Given a directed weighted graph and the source and destination vertex. The task is to find the sum of shortest distance on the path going from source to destination and and then from destination to source such that both the paths have at least a common vertex other than the source and the destination.
Note: On going from destination to source, all the directions of the edges are reversed.
Examples:
Input: src = 0, des = 1
Output: 17
Explanation:
Common vertex is 4 and path is 0 -> 4 -> 3 -> 1 -> 4 -> 0
Approach: The idea is to use Dijkstra’s algorithm. On finding the shortest path from source to destination and shortest path from destination to the source using Dijkstra’s algorithm, it may not result in a path where there is at least one node in common except the source and destination vertex.
- Let s be the source vertex and d be destination vertex and v be the intermediate node common in both the paths from source to destination and destination to source. The shortest pair of paths, so that v is in intersection of this two paths is a path: s -> v -> d -> v -> s and it’s length is
dis[s][v] + dis[v][d] + dis[d][v] + dis[v][s]
- Since s and d are fixed, just find v such that it gives shortest path.
- In order to find such v, follow the below steps:
- Find shortest distance from all vertices to s and d which gives us the values of dis[v][s] and dis[v][d]. For finding the shortest path from all the vertices to a given node refer Shortest paths from all vertices to a destination.
- Find shortest distance of all vertex from s and d which gives us d[s][v] and d[d][v].
- Iterate for all v and find minimum of d[s][v] + d[v][d] + d[d][v] + d[v][s].
Below is the implementation of the above approach:
CPP
// CPP implementation of the approach #include <bits/stdc++.h> using namespace std; #define INF 0x3f3f3f3f // iPair represents the Integer Pair typedef pair< int , int > iPair; // This class represents // a directed graph using // adjacency list representation class Graph { // Number of vertices int V; // In a weighted graph, store vertex // and weight pair for every edge list<pair< int , int > >* adj; public : // Constructor Graph( int V); // Function to add an edge to graph void addEdge( int u, int v, int w); // Find shortest path from // source vertex to all vertex void shortestPath( int src, vector< int >& dist); }; // Allocates memory for adjacency list Graph::Graph( int V) { this ->V = V; adj = new list<iPair>[V]; } // Function to add an edge to the graph void Graph::addEdge( int u, int v, int w) { adj[v].push_back(make_pair(u, w)); } // Function to find the shortest paths // from source to all other vertices void Graph::shortestPath( int src, vector< int >& dist) { // Create a priority queue to // store vertices that // are being preprocessed priority_queue<iPair, vector<iPair>, greater<iPair> > pq; // Insert source itself in priority // queue and initialize // its distance as 0 pq.push(make_pair(0, src)); dist[src] = 0; // Loop till priority queue // becomes empty (or all // distances are not finalized) while (!pq.empty()) { // The first vertex in pair // is the minimum distance // vertex, extract it from // priority queue int u = pq.top().second; pq.pop(); // 'i' is used to get all // adjacent vertices of a vertex list<pair< int , int > >::iterator i; for (i = adj[u].begin(); i != adj[u].end(); ++i) { // Get vertex label and // weight of current // adjacent of u int v = (*i).first; int weight = (*i).second; // If there is shorted // path to v through u if (dist[v] > dist[u] + weight) { // Updating distance of v dist[v] = dist[u] + weight; pq.push(make_pair(dist[v], v)); } } } } // Function to return the // required minimum path int minPath( int V, int src, int des, Graph g, Graph r) { // Create a vector for // distances and // initialize all distances // as infinite (INF) // To store distance of all // vertex from source vector< int > dist(V, INF); // To store distance of all // vertex from destination vector< int > dist2(V, INF); // To store distance of source // from all vertex vector< int > dist3(V, INF); // To store distance of // destination from all vertex vector< int > dist4(V, INF); // Computing shortest path from // source vertex to all vertices g.shortestPath(src, dist); // Computing shortest path from // destination vertex to all vertices g.shortestPath(des, dist2); // Computing shortest path from // all the vertices to source r.shortestPath(src, dist3); // Computing shortest path from // all the vertices to destination r.shortestPath(des, dist4); // Finding the intermediate node (IN) // such that the