Maximum cost path from source node to destination node via at most K intermediate nodes

Given a directed weighted graph consisting of N vertices and an array Edges[][], with each row representing two vertices connected by an edge and the weight of that edge, the task is to find the path with the maximum sum of weights from a given source vertex src to a given destination vertex dst, made up of at most K intermediate vertices. If no such path exists, then print -1.

Examples:

Input: N = 3, Edges[][] = {{0, 1, 100}, {1, 2, 100}, {0, 2, 500}}, src = 0, dst = 2, K = 0
Output: 500
Explanation:

Path 0 → 2: The path with maximum weight and at most 0 intermediate nodes is of weight 500.

Approach: The given problem can be solved by using BFS(Breadth-First Search) Traversal. Follow the steps below to solve the problem:

• Initialize the variable, say ans, to store the maximum distance between the source and the destination node having at most K intermediates nodes.
• Initialize an adjacency list of the graph using the edges.
• Initialize an empty queue and push the source vertex into it. Initialize a variable, say lvl, to store the number of nodes present in between src and dst.
• While the queue is not empty and lvl is less than K + 2 perform the following steps:
  • Store the size of the queue in a variable, say S.
  • Iterate over the range [1, S] and perform the following steps:
    • Pop the front element of the queue and store it in a variable, say T.
    • If T is the dst vertex, then update the value of ans as the maximum of ans and the current distance T.second.
    • Traverse through all the neighbors of the current popped node and check if the distance of its neighbor is greater than the current distance or not. If found to be true, then push it in the queue and update its distance.
  • Increase the value of lvl by 1.
• After completing the above steps, print the value of ans as the resultant maximum distance.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to find the longest distance
// from source to destination with at
// most K intermediate nodes
int findShortestPath(
    int n, vector<vector<int> >& edges,
    int src, int dst, int K)
{
    // Initialize the adjacency list
    vector<vector<pair<int, int> > > adjlist(
        n, vector<pair<int, int> >());

    // Initialize a queue to perform BFS
    queue<pair<int, int> > q;

    unordered_map<int, int> mp;

    // Store the maximum distance of
    // every node from source vertex
    int ans = INT_MIN;

    // Initialize adjacency list
    for (int i = 0; i < edges.size(); i++) {

        auto edge = edges[i];

        adjlist[edge[0]].push_back(
            make_pair(edge[1], edge[2]));
    }

    // Push the first element into queue
    q.push({ src, 0 });

    int level = 0;

    // Iterate until the queue becomes empty
    // and the number of nodes between src
    // and dst vertex is at most to K
    while (!q.empty() && level < K + 2) {

        // Current size of the queue
        int sz = q.size();

        for (int i = 0; i < sz; i++) {

            // Extract the front
            // element of the queue
            auto pr = q.front();

            // Pop the front element
            // of the queue
            q.pop();

            // If the dst vertex is reached
            if (pr.first == dst)
                ans = max(ans, pr.second);

            // Traverse the adjacent nodes
            for (auto pr2 : adjlist[pr.first]) {

                // If the distance is greater
                // than the current distance
                if (mp.find(pr2.first)
                        == mp.end()
                    || mp[pr2.first]
                           > pr.second
                                 + pr2.second) {

                    // Push it into the queue
                    q.push({ pr2.first,
                             pr.second
                                 + pr2.second });
                    mp[pr2.first] = pr.second
                                    + pr2.second;
                }
            }
        }

        // Increment the level by 1
        level++;
    }

    // Finally, return the maximum distance
    return ans != INT_MIN ? ans : -1;
}

// Driver Code
int main()
{
    int n = 3, src = 0, dst = 2, k = 1;
    vector<vector<int> > edges
        = { { 0, 1, 100 },
            { 1, 2, 100 },
            { 0, 2, 500 } };

    cout << findShortestPath(n, edges,
                             src, dst, k);

    return 0;
}

Java

import java.util.*;

class Main {

    // Function to find the longest distance
    // from source to destination with at
    // most K intermediate nodes
    public static int findShortestPath(int n, int[][] edges,
                                       int src, int dst,
                                       int K)
    {
        // Initialize the adjacency list
        List<List<int[]> > adjlist = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            adjlist.add(new ArrayList<int[]>());
        }

