Minimum cost path from source node to destination node via K intermediate nodes

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

Examples:

Input: N = 3, M = 3, src = 0, dst = 2, K = 1, edges[] = {{0, 1, 100}, {1, 2, 100}, {0, 2, 500}}
Output: 200
Explanation: The path 0 -> 1 -> 2 is the least weighted sum (= 100 + 100 = 200) path connecting src (= 0) and dst(= 2) with exactly K (= 1) intermediate vertex.

Input: N = 3, M = 3, src = 0, dst = 2, K = 0, edges[] = { { 0, 1, 100 }, { 1, 2, 100 }, { 0, 2, 500 } }
Output: 500
Explanation: The direct edge 0 -> 2 with weight 500 is the required path.

Approach: The given problem can be solved using Priority Queue and perform BFS. Follow the steps below to solve this problem:

Initialize a priority queue to store the tuples {cost to reach this vertex, vertex, number of stops}.
Push {0, src, k+1} as the first starting point.
Pop-out the top element of priority queue. If all stops are exhausted, then repeat this step.
If the destination is reached, then print the cost to reach the current vertex.
Otherwise, find the neighbor of this vertex which required the smallest cost to reach that vertex. Push it into the priority queue.
Repeat from step 2.
If no path is found after performing the above steps, print -1.

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 minimum cost path
// from the source vertex to destination
// vertex via K intermediate vertices
int leastWeightedSumPath(int n,
                         vector<vector<int> >& edges,
                         int src, int dst, int K)
{
    // Initialize the adjacency list
    unordered_map<int,
                  vector<pair<int, int> > >
        graph;
 
    // Generate the adjacency list
    for (vector<int>& edge : edges) {
        graph[edge[0]].push_back(
            make_pair(edge[1], edge[2]));
    }
 
    // Initialize the minimum priority queue
    priority_queue<vector<int>, vector<vector<int> >,
                   greater<vector<int> > >
        pq;
 
    // Stores the minimum cost to
    // travel between vertices via
    // K intermediate nodes
    vector<vector<int> > costs(n,
                               vector<int>(
                                   K + 2, INT_MAX));
 
    costs[src][K + 1] = 0;
 
    // Push the starting vertex,
    // cost to reach and the number
    // of remaining vertices
    pq.push({ 0, src, K + 1 });
 
    while (!pq.empty()) {
        // Pop the top element
        // of the stack
        auto top = pq.top();
 
        pq.pop();
 
        // If destination is reached
        if (top[1] == dst)
 
            // Return the cost
            return top[0];
 
        // If all stops are exhausted
        if (top[2] == 0)
            continue;
 
        // Find the neighbour with minimum cost
        for (auto neighbor : graph[top[1]]) {
 
            // Pruning
            if (costs[neighbor.first][top[2] - 1]
                < neighbor.second + top[0]) {
                continue;
            }
 
            // Update cost
            costs[neighbor.first][top[2] - 1]
                = neighbor.second + top[0];
 
            // Update priority queue
            pq.push({ neighbor.second + top[0],
                      neighbor.first, top[2] - 1 });
        }
    }
 
    // If no path exists
    return -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 } };
 
    // Function Call to find the path
    // from src to dist via k nodes
    // having least sum of weights
    cout << leastWeightedSumPath(n, edges,
                                 src, dst, k);
 
    return 0;
}

Java

// Java code to implement the approach
import java.util.ArrayList;
import java.util.Collections;
import java.util.List;
import java.util.PriorityQueue;
 
class GFG {
    // Function to find the minimum cost path
    // from the source vertex to destination
    // vertex via K intermediate vertices
    static int
    LeastWeightedSumPath(int n, List<List<Integer> > edges,
                         int src, int dst, int K)
    {
        // Initialize the adjacency list
        var graph = new ArrayList<List<int[]> >(
            Collections.nCopies(n, null));
 
        // Generate the adjacency list
        for (int i = 0; i < edges.size(); i++) {
            var edge = edges.get(i);
            if (graph.get(edge.get(0)) == null) {
                graph.set(edge.get(0),
                          new ArrayList<int[]>());
            }
            graph.get(edge.get(0))
                .add(
                    new int[] { edge.get(1), edge.get(2) });
        }
 
        // Initialize the minimum priority queue
        var pq = new PriorityQueue<int[]>(
            (i1, i2) -> i1[0] - i2[0]);
 
        // Stores the minimum cost to
        // travel between vertices via
        // K intermediate nodes
        var costs = new int[n][K + 2];
 
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < K + 2; j++) {
                costs[i][j] = Integer.MAX_VALUE;
            }
        }
 
        costs[src][K + 1] = 0;
 
