# Sum of shortest distance on source to destination and back having at least a common vertex

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:
1. 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.
2. Find shortest distance of all vertex from s and d which gives us d[s][v] and d[d][v].
3. 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 implementation of the approach ` ` `  `#include ` `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 >* 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[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, ` `                   ``greater > ` `        ``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 >::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; ` `} `

Output:

```17
```

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