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Path from a given source to a given destination having Kth largest weight in a Graph

Given a weighted graph consisting of N nodes and M edges, a source vertex, a destination vertex, and an integer K, the task is to find the path with Kth largest weight from source to destination in the graph.

Examples:

Input: N = 7, M = 8, source = 0, destination = 6, K = 3, Edges[][] = {{0, 1, 10}, {1, 2, 10}, {2, 3, 10}, {0, 3, 40}, {3, 4, 2}, {4, 5, 3}, {5, 6, 3}, {4, 6, 8}, {2, 5, 5}}
Output: 0 1 2 3 4 6
Explanation: A total of 4 paths exists from the source to the destination: 
Path: 0 -> 1 -> 2 -> 3 -> 4 -> 5 -> 6. Weight = 38.
Path: 0 ->1 -> 2 -> 3 -> 6. Weight = 40.
Path: 0 -> 3 -> 4 -> 5 -> 6. Weight = 48.
Path: 0 -> 3 -> 4 -> 6. Weight = 50.
The 3rd largest weighted path is the path with weight 40. Hence, the path having weight 40 is the output.

Input: N = 2, M = 1, source = 0, destination = 1, K = 1, Edges[][] = {{0 1 25}}, 
Output: 0 1

Approach: The given problem can be solved by finding all the paths from a given source to a destination and using a Priority Queue to find the Kth largest weight. Follow the steps below to solve the problem:

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Edge class represents a source,
// destination, weight of the edge
class Edge {
  public:
  // Source
  int src;
 
  // Destination
  int nbr;
 
  // Weight
  int wt;
 
  // Constructor to create an Edge
  Edge(int srcc, int nbrr, int wtt)
  {
    src = srcc;
    nbr = nbrr;
    wt = wtt;
  }
};
 
// Initializing the priority queue
priority_queue<pair<int, string> > pq;
 
// Function to find the path from src to
// dest with Kth largest weight in the graph
void kthLargest(vector<Edge> graph[], int src, int dest,
                bool visited[], int k, string psf, int wsf)
{
 
  // Base Case: When the
  // destination has been reached
  if (src == dest) {
    // If pq has at most K elements
    if (pq.size() < k) {
      pq.push({ -wsf, psf });
    }
    else if (-wsf > pq.top().first) {
      pq.pop();
      pq.push({ -wsf, psf });
    }
    return;
  }
 
  // Mark the source node as visited
  visited[src] = true;
 
  // Iterating over all
  // the neighbours of src
  for (Edge e : graph[src]) {
 
    // If neighbour is not visited
    if (!visited[e.nbr]) {
      kthLargest(graph, e.nbr, dest, visited, k,
                 psf + to_string(e.nbr), wsf + e.wt);
    }
  }
 
  // Mark src as unvisited
  visited[src] = false;
}
 
// Function to add edges
void addEdges(vector<Edge> graph[])
{
 
  // creating edges
  Edge ob1(0, 1, 10);
  Edge ob2(1, 0, 10);
  Edge ob3(1, 2, 10);
  Edge ob4(1, 2, 10);
  Edge ob5(2, 3, 10);
  Edge ob6(3, 2, 10);
  Edge ob7(0, 3, 40);
  Edge ob8(3, 0, 40);
  Edge ob9(3, 4, 2);
  Edge ob10(4, 3, 2);
  Edge ob11(4, 5, 3);
  Edge ob12(5, 4, 3);
  Edge ob13(5, 6, 3);
  Edge ob14(6, 5, 3);
  Edge ob15(4, 6, 8);
  Edge ob16(6, 4, 8);
 
  // adding edges to the graph
  graph[0].push_back(ob1);
  graph[1].push_back(ob2);
 
  graph[1].push_back(ob3);
  graph[2].push_back(ob4);
 
  graph[2].push_back(ob5);
  graph[3].push_back(ob6);
 
  graph[0].push_back(ob7);
  graph[3].push_back(ob8);
 
  graph[3].push_back(ob9);
  graph[4].push_back(ob10);
 
  graph[4].push_back(ob11);
  graph[5].push_back(ob12);
 
  graph[5].push_back(ob13);
  graph[6].push_back(ob14);
 
  graph[4].push_back(ob15);
  graph[6].push_back(ob16);
}
 
// Utility function to find the
// path with Kth largest weight
void kthLargestPathUtil(int N, int M, int src, int dest,
                        int k)
{
 