        // Initialize a queue to perform BFS
        Queue<int[]> q = new LinkedList<>();

        Map<Integer, Integer> mp = new HashMap<>();

        // Store the maximum distance of
        // every node from source vertex
        int ans = Integer.MIN_VALUE;

        // Initialize adjacency list
        for (int[] edge : edges) {
            adjlist.get(edge[0]).add(
                new int[] { edge[1], edge[2] });
        }

        // Push the first element into queue
        q.add(new int[] { src, 0 });

        int level = 0;

        // Iterate until the queue becomes empty
        // and the number of nodes between src
        // and dst vertex is at most to K
        while (!q.isEmpty() && level < K + 2) {

            // Current size of the queue
            int sz = q.size();

            for (int i = 0; i < sz; i++) {

                // Extract the front
                // element of the queue
                int[] pr = q.poll();

                // If the dst vertex is reached
                if (pr[0] == dst)
                    ans = Math.max(ans, pr[1]);

                // Traverse the adjacent nodes
                for (int[] pr2 : adjlist.get(pr[0])) {

                    // If the distance is greater
                    // than the current distance
                    if (!mp.containsKey(pr2[0])
                        || mp.get(pr2[0])
                               > pr[1] + pr2[1]) {

                        // Push it into the queue
                        q.add(new int[] { pr2[0],
                                          pr[1] + pr2[1] });
                        mp.put(pr2[0], pr[1] + pr2[1]);
                    }
                }
            }

            // Increment the level by 1
            level++;
        }

        // Finally, return the maximum distance
        return ans != Integer.MIN_VALUE ? ans : -1;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int n = 3, src = 0, dst = 2, k = 1;
        int[][] edges = { { 0, 1, 100 },
                          { 1, 2, 100 },
                          { 0, 2, 500 } };

        System.out.println(
            findShortestPath(n, edges, src, dst, k));
    }
}

Python3

# Python3 program for the above approach
from collections import deque

# Function to find the longest distance
# from source to destination with at
# most K intermediate nodes
def findShortestPath(n, edges, src, dst, K):

    # Initialize the adjacency list
    adjlist = [[] for i in range(n)]

    # Initialize a queue to perform BFS
    q = deque()

    mp = {}

    # Store the maximum distance of
    # every node from source vertex
    ans = -10**9

    # Initialize adjacency list
    for i in range(len(edges)):
        edge = edges[i]
        adjlist[edge[0]].append([edge[1],
                                 edge[2]])

    # Push the first element into queue
    q.append([src, 0])

    level = 0

    # Iterate until the queue becomes empty
    # and the number of nodes between src
    # and dst vertex is at most to K
    while (len(q) > 0 and level < K + 2):

        # Current size of the queue
        sz = len(q)

        for i in range(sz):

            # Extract the front
            # element of the queue
            pr = q.popleft()

            # Pop the front element
            # of the queue
            # q.pop()

            # If the dst vertex is reached
            if (pr[0] == dst):
                ans = max(ans, pr[1])

            # Traverse the adjacent nodes
            for pr2 in adjlist[pr[0]]:

                # If the distance is greater
                # than the current distance
                if ((pr2[0] not in mp) or
                  mp[pr2[0]] > pr[1] + pr2[1]):

                    # Push it into the queue
                    q.append([pr2[0], pr[1] + pr2[1]])
                    mp[pr2[0]] = pr[1] + pr2[1]

        # Increment the level by 1
        level += 1

    # Finally, return the maximum distance
    return ans if ans != -10**9 else -1

# Driver Code
if __name__ == '__main__':

    n, src, dst, k = 3, 0, 2, 1

    edges= [ [ 0, 1, 100 ],
             [ 1, 2, 100 ],
             [ 0, 2, 500 ] ]

    print(findShortestPath(n, edges,src, dst, k))

