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 from both direction simultaneously.
Bidirectional search is a graph search algorithm which find smallest path from source to goal vertex. It runs two simultaneous search -
- Forward search from source/initial vertex toward goal vertex
- Backward search from 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 O(bd). On the other hand, if we execute two search operation then the complexity would be O(bd/2) for each search and total complexity would be O(bd/2 +bd/2) which is far less than O(bd).
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 O(bd/2).
Below is very simple implementation representing the concept of bidirectional search using BFS. This implementation considers undirected paths without any weight.
C++ // 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; } // Java program for Bidirectional BFS search // to check path between two vertices import java.io.*; import java.util.*; // class representing undirected graph // using adjacency list class Graph { // number of nodes in graph private int V; // Adjacency list private LinkedList<Integer>[] adj; // Constructor @SuppressWarnings("unchecked") public Graph(int v) { V = v; adj = new LinkedList[v]; for (int i = 0; i < v; i++) adj[i] = new LinkedList<Integer>(); } // Method for adding undirected edge public void addEdge(int u, int v) { adj[u].add(v); adj[v].add(u); } // Method for Breadth First Search public void bfs(Queue<Integer> queue, Boolean[] visited, int[] parent) { int current = queue.poll(); for (int i : adj[current]) { // 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.add(i); } } } // check for intersecting vertex public int isIntersecting(Boolean[] s_visited, Boolean[] t_visited) { 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 public void printPath(int[] s_parent, int[] t_parent, int s, int t, int intersectNode) { LinkedList<Integer> path = new LinkedList<Integer>(); path.add(intersectNode); int i = intersectNode; while (i != s) { path.add(s_parent[i]); i = s_parent[i]; } Collections.reverse(path); i = intersectNode; while (i != t) { path.add(t_parent[i]); i = t_parent[i]; } System.out.println("*****Path*****"); for (int it : path) System.out.print(it + " "); System.out.println(); } // Method for bidirectional searching public int biDirSearch(int s, int t) { // Booleanean array for BFS started from // source and target(front and backward BFS) // for keeping track on visited nodes Boolean[] s_visited = new Boolean[V]; Boolean[] t_visited = new Boolean[V]; // Keep track on parents of nodes // for front and backward search int[] s_parent = new int[V]; int[] t_parent = new int[V]; // queue for front and backward search Queue<Integer> s_queue = new LinkedList<Integer>(); Queue<Integer> t_queue = new LinkedList<Integer>(); int intersectNode = -1; // necessary initialization for (int i = 0; i < V; i++) { s_visited[i] = false; t_visited[i] = false; } s_queue.add(s); s_visited[s] = true; // parent of source is set to -1 s_parent[s] = -1; t_queue.add(t); t_visited[t] = true; // parent of target is set to -1 t_parent[t] = -1; while (!s_queue.isEmpty() && !t_queue.isEmpty()) { // 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) { System.out.printf( "Path exist between %d and %d\n", s, t); System.out.printf("Intersection at: %d\n", intersectNode); // print the path and exit the program printPath(s_parent, t_parent, s, t, intersectNode); System.exit(0); } } return -1; } } public class GFG { // Driver code public static void main(String[] args) { // 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 = new Graph(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) System.out.printf( "Path don't exist between %d and %d", s, t); } } // This code is contributed by cavi4762. # Python3 program for Bidirectional BFS # Search to check path between two vertices # Class definition for node to # be added to graph class AdjacentNode: def __init__(self, vertex): self.vertex = vertex self.next = None # BidirectionalSearch implementation class BidirectionalSearch: def __init__(self, vertices): # Initialize vertices and # graph with vertices self.vertices = vertices self.graph = [None] * self.vertices # Initializing queue for forward # and backward search self.src_queue = list() self.dest_queue = list() # Initializing source and # destination visited nodes as False self.src_visited = [False] * self.vertices self.dest_visited = [False] * self.vertices # Initializing source and destination # parent nodes self.src_parent = [None] * self.vertices self.dest_parent = [None] * self.vertices # Function for adding undirected edge def add_edge(self, src, dest): # Add edges to graph # Add source to destination node = AdjacentNode(dest) node.next = self.graph[src] self.graph[src] = node # Since graph is undirected add # destination to source node = AdjacentNode(src) node.next = self.graph[dest] self.graph[dest] = node # Function for Breadth First Search def bfs(self, direction = 'forward'): if direction == 'forward': # BFS in forward direction current = self.src_queue.pop(0) connected_node = self.graph[current] while connected_node: vertex = connected_node.vertex if not self.src_visited[vertex]: self.src_queue.append(vertex) self.src_visited[vertex] = True self.src_parent[vertex] = current connected_node = connected_node.next else: # BFS in backward direction current = self.dest_queue.pop(0) connected_node = self.graph[current] while connected_node: vertex = connected_node.vertex if not self.dest_visited[vertex]: self.dest_queue.append(vertex) self.dest_visited[vertex] = True self.dest_parent[vertex] = current connected_node = connected_node.next # Check for intersecting vertex def is_intersecting(self): # Returns intersecting node # if present else -1 for i in range(self.vertices): if (self.src_visited[i] and self.dest_visited[i]): return i return -1 # Print the path from source to target def print_path(self, intersecting_node, src, dest): # Print final path from # source to destination path = list() path.append(intersecting_node) i = intersecting_node while i != src: path.append(self.src_parent[i]) i = self.src_parent[i] path = path[::-1] i = intersecting_node while i != dest: path.append(self.dest_parent[i]) i = self.dest_parent[i] print("*****Path*****") path = list(map(str, path)) print(' '.join(path)) # Function for bidirectional searching def bidirectional_search(self, src, dest): # Add source to queue and mark # visited as True and add its # parent as -1 self.src_queue.append(src) self.src_visited[src] = True self.src_parent[src] = -1 # Add destination to queue and # mark visited as True and add # its parent as -1 self.dest_queue.append(dest) self.dest_visited[dest] = True self.dest_parent[dest] = -1 while self.src_queue and self.dest_queue: # BFS in forward direction from # Source Vertex self.bfs(direction = 'forward') # BFS in reverse direction # from Destination Vertex self.bfs(direction = 'backward') # Check for intersecting vertex intersecting_node = self.is_intersecting() # If intersecting vertex exists # then path from source to # destination exists if intersecting_node != -1: print(f"Path exists between {src} and {dest}") print(f"Intersection at : {intersecting_node}") self.print_path(intersecting_node, src, dest) exit(0) return -1 # Driver code if __name__ == '__main__': # Number of Vertices in graph n = 15 # Source Vertex src = 0 # Destination Vertex dest = 14 # Create a graph graph = BidirectionalSearch(n) graph.add_edge(0, 4) graph.add_edge(1, 4) graph.add_edge(2, 5) graph.add_edge(3, 5) graph.add_edge(4, 6) graph.add_edge(5, 6) graph.add_edge(6, 7) graph.add_edge(7, 8) graph.add_edge(8, 9) graph.add_edge(8, 10) graph.add_edge(9, 11) graph.add_edge(9, 12) graph.add_edge(10, 13) graph.add_edge(10, 14) out = graph.bidirectional_search(src, dest) if out == -1: print(f"Path does not exist between {src} and {dest}") # This code is contributed by Nirjhari Jankar // C# program for Bidirectional BFS search // to check path between two vertices using System; using System.Collections.Generic; // class representing undirected graph // using adjacency list public class Graph { // number of nodes in graph private int V; // Adjacency list private List<int>[] adj; // Constructor public Graph(int v) { V = v; adj = new List<int>[ v ]; for (int i = 0; i < v; i++) adj[i] = new List<int>(); } // Method for adding undirected edge public void AddEdge(int u, int v) { adj[u].Add(v); adj[v].Add(u); } // Method for Breadth First Search public void BFS(Queue<int> queue, bool[] visited, int[] parent) { int current = queue.Dequeue(); foreach(int i in adj[current]) { // 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.Enqueue(i); } } } // check for intersecting vertex public int IsIntersecting(bool[] s_visited, bool[] t_visited) { 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 public void