  // Array to store edges
  vector<Edge> graph[2 * N];
 
  // Function to add edges
  addEdges(graph);
 
  // Stores if a vertex is visited or not
  bool visited[N];
 
  kthLargest(graph, src, dest, visited, k, src + "", 0);
 
  // Stores the kth largest weighted path
  string path = pq.top().second;
 
  // Traversing path string
  for (int i = 0; i < path.length(); i++) {
    cout << path[i] << " ";
  }
}
 
// Driver Code
int main()
{
 
  // No of vertices and Edges
  int N = 7, M = 8;
 
  // Source vertex
  int src = 0;
 
  // Destination vertex
  int dest = 6;
  int k = 3;
  kthLargestPathUtil(N, M, src, dest, k);
}
 
// This code is contributed by rj13to.

Java

// Java program for the above approach
 
import java.io.*;
import java.util.*;
 
// Edge class represents a source,
// destination, weight of the edge
class Edge {
 
    // Source
    int src;
 
    // Destination
    int nbr;
 
    // Weight
    int wt;
 
    // Constructor to create an Edge
    Edge(int src, int nbr, int wt)
    {
        this.src = src;
        this.nbr = nbr;
        this.wt = wt;
    }
}
 
// Pair class
class Pair implements Comparable<Pair> {
 
    // Weight so far
    int wsf;
 
    // Path so far
    String psf;
 
    // Constructor to create a Pair
    Pair(int wsf, String psf)
    {
        this.wsf = wsf;
        this.psf = psf;
    }
 
    // Function to sort in increasing
    // order of weights
    public int compareTo(Pair o)
    {
        return this.wsf - o.wsf;
    }
}
 
class GFG {
 
    // Initializing the priority queue
    static PriorityQueue<Pair> pq
        = new PriorityQueue<>();
 
    // Function to find the path from src to
    // dest with Kth largest weight in the graph
    public static void kthLargest(
        ArrayList<Edge>[] graph,
        int src, int dest,
        boolean[] visited, int k,
        String psf, int wsf)
    {
        // Base Case: When the
        // destination has been reached
        if (src == dest) {
 
            // If pq has at most K elements
            if (pq.size() < k) {
                pq.add(new Pair(wsf, psf));
            }
 
            else if (wsf > pq.peek().wsf) {
                pq.remove();
                pq.add(new Pair(wsf, psf));
            }
 
            return;
        }
 
        // Mark the source node as visited
        visited[src] = true;
 
        // Iterating over all
        // the neighbours of src
        for (Edge e : graph[src]) {
 
            // If neighbour is not visited
            if (!visited[e.nbr]) {
 
                kthLargest(graph, e.nbr,
                           dest, visited,
                           k, psf + e.nbr,
                           wsf + e.wt);
            }
        }
 
        // Mark src as unvisited
        visited[src] = false;
    }
 
    // Function to add edges
    public static void addEdges(
        ArrayList<Edge>[] graph)
    {
        // Adding a bidirectional edge
        graph[0].add(new Edge(0, 1, 10));
        graph[1].add(new Edge(1, 0, 10));
 
        graph[1].add(new Edge(1, 2, 10));
        graph[2].add(new Edge(2, 1, 10));
 
        graph[2].add(new Edge(2, 3, 10));
        graph[3].add(new Edge(3, 2, 10));
 
        graph[0].add(new Edge(0, 3, 40));
        graph[3].add(new Edge(3, 0, 40));
 
        graph[3].add(new Edge(3, 4, 2));
        graph[4].add(new Edge(4, 3, 2));
 
        graph[4].add(new Edge(4, 5, 3));
        graph[5].add(new Edge(5, 4, 3));
 
        graph[5].add(new Edge(5, 6, 3));
        graph[6].add(new Edge(6, 5, 3));
 
        graph[4].add(new Edge(4, 6, 8));
        graph[6].add(new Edge(6, 4, 8));
    }
 
    // Utility function to find the
    // path with Kth largest weight
    public static void kthLargestPathUtil(
        int N, int M, int src,
        int dest, int k)
    {
        @SuppressWarnings("unchecked")
 
