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 –

1. Forward search form source/initial vertex toward goal vertex
2. 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-

1. Both initial and goal states are unique and completely defined.
2. 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

• https://en.wikipedia.org/wiki/Bidirectional_search

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