# This code is contributed by mohit kumar 29

Javascript

// JavaScript implementation of the above C++ code

function findShortestPath(n, edges, src, dst, k) {
    // Initialize the adjacency list
    var adjlist = new Array(n).fill(null).map(() => []);

    // Initialize a queue to perform BFS
    var q = [];

    var mp = new Map();

    // Store the maximum distance of
    // every node from source vertex
    var ans = Number.MIN_SAFE_INTEGER;

    // Initialize adjacency list
    for (var i = 0; i < edges.length; i++) {
        var edge = edges[i];
        adjlist[edge[0]].push([edge[1], edge[2]]);
    }

    // Push the first element into queue
    q.push([src, 0]);

    var level = 0;

    // Iterate until the queue becomes empty
    // and the number of nodes between src
    // and dst vertex is at most to K
    while (q.length > 0 && level < k + 2) {
        // Current size of the queue
        var sz = q.length;

        for (var i = 0; i < sz; i++) {
            // Extract the front
            // element of the queue
            var pr = q.shift();

            // If the dst vertex is reached
            if (pr[0] === dst) {
                ans = Math.max(ans, pr[1]);
            }

            // Traverse the adjacent nodes
            for (var j = 0; j < adjlist[pr[0]].length; j++) {
                var pr2 = adjlist[pr[0]][j];

                // If the distance is greater
                // than the current distance
                if (mp.get(pr2[0]) === undefined || mp.get(pr2[0]) > pr[1] + pr2[1]) {
                    // Push it into the queue
                    q.push([pr2[0], pr[1] + pr2[1]]);
                    mp.set(pr2[0], pr[1] + pr2[1]);
                }
            }
        }

        // Increment the level by 1
        level++;
    }

    // Finally, return the maximum distance
    return ans !== Number.MIN_SAFE_INTEGER ? ans : -1;
}

// Example usage
var n = 3, src = 0, dst = 2, k = 1;
var edges = [[0, 1, 100], [1, 2, 100], [0, 2, 500]];

console.log(findShortestPath(n, edges, src, dst, k));

C#

// C# program for the above approach

using System;
using System.Collections.Generic;

class GFG {
    // Function to find the longest distance
    // from source to destination with at
    // most K intermediate nodes
    static int FindShortestPath(int n, int[][] edges, int src, int dst, int K)
    {
        // Initialize the adjacency list
        List<int[]>[] adjlist = new List<int[]>[n];
        for (int i = 0; i < n; i++) {
            adjlist[i] = new List<int[]>();
        }

        // Initialize a queue to perform BFS
        Queue<int[]> q = new Queue<int[]>();

        Dictionary<int, int> mp = new Dictionary<int, int>();

        // Store the maximum distance of
        // every node from source vertex
        int ans = -1000000000;

        // Initialize adjacency list
        for (int i = 0; i < edges.Length; i++) {
            int[] edge = edges[i];
            adjlist[edge[0]].Add(new int[] {edge[1], edge[2]});
        }

        // Push the first element into queue
        q.Enqueue(new int[] {src, 0});

        int level = 0;

        // Iterate until the queue becomes empty
        // and the number of nodes between src
        // and dst vertex is at most to K
        while (q.Count > 0 && level < K + 2) {

            // Current size of the queue
            int sz = q.Count;

            for (int i = 0; i < sz; i++) {

                // Extract the front
                // element of the queue
                int[] pr = q.Dequeue();

                // If the dst vertex is reached
                if (pr[0] == dst) {
                    ans = Math.Max(ans, pr[1]);
                }

                // Traverse the adjacent nodes
                foreach (int[] pr2 in adjlist[pr[0]]) {