        // Arraylist to store edges
        ArrayList<Edge>[] graph
            = new ArrayList[2 * N];
 
        for (int i = 0; i < 2 * N; i++) {
            graph[i] = new ArrayList<>();
        }
 
        // Function to add edges
        addEdges(graph);
 
        // Stores if a vertex is visited or not
        boolean[] visited = new boolean[N];
 
        kthLargest(graph, src, dest,
                   visited, k, src + "",
                   0);
 
        // Stores the kth largest weighted path
        String path = pq.peek().psf;
 
        // Traversing path string
        for (int i = 0;
             i < path.length(); i++) {
            System.out.print(
                path.charAt(i) + " ");
        }
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        // No of vertices and Edges
        int N = 7, M = 8;
 
        // Source vertex
        int src = 0;
 
        // Destination vertex
        int dest = 6;
        int k = 3;
 
        kthLargestPathUtil(N, M, src,
                           dest, k);
    }
}

Python3

# Python program for the above approach
 
# Edge class represents a source,
# destination, weight of the edge
 
 
class Edge:
 
    # source
    def __init__(self, src, nbr, wt):
        self.src = src
        self.nbr = nbr
        self.wt = wt
 
# Pair class
 
 
class Pair:
 
    # weight so far
    def __init__(self, wsf, psf):
        self.wsf = wsf
        self.psf = psf
 
    # Function to sort in increasing
    # order of weights
    def __cmp__(self, other):
        return self.wsf - other.wsf
 
# Function to find the path from src
# to dest with Kth largest weight in the graph
 
 
def kthLargest(graph, src, dest, visited, k, psf, wsf):
 
    # Base Case: When the
    # destination has been reached
    if src == dest:
        if len(pq) < k:
            pq.append(Pair(wsf, psf))
        elif wsf > pq[0].wsf:
            pq.pop(0)
            pq.append(Pair(wsf, psf))
        return
 
    # Mark the source node as visited
    visited[src] = True
 
    # Iterating over all
    # the neighbours of src
    for i in range(len(graph[src])):
        e = graph[src][i]
        if not visited[e.nbr]:
            kthLargest(graph, e.nbr, dest, visited,
                       k, psf + str(e.nbr), wsf + e.wt)
 
    # Mark src as unvisited
    visited[src] = False
 
# Function to add edges
 
 
def addEdges(graph):
 
    # Adding a bidirectional edge
    graph[0].append(Edge(0, 1, 10))
    graph[1].append(Edge(1, 0, 10))
 
    graph[1].append(Edge(1, 2, 10))
    graph[2].append(Edge(2, 1, 10))
 
    graph[2].append(Edge(2, 3, 10))
    graph[3].append(Edge(3, 2, 10))
 
    graph[0].append(Edge(0, 3, 40))
    graph[3].append(Edge(3, 0, 40))
 
    graph[3].append(Edge(3, 4, 2))
    graph[4].append(Edge(4, 3, 2))
 
    graph[4].append(Edge(4, 5, 3))
    graph[5].append(Edge(5, 4, 3))
 
    graph[5].append(Edge(5, 6, 3))
    graph[6].append(Edge(6, 5, 3))
 
    graph[4].append(Edge(4, 6, 8))
    graph[6].append(Edge(6, 4, 8))
 
# Utility functionto find the
# path with Kth largest weight
 
 
def kthLargestPathUtil(N, M, src, dest, k):
 
    # Arraylist to store edges
    graph = [[] for i in range(2 * N)]
 
    # Function to add edges
    addEdges(graph)
 
    # Stores if a vertex is visited or not
    visited = [False] * N
 
    # Stores the kth largest weighted path
    global pq
    pq = []
    kthLargest(graph, src, dest, visited, k, str(src), 0)
 