        // Push the starting vertex,
        // cost to reach and the number
        // of remaining vertices
        pq.offer(new int[] { 0, src, K + 1 });
 
        while (!pq.isEmpty()) {
            // Pop the top element
            // of the queue
            var top = pq.poll();
 
            // If destination is reached
            if (top[1] == dst) {
                // Return the cost
                return top[0];
            }
 
            // If all stops are exhausted
            if (top[2] == 0) {
                continue;
            }
 
            // Find the neighbour with minimum cost
            if (graph.get(top[1]) != null) {
                for (var neighbor : graph.get(top[1])) {
 
                    // Pruning
                    if (costs[neighbor[0]][top[2] - 1]
                        < neighbor[1] + top[0]) {
                        continue;
                    }
 
                    // Update cost
                    costs[neighbor[0]][top[2] - 1]
                        = neighbor[1] + top[0];
 
                    // Update priority queue
                    pq.offer(new int[] {
                        neighbor[1] + top[0], neighbor[0],
                        top[2] - 1 });
                }
            }
        }
 
        // If no path exists
        return -1;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int n = 3, src = 0, dst = 2, k = 1;
        var edges = new ArrayList<List<Integer> >();
        edges.add(List.of(0, 1, 100));
        edges.add(List.of(1, 2, 100));
        edges.add(List.of(0, 2, 500));
 
        // Function Call to find the path
        // from src to dist via k nodes
        // having least sum of weights
        System.out.println(
            LeastWeightedSumPath(n, edges, src, dst, k));
    }
}
 
// This code is contributed by phasing17

Python3

# Python3 program for the above approach
 
# Function to find the minimum cost path
# from the source vertex to destination
# vertex via K intermediate vertices
def leastWeightedSumPath(n, edges, src, dst, K):
    graph = [[] for i in range(3)]
 
    # Generate the adjacency list
    for edge in edges:
        graph[edge[0]].append([edge[1], edge[2]])
 
    # Initialize the minimum priority queue
    pq = []
 
    # Stores the minimum cost to
    # travel between vertices via
    # K intermediate nodes
    costs = [[10**9 for i in range(K + 2)] for i in range(n)]
 
    costs[src][K + 1] = 0
 
    # Push the starting vertex,
    # cost to reach and the number
    # of remaining vertices
    pq.append([ 0, src, K + 1 ])
 
    pq = sorted(pq)[::-1]
 
    while (len(pq) > 0):
       
        # Pop the top element
        # of the stack
        top = pq[-1]
 
        del pq[-1]
 
        # If destination is reached
        if (top[1] == dst):
           
            # Return the cost
            return top[0]
 
        # If all stops are exhausted
        if (top[2] == 0):
            continue
 
        # Find the neighbour with minimum cost
        for neighbor in graph[top[1]]:
 
            # Pruning
            if (costs[neighbor[0]][top[2] - 1] < neighbor[1] + top[0]):
                continue
 
            # Update cost
            costs[neighbor[0]][top[2] - 1] = neighbor[1]+ top[0]
 
            # Update priority queue
            pq.append([neighbor[1]+ top[0],neighbor[0], top[2] - 1])
        pq = sorted(pq)[::-1]
 
    # If no path exists
    return -1
 
# Driver Code
if __name__ == '__main__':
    n, src, dst, k = 3, 0, 2, 1
    edges = [ [ 0, 1, 100 ],
            [ 1, 2, 100 ],
            [ 0, 2, 500 ] ]
 
    # Function Call to find the path
    # from src to dist via k nodes
    # having least sum of weights
    print (leastWeightedSumPath(n, edges, src, dst, k))
 
# This code is contributed by mohit kumar 29.

C#

using System;
using System.Collections.Generic;
 
class GFG {
    // Function to find the minimum cost path
    // from the source vertex to destination
    // vertex via K intermediate vertices
    static int LeastWeightedSumPath(int n, List<List<int>> edges, 
                                    int src, int dst, int K) {
        // Initialize the adjacency list
        var graph = new Dictionary<int, List<Tuple<int, int>>>();
 
        // Generate the adjacency list
        for (int i = 0; i < edges.Count; i++) {
            var edge = edges[i];
            if (!graph.ContainsKey(edge[0])) {
                graph[edge[0]] = new List<Tuple<int, int>>();
            }
            graph[edge[0]].Add(new Tuple<int, int>(edge[1], edge[2]));
        }
 
        // Initialize the minimum priority queue
        var pq = new SortedSet<Tuple<int, int, int>>(
            Comparer<Tuple<int, int, int>>.Create((i1, i2) => 
                                                  i1.Item1.CompareTo(i2.Item1)));
 