                    // If the distance is greater
                    // than the current distance
                    if (!mp.ContainsKey(pr2[0]) || mp[pr2[0]] > pr[1] + pr2[1]) {
                        // Push it into the queue
                        q.Enqueue(new int[] {pr2[0], pr[1] + pr2[1]});
                        mp[pr2[0]] = pr[1] + pr2[1];
                    }
                }
            }

            // Increment the level by 1
            level++;
        }

        // Finally, return the maximum distance
        return ans != -1000000000 ? ans : -1;
    }

    // Driver Code
    public static void Main()
    {
        int n = 3, src = 0, dst = 2, k = 1;

        int[][] edges = new int[][] {
            new int[] {0, 1, 100},
            new int[] {1, 2, 100},
            new int[] {0, 2, 500}
        };

        Console.WriteLine(FindShortestPath(n, edges, src, dst, k));
    }
}


// This code is contributed by codebraxnzt

Output

500

Time Complexity: O(N + E)
Auxiliary Space: O(N)

Alternate approach: Modification of Bellman Ford algorithm after modifying the weights

If all the weights of the given graph are made negative of the original weights, the path taken to minimize the sum of weights with at most k nodes in middle will give us the path we need. Hence this question is similar to this problem. Below is the code implementation of the problem

C++

#include <bits/stdc++.h>
using namespace std;

int max_cost(int n, vector<vector<int> >& edges, int src,
             int dst, int k)
{
    // We use 2 arrays for this algorithm
    // temp is the shortest distances array in current pass
    vector<int> temp(n, INT_MAX);
    temp[src] = 0;
    for (int i = 0; i <= k; i++) {
        // c is the shortest distances array in previous
        // pass For every iteration current pass becomes the
        // previous
        vector<int> c(temp);
        for (auto edge : edges) {
            int a = edge[0], b = edge[1], d = edge[2];
            // Updating the current array using previous
            // array Subtracting d is same as adding -d
            temp[b]
                = min(temp[b],
                      c[a] == INT_MAX ? INT_MAX : c[a] - d);
        }
    }
    // Checking is dst is reachable from src or not
    if (temp[dst] != INT_MAX) {
        // Returning the negative value of the shortest
        // distance to return the longest distance
        return -temp[dst];
    }
    return -1;
}

int main()
{
    vector<vector<int> > edges = {
        { 0, 1, 100 },
        { 1, 2, 100 },
        { 0, 2, 500 },
    };
    int src = 0;
    int dst = 2;
    int k = 1;
    int n = 3;

    cout << max_cost(n, edges, src, dst, k) << endl;
    return 0;
}
// This code was contributed Prajwal Kandekar

Java

import java.util.*;

public class Main {
    static int max_cost(int n, List<List<Integer>> edges, int src,
             int dst, int k)
    {

        // We use 2 arrays for this algorithm
        // temp is the shortest distances array in current pass
        int[] temp = new int[n];
        Arrays.fill(temp, Integer.MAX_VALUE);
        temp[src] = 0;
        for (int i = 0; i <= k; i++)
        {

            // c is the shortest distances array in previous
            // pass For every iteration current pass becomes the
            // previous
            int[] c = temp.clone();
            for (List<Integer> edge : edges) {
                int a = edge.get(0), b = edge.get(1), d = edge.get(2);

                // Updating the current array using previous
                // array Subtracting d is same as adding -d
                temp[b] = Math.min(temp[b], c[a] == Integer.MAX_VALUE ? Integer.MAX_VALUE : c[a] - d);
            }
        }

        // Checking if dst is reachable from src or not
        if (temp[dst] != Integer.MAX_VALUE)
        {

            // Returning the negative value of the shortest
            // distance to return the longest distance
            return -temp[dst];
        }
        return -1;
    }

    public static void main(String[] args) {
        List<List<Integer>> edges = Arrays.asList(
            Arrays.asList(0, 1, 100),
            Arrays.asList(1, 2, 100),
            Arrays.asList(0, 2, 500)
        );
        int src = 0;
        int dst = 2;
        int k = 1;
        int n = 3;