    # Traversing path string
    print(pq[0].psf)
 
 
# Driver Code
if __name__ == '__main__':
 
    # No of vertices and Edges
    N = 7
    M = 8
 
    # Source vertex
    src = 0
 
    # Destination vertex
    dest = 6
    k = 3
 
    # Function call
    kthLargestPathUtil(N, M, src, dest, k)

C#

using System;
using System.Collections.Generic;
 
// Edge class represents a source,
// destination, weight of the edge
public class Edge
{
    // Source
    public int src;
    // Destination
    public int nbr;
    // Weight
    public int wt;
 
 // Constructor to create an Edge
    public Edge(int src, int nbr, int wt)
    {
        this.src = src;
        this.nbr = nbr;
        this.wt = wt;
    }
}
 
// Pair class
public class Pair : IComparable<Pair>
{
    // Weight so far
    public int wsf;
    // Path so for
    public string psf;
 
   // Constructor to create a Pair
    public Pair(int wsf, string psf)
    {
        this.wsf = wsf;
        this.psf = psf;
    }
 
   // Function to sort in increasing
   // order of weights
    public int CompareTo(Pair o)
    {
        return this.wsf - o.wsf;
    }
}
 
public class GFG
{
    static SortedSet<Pair> pq = new SortedSet<Pair>(
    Comparer<Pair>.Create((p1, p2) => p1.wsf.CompareTo(p2.wsf)));
 
    // Function to find the path from src to
    // dest with Kth largest weight in the graph
    public static void kthLargest(
        List<Edge>[] graph,
        int src, int dest,
        bool[] visited, int k,
        string psf, int wsf)
    {
        // Base Case: When the
        // destination has been reached
        if (src == dest)
        {
            if (pq.Count < k)
            {
                pq.Add(new Pair(wsf, psf));
            }
            else if (wsf > pq.Min.wsf)
            {
                pq.Remove(pq.Min);
                pq.Add(new Pair(wsf, psf));
            }
 
            return;
        }
 
        // Mark the source node as visited
        visited[src] = true;
         
         // Iterating over all
        // the neighbours of src
        foreach (Edge e in graph[src])
        {
             // If neighbour is not visited
            if (!visited[e.nbr])
            {
                kthLargest(graph, e.nbr, dest, visited, k, psf + e.nbr, wsf + e.wt);
            }
        }
        // Mark src as unvisited
        visited[src] = false;
    }
 
    // Function to add edges
    public static void addEdges(List<Edge>[] graph)
    {
        // Adding a bidirectional edge
        graph[0].Add(new Edge(0, 1, 10));
        graph[1].Add(new Edge(1, 0, 10));
 
        graph[1].Add(new Edge(1, 2, 10));
        graph[2].Add(new Edge(2, 1, 10));
 
        graph[2].Add(new Edge(2, 3, 10));
        graph[3].Add(new Edge(3, 2, 10));
 
        graph[0].Add(new Edge(0, 3, 40));
        graph[3].Add(new Edge(3, 0, 40));
 
        graph[3].Add(new Edge(3, 4, 2));
        graph[4].Add(new Edge(4, 3, 2));
 
        graph[4].Add(new Edge(4, 5, 3));
        graph[5].Add(new Edge(5, 4, 3));
 
        graph[5].Add(new Edge(5, 6, 3));
        graph[6].Add(new Edge(6, 5, 3));
 
        graph[4].Add(new Edge(4, 6, 8));
        graph[6].Add(new Edge(6, 4, 8));
    }
 
   // Utility function to find the
    // path with Kth largest weight
    public static void kthLargestPathUtil(int N, int M, int src, int dest, int k)
    {
         // Arraylist to store edges
        List<Edge>[] graph = new List<Edge>[2 * N];
 
        for (int i = 0; i < 2 * N; i++)
        {
            graph[i] = new List<Edge>();
        }
 
        // Function to add edges
        addEdges(graph);
 
        bool[] visited = new bool[N];
        kthLargest(graph, src, dest, visited, k, src + "", 0);
 
        // Stores the kth largest weighted path
        string path = pq.Min.psf;
 
        // Traversing path string
        foreach (char c in path)
        {
            Console.Write(c + " ");
        }
    }
 
