Searching a graph is quite famous problem and have a lot of practical use. We have already discussed here how to search for a goal vertex starting from a source vertex using BFS. In normal graph search using BFS/DFS we begin our search in one direction usually from source vertex toward the goal vertex, **but what if we start search form both direction simultaneously.**

Bidirectional search is a graph search algorithm which find smallest path form source to goal vertex. It runs two simultaneous search –

- Forward search form source/initial vertex toward goal vertex
- Backward search form goal/target vertex toward source vertex

Bidirectional search replaces single search graph(which is likely to grow exponentially) with two smaller sub graphs – one starting from initial vertex and other starting from goal vertex. **The search terminates when two graphs intersect.**

Just like A* algorithm, bidirectional search can be guided by a heuristic estimate of remaining distance from source to goal and vice versa for finding shortest path possible.

Consider following simple example-

Suppose we want to find if there exists a path from vertex 0 to vertex 14. Here we can execute two searches, one from vertex 0 and other from vertex 14. When both forward and backward search meet at vertex 7, we know that we have found a path from node 0 to 14 and search can be terminated now. We can clearly see that we have successfully avoided unnecessary exploration.

**Why bidirectional approach?**

Because in many cases it is faster, it dramatically reduce the amount of required exploration.

Suppose if branching factor of tree is **b** and distance of goal vertex from source is **d**, then the normal BFS/DFS searching complexity would be . On the other hand, if we execute two search operation then the complexity would be for each search and total complexity would be which is far less than .

**When to use bidirectional approach?**

We can consider bidirectional approach when-

- Both initial and goal states are unique and completely defined.
- The branching factor is exactly the same in both directions.

**Performance measures**

- Completeness : Bidirectional search is complete if BFS is used in both searches.
- Optimality : It is optimal if BFS is used for search and paths have uniform cost.
- Time and Space Complexity : Time and space complexity is

Below is very simple implementation representing the concept of bidirectional search using BFS. This implementation considers undirected paths without any weight.

// C++ program for Bidirectional BFS search // to check path between two vertices #include <bits/stdc++.h> using namespace std; // class representing undirected graph // using adjacency list class Graph { //number of nodes in graph int V; // Adjacency list list<int> *adj; public: Graph(int V); int isIntersecting(bool *s_visited, bool *t_visited); void addEdge(int u, int v); void printPath(int *s_parent, int *t_parent, int s, int t, int intersectNode); void BFS(list<int> *queue, bool *visited, int *parent); int biDirSearch(int s, int t); }; Graph::Graph(int V) { this->V = V; adj = new list<int>[V]; }; // Method for adding undirected edge void Graph::addEdge(int u, int v) { this->adj[u].push_back(v); this->adj[v].push_back(u); }; // Method for Breadth First Search void Graph::BFS(list<int> *queue, bool *visited, int *parent) { int current = queue->front(); queue->pop_front(); list<int>::iterator i; for (i=adj[current].begin();i != adj[current].end();i++) { // If adjacent vertex is not visited earlier // mark it visited by assigning true value if (!visited[*i]) { // set current as parent of this vertex parent[*i] = current; // Mark this vertex visited visited[*i] = true; // Push to the end of queue queue->push_back(*i); } } }; // check for intersecting vertex int Graph::isIntersecting(bool *s_visited, bool *t_visited) { int intersectNode = -1; for(int i=0;i<V;i++) { // if a vertex is visited by both front // and back BFS search return that node // else return -1 if(s_visited[i] && t_visited[i]) return i; } return -1; }; // Print the path from source to target void Graph::printPath(int *s_parent, int *t_parent, int s, int t, int intersectNode) { vector<int> path; path.push_back(intersectNode); int i = intersectNode; while (i != s) { path.push_back(s_parent[i]); i = s_parent[i]; } reverse(path.begin(), path.end()); i = intersectNode; while(i != t) { path.push_back(t_parent[i]); i = t_parent[i]; } vector<int>::iterator it; cout<<"*****Path*****\n"; for(it = path.begin();it != path.end();it++) cout<<*it<<" "; cout<<"\n"; }; // Method for bidirectional searching int Graph::biDirSearch(int s, int t) { // boolean array for BFS started from // source and target(front and backward BFS) // for keeping track on visited nodes bool s_visited[V], t_visited[V]; // Keep track on parents of nodes // for front and backward search int s_parent[V], t_parent[V]; // queue for front and backward search list<int> s_queue, t_queue; int intersectNode = -1; // necessary initialization for(int i=0; i<V; i++) { s_visited[i] = false; t_visited[i] = false; } s_queue.push_back(s); s_visited[s] = true; // parent of source is set to -1 s_parent[s]=-1; t_queue.push_back(t); t_visited[t] = true; // parent of target is set to -1 t_parent[t] = -1; while (!s_queue.empty() && !t_queue.empty()) { // Do BFS from source and target vertices BFS(&s_queue, s_visited, s_parent); BFS(&t_queue, t_visited, t_parent); // check for intersecting vertex intersectNode = isIntersecting(s_visited, t_visited); // If intersecting vertex is found // that means there exist a path if(intersectNode != -1) { cout << "Path exist between " << s << " and " << t << "\n"; cout << "Intersection at: " << intersectNode << "\n"; // print the path and exit the program printPath(s_parent, t_parent, s, t, intersectNode); exit(0); } } return -1; } // Driver code int main() { // no of vertices in graph int n=15; // source vertex int s=0; // target vertex int t=14; // create a graph given in above diagram Graph g(n); g.addEdge(0, 4); g.addEdge(1, 4); g.addEdge(2, 5); g.addEdge(3, 5); g.addEdge(4, 6); g.addEdge(5, 6); g.addEdge(6, 7); g.addEdge(7, 8); g.addEdge(8, 9); g.addEdge(8, 10); g.addEdge(9, 11); g.addEdge(9, 12); g.addEdge(10, 13); g.addEdge(10, 14); if (g.biDirSearch(s, t) == -1) cout << "Path don't exist between " << s << " and " << t << "\n"; return 0; }

Output:

Path exist between 0 and 14 Intersection at: 7 *****Path***** 0 4 6 7 8 10 14

**References**

This article is contributed by **Atul Kumar**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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