        // Stores the minimum cost to
        // travel between vertices via
        // K intermediate nodes
        var costs = new int[n, K + 2];
 
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < K + 2; j++) {
                costs[i, j] = int.MaxValue;
            }
        }
 
        costs[src, K + 1] = 0;
 
        // Push the starting vertex,
        // cost to reach and the number
        // of remaining vertices
        pq.Add(new Tuple<int, int, int>(0, src, K + 1));
 
        while (pq.Count > 0) {
            // Pop the top element
            // of the stack
            var top = pq.Min;
            pq.Remove(top);
 
            // If destination is reached
            if (top.Item2 == dst) {
                // Return the cost
                return top.Item1;
            }
 
            // If all stops are exhausted
            if (top.Item3 == 0) {
                continue;
            }
 
            // Find the neighbour with minimum cost
            if (graph.ContainsKey(top.Item2)) {
                foreach (var neighbor in graph[top.Item2]) {
 
                    // Pruning
                    if (costs[neighbor.Item1, top.Item3 - 1] < neighbor.Item2 + top.Item1) {
                        continue;
                    }
 
                    // Update cost
                    costs[neighbor.Item1, top.Item3 - 1] = neighbor.Item2 + top.Item1;
 
                    // Update priority queue
                    pq.Add(new Tuple<int, int, int>(neighbor.Item2 + top.Item1, neighbor.Item1, top.Item3 - 1));
                }
            }
        }
 
        // If no path exists
        return -1;
    }
 
    // Driver Code
    static void Main() {
        int n = 3, src = 0, dst = 2, k = 1;
        var edges = new List<List<int>>() {
            new List<int> { 0, 1, 100 },
            new List<int> { 1, 2, 100 },
            new List<int> { 0, 2, 500 }
        };
 
        // Function Call to find the path
        // from src to dist via k nodes
        // having least sum of weights
        Console.WriteLine(LeastWeightedSumPath(n, edges, src, dst, k));
    }
}

Javascript

<script>
// Javascript program for the above approach
 
// Function to find the minimum cost path
// from the source vertex to destination
// vertex via K intermediate vertices
function leastWeightedSumPath(n, edges, src, dst, K)
{
 
    // Initialize the adjacency list
    var graph = new Map();
 
    // Generate the adjacency list
    for (var edge of edges) {
        if(!graph.has(edge[0]))
            graph.set(edge[0], new Array())
        var tmp = graph.get(edge[0]);
        tmp.push([edge[1], edge[2]]);
        graph.set(edge[0],tmp);
    }
 
    // Initialize the minimum priority queue
    var pq = [];
 
    // Stores the minimum cost to
    // travel between vertices via
    // K intermediate nodes
    var costs = Array.from(Array(n+1), ()=>Array(K+2).fill(1000000000));
    costs[src][K + 1] = 0;
 
    // Push the starting vertex,
    // cost to reach and the number
    // of remaining vertices
    pq.push([ 0, src, K + 1 ]);
 
    while (pq.length!=0) {
        // Pop the top element
        // of the stack
        var top = pq[pq.length-1];
 
        pq.pop();
 
        // If destination is reached
        if (top[1] == dst)
 
            // Return the cost
            return top[0];
 
        // If all stops are exhausted
        if (top[2] == 0)
            continue;
 
        if(graph.has(top[1]))
        {
        // Find the neighbour with minimum cost
        for (var neighbor of graph.get(top[1])) {
            
            // Pruning
            if (costs[neighbor[0]][top[2] - 1]
                < neighbor[1] + top[0]) {
                continue;
            }
 
            // Update cost
            costs[neighbor[0]][top[2] - 1]
                = neighbor[1] + top[0];
 
            // Update priority queue
            pq.push([neighbor[1] + top[0],
                      neighbor[0], top[2] - 1 ]);
        }
        }
        pq.sort((a,b)=>{
            if(a[0]==b[0])
             return b[1]-a[1]
            return b[0]-a[0];
        });
    }
 
    // If no path exists
    return -1;
}
 
// Driver Code
var n = 3, src = 0, dst = 2, k = 1;
var edges
    = [ [ 0, 1, 100 ],
        [ 1, 2, 100 ],
        [ 0, 2, 500 ] ];
// Function Call to find the path
// from src to dist via k nodes
// having least sum of weights
document.write(leastWeightedSumPath(n, edges,
                             src, dst, k));
 
// This code is contributed by rrrtnx.
</script>

Output:

Time Complexity : O(N * log N)
Auxiliary Space: O(N)

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