PrintPath(int[] s_parent, int[] t_parent, int s, int t, int intersectNode) { List<int> path = new List<int>(); path.Add(intersectNode); int i = intersectNode; while (i != s) { path.Add(s_parent[i]); i = s_parent[i]; } path.Reverse(); i = intersectNode; while (i != t) { path.Add(t_parent[i]); i = t_parent[i]; } Console.WriteLine("*****Path*****"); foreach(int it in path) Console.Write(it + " "); Console.WriteLine(); } // Method for bidirectional searching public int 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 = new bool[V]; bool[] t_visited = new bool[V]; // Keep track on parents of nodes // for front and backward search int[] s_parent = new int[V]; int[] t_parent = new int[V]; // queue for front and backward search Queue<int> s_queue = new Queue<int>(); Queue<int> t_queue = new Queue<int>(); int intersectNode = -1; // necessary initialization for (int i = 0; i < V; i++) { s_visited[i] = false; t_visited[i] = false; } s_queue.Enqueue(s); s_visited[s] = true; // parent of source is set to -1 s_parent[s] = -1; t_queue.Enqueue(t); t_visited[t] = true; // parent of target is set to -1 t_parent[t] = -1; while (s_queue.Count > 0 && t_queue.Count > 0) { // 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) { Console.WriteLine( "Path exist between {0} and {1}", s, t); Console.WriteLine("Intersection at: {0}", intersectNode); // print the path and exit the program PrintPath(s_parent, t_parent, s, t, intersectNode); Environment.Exit(0); } } return -1; } } public class GFG { // Driver code static void Main(string[] args) { // 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 = new Graph(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) Console.WriteLine( "Path don't exist between {0} and {1}", s, t); } } // This code is contributed by cavi4762. class Graph { // Constructor to initialize the graph with V vertices. constructor(V) { this.V = V; // Number of vertices // Adjacency list representation this.adj = new Array(V).fill().map(() => []); } // Function to add an edge between vertices u and v. addEdge(u, v) { this.adj[u].push(v); this.adj[v].push(u); } // Breadth-First Search (BFS) starting from a given source node. BFS(queue, visited, parent) { const current = queue.shift(); for (const neighbor of this.adj[current]) { if (!visited[neighbor]) { parent[neighbor] = current; visited[neighbor] = true; queue.push(neighbor); } } } // Check if there is an intersection between two BFS searches. isIntersecting(s_visited, t_visited) { for (let i = 0; i < this.V; i++) { if (s_visited[i] && t_visited[i]) { return i; // Return the intersecting node } } return -1; // No intersection found } // Print the path from source 's' to target 't' through the intersection node. printPath(s_parent, t_parent, s, t, intersectNode) { const path = []; path.push(intersectNode); let i = intersectNode; while (i !== s) { path.push(s_parent[i]); i = s_parent[i]; } path.reverse(); i = intersectNode; while (i !== t) { path.push(t_parent[i]); i = t_parent[i]; } console.log("*****Path*****"); console.log(path.join(' ')); } // Bidirectional search to find the shortest path between 's' and 't'. biDirSearch(s, t) { const s_visited = new Array(this.V).fill(false); const t_visited = new Array(this.V).fill(false); const s_parent = new Array(this.V).fill(-1); const t_parent = new Array(this.V).fill(-1); const s_queue = []; const t_queue = []; let intersectNode = -1; // Start BFS from the source node 's'. s_queue.push(s); s_visited[s] = true; s_parent[s] = -1; // Start BFS from the target node 't'. t_queue.push(t); t_visited[t] = true; t_parent[t] = -1; // Continue BFS until an intersection is found or both searches are exhausted. while (s_queue.length > 0 && t_queue.length > 0) { this.BFS(s_queue, s_visited, s_parent); this.BFS(t_queue, t_visited, t_parent); intersectNode = this.isIntersecting(s_visited, t_visited); if (intersectNode !== -1) { console.log(`Path exists between ${s} and ${t}`); console.log(`Intersection at: ${intersectNode}`); this.printPath(s_parent, t_parent, s, t, intersectNode); return; } } console.log(`Path does not exist between ${s} and ${t}`); } } // Driver code const n = 15; const s = 0; const t = 14; const g = new Graph(n); // Adding edges to the graph. 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); // Perform bidirectional search to find the path from 's' to 't'. g.biDirSearch(s, t); Output:
Path exist between 0 and 14
Intersection at: 7
*****Path*****
0 4 6 7 8 10 14
References