Best First Search (Informed Search)

Prerequisites : BFS, DFS 
In BFS and DFS, when we are at a node, we can consider any of the adjacent as next node. So both BFS and DFS blindly explore paths without considering any cost function. The idea of Best First Search is to use an evaluation function to decide which adjacent is most promising and then explore. Best First Search falls under the category of Heuristic Search or Informed Search. 

We use a priority queue to store costs of nodes. So the implementation is a variation of BFS, we just need to change Queue to PriorityQueue. 

// This pseudocode is adapted from below 
// source:
// https://courses.cs.washington.edu/
Best-First-Search(Grah g, Node start)
    1) Create an empty PriorityQueue
       PriorityQueue pq;
    2) Insert "start" in pq.
       pq.insert(start)
    3) Until PriorityQueue is empty
          u = PriorityQueue.DeleteMin
          If u is the goal
             Exit
          Else
             Foreach neighbor v of u
                If v "Unvisited"
                    Mark v "Visited"                    
                    pq.insert(v)
             Mark u "Examined"                    
End procedure

Let us consider the below example. 
 

BFS

We start from source "S" and search for
goal "I" using given costs and Best
First search.

pq initially contains S
We remove s from and process unvisited
neighbors of S to pq.
pq now contains {A, C, B} (C is put
before B because C has lesser cost)

We remove A from pq and process unvisited
neighbors of A to pq.
pq now contains {C, B, E, D}

We remove C from pq and process unvisited
neighbors of C to pq.
pq now contains {B, H, E, D}

We remove B from pq and process unvisited
neighbors of B to pq.
pq now contains {H, E, D, F, G}

We remove H from pq.  Since our goal
"I" is a neighbor of H, we return.

Below is the implementation of the above idea:



C++

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// C++ program to implement Best First Search using priority
// queue
#include <bits/stdc++.h>
using namespace std;
typedef pair<int, int> pi;
 
vector<vector<pi> > graph;
 
// Function for adding edges to graph
void addedge(int x, int y, int cost)
{
    graph[x].push_back(make_pair(cost, y));
    graph[y].push_back(make_pair(cost, x));
}
 
// Function For Implementing Best First Search
// Gives output path having lowest cost
void best_first_search(int source, int target, int n)
{
    vector<bool> visited(n, false);
    // MIN HEAP priority queue
    priority_queue<pi, vector<pi>, greater<pi> > pq;
    // sorting in pq gets done by first value of pair
    pq.push(make_pair(0, source));
    visited = true;
 
    while (!pq.empty()) {
        int x = pq.top().second;
        // Displaying the path having lowest cost
        cout << x << " ";
        pq.pop();
        if (x == target)
            break;
 
        for (int i = 0; i < graph[x].size(); i++) {
            if (!visited[graph[x][i].second]) {
                visited[graph[x][i].second] = true;
                pq.push(graph[x][i]);
            }
        }
    }
}
 
// Driver code to test above methods
int main()
{
    // No. of Nodes
    int v = 14;
    graph.resize(v);
 
    // The nodes shown in above example(by alphabets) are
    // implemented using integers addedge(x,y,cost);
    addedge(0, 1, 3);
    addedge(0, 2, 6);
    addedge(0, 3, 5);
    addedge(1, 4, 9);
    addedge(1, 5, 8);
    addedge(2, 6, 12);
    addedge(2, 7, 14);
    addedge(3, 8, 7);
    addedge(8, 9, 5);
    addedge(8, 10, 6);
    addedge(9, 11, 1);
    addedge(9, 12, 10);
    addedge(9, 13, 2);
 
    int source = 0;
    int target = 9;
 
    // Function call
    best_first_search(source, target, v);
 
    return 0;
}

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Output

0 1 3 2 8 9 

Analysis : 

  • worst case time complexity for Best First Search is O(n * Log n) where n is number of nodes. In worst case, we may have to visit all nodes before we reach goal. Note that priority queue is implemented using Min(or Max) Heap, and insert and remove operations take O(log n) time.
  • Performance of the algorithm depends on how well the cost or evaluation function is designed.

Related Article: 
A* Search Algorithm 
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