add(s); // Since the graph is undirected
    }

    // method returns farthest node and
    // its distance from node u
    public Pair<int, int> bfs(int u)
    {
        int[] dis = new int[V];

        // mark all distance with -1
        for(int i = 0; i < V; i++)
            dis[i] = -1;

        Queue<int> q = new Queue<int>();

        q.Enqueue(u);

        // distance of u from u will be 0
        dis[u] = 0;
        while (q.Count != 0)
        {
            int t = q.Dequeue();

            // loop for all adjacent nodes of node-t
            for(int i = 0; i < adj[t].Count; ++i)
            {
                int v = adj[t][i];

                // push node into queue only if
                // it is not visited already
                if(dis[v] == -1)
                {
                    q.Enqueue(v);

                    // make distance of v, one more
                    // than distance of t
                    dis[v] = dis[t] + 1;
                }
            }
        }
        int maxDis = 0;
        int nodeIdx = 0;

        // get farthest node distance and its index
        for(int i = 0; i < V; ++i)
        {
            if(dis[i] > maxDis)
            {
                maxDis = dis[i];
                nodeIdx = i;
            }
        }
        return new Pair<int, int>(nodeIdx, maxDis);
    }

    // method prints longest path of given tree
    public void longestPathLength()
    {
        Pair<int, int> t1, t2;

        // first bfs to find one end point of
        // longest path
        t1 = bfs(0);

        // second bfs to find actual longest path
        t2 = bfs(t1.first);

        Console.WriteLine("longest path is from " + t1.first +
                " to " + t2.first + " of length " + t2.second);
    }
}

// Driver Code
public static void Main(String[] args)
{
    // Create a graph given in the example
    Graph graph = new Graph(10);
    graph.addEdge(0, 1);
    graph.addEdge(1, 2);
    graph.addEdge(2, 3);
    graph.addEdge(2, 9);
    graph.addEdge(2, 4);
    graph.addEdge(4, 5);
    graph.addEdge(1, 6);
    graph.addEdge(6, 7);
    graph.addEdge(6, 8);

    graph.longestPathLength();
}
}

// This code is contributed by Rajput-Ji

Python3

# Python program to find the Longest Path of the Tree
# By Aaditya Upadhyay

from collections import deque


class Graph:

    # Initialisation of graph
    def __init__(self, vertices):

        # No. of vertices
        self.vertices = vertices

        # adjacency list
        self.adj = {i: [] for i in range(self.vertices)}

    def addEdge(self, u, v):
        # add u to v's list
        self.adj[u].append(v)
        # since the graph is undirected
        self.adj[v].append(u)

    # method return farthest node and its distance from node u
    def BFS(self, u):
        # marking all nodes as unvisited
        visited = [False for i in range(self.vertices + 1)]
        # mark all distance with -1
        distance = [-1 for i in range(self.vertices + 1)]

        # distance of u from u will be 0
        distance[u] = 0
        # in-built library for queue which performs fast operations on both the ends
        queue = deque()
        queue.append(u)
        # mark node u as visited
        visited[u] = True

        while queue:

            # pop the front of the queue(0th element)
            front = queue.popleft()
            # loop for all adjacent nodes of node front

            for i in self.adj[front]:
                if not visited[i]:
                    # mark the ith node as visited
                    visited[i] = True
                    # make distance of i , one more than distance of front
                    distance[i] = distance[front]+1
                    # Push node into the stack only if it is not visited already
                    queue.append(i)

        maxDis = 0

        # get farthest node distance and its index
        for i in range(self.vertices):
            if distance[i] > maxDis:

                maxDis = distance[i]
                nodeIdx = i

        return nodeIdx, maxDis

    # method prints longest path of given tree
    def LongestPathLength(self):

        # first DFS to find one end point of longest path
        node, Dis = self.BFS(0)

        # second DFS to find the actual longest path
        node_2, LongDis = self.BFS(node)

        print('Longest path is from', node, 'to', node_2, 'of length', LongDis)


# create a graph given in the example

G = Graph(10)
G.addEdge(0, 1)
G.addEdge(1, 2)
G.addEdge(2, 3)
G.addEdge(2, 9)
G.addEdge(2, 4)
G.addEdge(4, 5)
G.addEdge(1, 6)
G.addEdge(6, 7)
G.addEdge(6, 8)

G.LongestPathLength()

Javascript

<script>

// Javascript program to find longest path of the tree


// Utility Pair class for storing
// maximum distance Node with its distance
class Pair
{
    // Constructor
    constructor(first, second)
    {
        this.first = first;
        this.second = second;
    }
}

// This class represents a undirected graph
// using adjacency list
var V; // No. of vertices
var adj; //Adjacency List

// Constructor
function initialize(V)
{
    this.V = V;

    // Initializing Adjacency List
    adj = Array.from(Array(V), ()=>Array());
}

// function to add an edge to graph
function addEdge(s, d)
{
    adj[s].push(d); // push d to s's list.
    adj[d].push(s); // Since the graph is undirected
}

// method returns farthest node and
// its distance from node u
function bfs(u)
{
    var dis = Array(V);

    // mark all distance with -1
    for(var i = 0; i < V; i++)
        dis[i] = -1;
    var q = [];
    q.push(u);

    // distance of u from u will be 0
    dis[u] = 0;
    while (q.length != 0)
    {
        var t = q.shift();

        // loop for all adjacent nodes of node-t
        for(var i = 0; i < adj[t].length; ++i)
        {
            var v = adj[t][i];

            // push node into queue only if
            // it is not visited already
            if(dis[v] == -1)
            {
                q.push(v);

                // make distance of v, one more
                // than distance of t
                dis[v] = dis[t] + 1;
            }
        }
    }
    var maxDis = 0;
    var nodeIdx = 0;

    // get farthest node distance and its index
    for(var i = 0; i < V; ++i)
    {
        if(dis[i] > maxDis)
        {
            maxDis = dis[i];
            nodeIdx = i;
        }
    }
    return new Pair(nodeIdx, maxDis);
}

// method prints longest path of given tree
function longestPathLength()
{
    var t1, t2;

    // first bfs to find one end point of
    // longest path
    t1 = bfs(0);

    // second bfs to find actual longest path
    t2 = bfs(t1.first);
    document.write("longest path is from " + t1.first +
            " to " + t2.first + " of length " + t2.second);
}


// Create a graph given in the example
initialize(10)
addEdge(0, 1);
addEdge(1, 2);
addEdge(2, 3);
addEdge(2, 9);
addEdge(2, 4);
addEdge(4, 5);
addEdge(1, 6);
addEdge(6, 7);
addEdge(6, 8);
longestPathLength();

// This code is contributed by famously.
</script>

Output

Longest path is from 5 to 7 of length 5

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

This article is contributed by Utkarsh Trivedi. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.