        System.out.println(max_cost(n, edges, src, dst, k));
    }
}

Python3

def max_cost(n, edges, src, dst, k):
      # We use 2 arrays for this algorithm
    # temp is the shortest distances array in current pass
    temp=[0 if i==src else float("inf") for i in range(n)]
    for _ in range(k+1):
          # c is the shortest distances array in previous pass
        # For every iteration current pass becomes the previous
        c=temp.copy()
        for a,b,d in edges:
              # Updating the current array using previous array
            # Subtracting d is same as adding -d
            temp[b]=min(temp[b],c[a]-d)
    # Checking is dst is reachable from src or not
    if temp[dst]!=float("inf"):
          # Returning the negative value of the shortest distance to return the longest distance
        return -temp[dst]
    return -1

edges = [
    [0, 1, 100],
    [1, 2, 100],
    [0, 2, 500],
]

src = 0
dst = 2
k = 1
n = 3

print(max_cost(n,edges,src,dst,k))

# This code was contributed by Akshayan Muralikrishnan

Javascript

function max_cost(n, edges, src, dst, k) {
    // We use 2 arrays for this algorithm
    // temp is the shortest distances array in current pass
    let temp = new Array(n).fill(Number.MAX_SAFE_INTEGER);
    temp[src] = 0;

    for (let i = 0; i <= k; i++) {
        // c is the shortest distances array in previous
        // pass For every iteration current pass becomes the
        // previous
        let c = [...temp];
        for (let j = 0; j < edges.length; j++) {
            let [a, b, d] = edges[j];
            // Updating the current array using previous
            // array Subtracting d is same as adding -d
            temp[b] = Math.min(temp[b], (c[a] === Number.MAX_SAFE_INTEGER) ? Number.MAX_SAFE_INTEGER : c[a] - d);
        }
    }

    // Checking is dst is reachable from src or not
    if (temp[dst] !== Number.MAX_SAFE_INTEGER) {
        // Returning the negative value of the shortest
        // distance to return the longest distance
        return -temp[dst];
    }
    return -1;
}

let edges = [    [0, 1, 100],
    [1, 2, 100],
    [0, 2, 500]
];
let src = 0;
let dst = 2;
let k = 1;
let n = 3;

console.log(max_cost(n, edges, src, dst, k));

C#

using System;
using System.Collections.Generic;
using System.Linq;

public class MainClass
{
    static int MaxCost(int n, List<List<int>> edges, int src, int dst, int k)
    {

        // We use 2 arrays for this algorithm
        // temp is the shortest distances array in current pass
        int[] temp = new int[n];
        Array.Fill(temp, int.MaxValue);
        temp[src] = 0;
        for (int i = 0; i <= k; i++)
        {

            // c is the shortest distances array in previous
            // pass For every iteration current pass becomes the
            // previous
            int[] c = (int[])temp.Clone();
            foreach (var edge in edges)
            {
                int a = edge[0], b = edge[1], d = edge[2];

                // Updating the current array using previous
                // array Subtracting d is same as adding -d
                temp[b] = Math.Min(temp[b], c[a] == int.MaxValue ? int.MaxValue : c[a] - d);
            }
        }

        // Checking if dst is reachable from src or not
        if (temp[dst] != int.MaxValue)
        {

            // Returning the negative value of the shortest
            // distance to return the longest distance
            return -temp[dst];
        }
        return -1;
    }

    public static void Main()
    {
        List<List<int>> edges = new List<List<int>>()
        {
            new List<int>(){0, 1, 100},
            new List<int>(){1, 2, 100},
            new List<int>(){0, 2, 500}
        };
        int src = 0;
        int dst = 2;
        int k = 1;
        int n = 3;

        Console.WriteLine(MaxCost(n, edges, src, dst, k));
    }
}

Output

500

Time Complexity: O(E*k) where E is the number of edges
Auxiliary Space: O(n)