# Bidirectional Search

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; ` `} ` |

*chevron_right*

*filter_none*

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.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

## Recommended Posts:

- Best First Search (Informed Search)
- Iterative Deepening Search(IDS) or Iterative Deepening Depth First Search(IDDFS)
- Applications of Depth First Search
- Breadth First Search or BFS for a Graph
- Depth First Search or DFS for a Graph
- Top 10 Interview Questions on Depth First Search (DFS)
- Uniform-Cost Search (Dijkstra for large Graphs)
- Finding minimum vertex cover size of a graph using binary search
- Count number of ways to reach destination in a Maze using BFS
- Print the path between any two nodes of a tree | DFS
- Product of minimum edge weight between all pairs of a Tree
- Find the maximum cost path from the bottom-left corner to the top-right corner
- Implementing Generic Graph in Java
- Find the winner of the match | Multiple Queries