distance of path // src -> IN -> des -> IN -> src is minimum // To store the shortest distance int ans = INT_MAX; for ( int i = 0; i < V; i++) { // Intermediate node should not be // the source and destination if (i != des && i != src) ans = min( ans, dist[i] + dist2[i] + dist3[i] + dist4[i]); } // Return the minimum path required return ans; } // Driver code int main() { // Create the graph int V = 5; int src = 0, des = 1; // To store the original graph Graph g(V); // To store the reverse graph // and compute distance from all // vertex to a particular vertex Graph r(V); // Adding edges g.addEdge(0, 2, 1); g.addEdge(0, 4, 5); g.addEdge(1, 4, 1); g.addEdge(2, 0, 10); g.addEdge(2, 3, 5); g.addEdge(3, 1, 1); g.addEdge(4, 0, 5); g.addEdge(4, 2, 100); g.addEdge(4, 3, 5); // Adding edges in reverse direction r.addEdge(2, 0, 1); r.addEdge(4, 0, 5); r.addEdge(4, 1, 1); r.addEdge(0, 2, 10); r.addEdge(3, 2, 5); r.addEdge(1, 3, 1); r.addEdge(0, 4, 5); r.addEdge(2, 4, 100); r.addEdge(3, 4, 5); cout << minPath(V, src, des, g, r); return 0; } |
Python3
# Python implementation of the approach from typing import List from queue import PriorityQueue from sys import maxsize as INT_MAX INF = 0x3f3f3f3f # This class represents # a directed graph using # adjacency list representation class Graph: def __init__( self , V: int ) - > None : # Number of vertices self .V = V # In a weighted graph, store vertex # and weight pair for every edge self .adj = [[] for _ in range (V)] # Function to add an edge to the graph def addEdge( self , u: int , v: int , w: int ) - > None : self .adj[v].append((u, w)) # Function to find the shortest paths # from source to all other vertices def shortestPath( self , src: int , dist: List [ int ]) - > None : # Create a priority queue to # store vertices that # are being preprocessed pq = PriorityQueue() # Insert source itself in priority # queue and initialize # its distance as 0 pq.put(( 0 , src)) dist[src] = 0 # Loop till priority queue # becomes empty (or all # distances are not finalized) while not pq.empty(): # The first vertex in pair # is the minimum distance # vertex, extract it from # priority queue u = pq.get()[ 1 ] # 'i' is used to get all # adjacent vertices of a vertex for i in self .adj[u]: # Get vertex label and # weight of current # adjacent of u v = i[ 0 ] weight = i[ 1 ] # If there is shorted # path to v through u if dist[v] > dist[u] + weight: # Updating distance of v dist[v] = dist[u] + weight pq.put((dist[v], v)) # Function to return the # required minimum path def minPath(V: int , src: int , des: int , g: Graph, r: Graph) - > int : # Create a vector for # distances and # initialize all distances # as infinite (INF) # To store distance of all # vertex from source dist = [INF for _ in range (V)] # To store distance of all # vertex from destination dist2 = [INF for _ in range (V)] # To store distance of source # from all vertex dist3 = [INF for _ in range (V)] # To store distance of # destination from all vertex dist4 = [INF for _ in range (V)] # Computing shortest path from # source vertex to all vertices g.shortestPath(src, dist) # Computing shortest path from # destination vertex to all vertices g.shortestPath(des, dist2) # Computing shortest path from # all the vertices to source r.shortestPath(src, dist3) # Computing shortest path from # all the vertices to destination r.shortestPath(des, dist4) # Finding the intermediate node (IN) # such that the distance of path # src -> IN -> des -> IN -> src is minimum # To store the shortest distance ans = INT_MAX for i in range (V): # Intermediate node should not be # the source and destination if (i ! = des and i ! = src): ans = min (ans, dist[i] + dist2[i] + dist3[i] + dist4[i]) # Return the minimum path required return ans # Driver code if __name__ = = "__main__" : # Create the graph V = 5 src = 0 des = 1 # To store the original graph g = Graph(V) # To store the reverse graph # and compute distance from all # vertex to a particular vertex r = Graph(V) # Adding edges g.addEdge( 0 , 2 , 1 ) g.addEdge( 0 , 4 , 5 ) g.addEdge( 1 , 4 , 1 ) g.addEdge( 2 , 0 , 10 ) g.addEdge( 2 , 3 , 5 ) g.addEdge( 3 , 1 , 1 ) g.addEdge( 4 , 0 , 5 ) g.addEdge( 4 , 2 , 100 ) g.addEdge( 4 , 3 , 5 ) # Adding edges in reverse direction r.addEdge( 2 , 0 , 1 ) r.addEdge( 4 , 0 , 5 ) r.addEdge( 4 , 1 , 1 ) r.addEdge( 0 , 2 , 10 ) r.addEdge( 3 , 2 , 5 ) r.addEdge( 1 , 3 , 1 ) r.addEdge( 0 , 4 , 5 ) r.addEdge( 2 , 4 , 100 ) r.addEdge( 3 , 4 , 5 ) print (minPath(V, src, des, g, r)) # This code is contributed by sanjeev2552 |
17
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.