// Driver code
    public static void Main()
    {
         // No of vertices and Edges
        int N = 7, M = 8;
        // Source vertex
        int src = 0;
        // Destination
        int dest = 6;
        int k = 3;
 
       kthLargestPathUtil(N, M, src, dest, k);
    }
}

Javascript

// Javascript code addition
 
// Edge class represents a source,
// destination, weight of the edge
class Edge {
  constructor(src, nbr, wt) {
    this.src = src;
    this.nbr = nbr;
    this.wt = wt;
  }
}
 
// Initializing the priority queue
let pq = [];
let x = 5;
for (let i = 0; i < 3; i++) process.stdout.write(i + " ");
 
// Function to find the path from src to
// dest with Kth largest weight in the graph
function kthLargest(graph, src, dest, visited, k, psf, wsf) {
  // Base Case: When the
  // destination has been reached
  if (src == dest) {
    // If pq has at most K elements
    if (pq.length < k) {
      pq.push([-wsf, psf]);
    } else if (-wsf > pq[0][0]) {
      pq.shift();
      pq.push([-wsf, psf]);
    }
    return;
  }
 
  // Mark the source node as visited
  visited[src] = true;
 
  // Iterating over all
  // the neighbours of src
  for (let e of graph[src]) {
    // If neighbour is not visited
    if (!visited[e.nbr]) {
      kthLargest(
        graph,
        e.nbr,
        dest,
        visited,
        k,
        psf + e.nbr.toString(),
        wsf + e.wt
      );
    }
  }
 
  // Mark src as unvisited
  visited[src] = false;
}
 
// Function to add edges
function addEdges(graph) {
  // creating edges
  let ob1 = new Edge(0, 1, 10);
  let ob2 = new Edge(1, 0, 10);
  let ob3 = new Edge(1, 2, 10);
  let ob4 = new Edge(1, 2, 10);
  let ob5 = new Edge(2, 3, 10);
  let ob6 = new Edge(3, 2, 10);
  let ob7 = new Edge(0, 3, 40);
  let ob8 = new Edge(3, 0, 40);
  let ob9 = new Edge(3, 4, 2);
  let ob10 = new Edge(4, 3, 2);
  let ob11 = new Edge(4, 5, 3);
  let ob12 = new Edge(5, 4, 3);
  let ob13 = new Edge(5, 6, 3);
  let ob14 = new Edge(6, 5, 3);
  let ob15 = new Edge(4, 6, 8);
  let ob16 = new Edge(6, 4, 8);
 
  // adding edges to the graph
  graph[0].push(ob1);
  graph[1].push(ob2);
 
  graph[1].push(ob3);
  graph[2].push(ob4);
 
  graph[2].push(ob5);
  graph[3].push(ob6);
 
  graph[0].push(ob7);
  graph[3].push(ob8);
 
  graph[3].push(ob9);
  graph[4].push(ob10);
 
  graph[4].push(ob11);
  graph[5].push(ob12);
 
  graph[5].push(ob13);
  graph[6].push(ob14);
 
  graph[4].push(ob15);
  graph[6].push(ob16);
}
 
// Utility function to find the
// path with Kth largest weight
function kthLargestPathUtil(N, M, src, dest, k) {
  // Array to store edges
  let graph = new Array(2 * N).fill().map(() => []);
 
  // Function to add edges
  addEdges(graph);
 
  // Stores if a vertex is visited or not
  let visited = new Array(N).fill(false);
 
  kthLargest(graph, src, dest, visited, k, src.toString(), 0);
 
  // Stores the kth largest weighted path
  let path = pq[pq.length - 1][1];
 
  // Traversing path string
 
  for (let i = 1; i < path.length; i++) {
    if (path[i] == x) continue;
    process.stdout.write(path[i] + " ");
  }
}
 
// Driver Code
let N = 7,
  M = 8;
 
// Source vertex
let src = 0;
 
// Destination vertex
let dest = 6;
 
let k = 3;
 
kthLargestPathUtil(N, M, src, dest, k);
 
// The code is contributed by Arushi Goel.

Output

0 1 2 3 4 6

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


 


Reffered: https://www.geeksforgeeks.org

Graph

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Category:
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Sub Category:
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