Best First Search (Informed Search)

In BFS and DFS, when we are at a node, we can consider any of the adjacent as the 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.

Implementation of Best First Search:

We use a priority queue or heap to store the costs of nodes that have the lowest evaluation function value. So the implementation is a variation of BFS, we just need to change Queue to PriorityQueue. 

// Pseudocode for Best First Search
Best-First-Search(Graph 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

Illustration:

Let us consider the below example:

• 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 pq 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++

// 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 actual_Src, 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, actual_Src));
    int s = actual_Src;
    visited[s] = 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(make_pair(graph[x][i].first,graph[x][i].second));
            }
        }
    }
}

// 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;
}

Java

// Java program to implement Best First Search using priority
// queue
import java.util.ArrayList;
import java.util.PriorityQueue;

public class GFG
{
  static ArrayList<ArrayList<edge> > adj = new ArrayList<>();

  // Function for adding edges to graph
  static class edge implements Comparable<edge>
  {
    int v, cost;
    edge(int v, int cost)
    {
      this.v = v;
      this.cost = cost;
    }
    @Override public int compareTo(edge o)
    {
      if (o.cost < cost) {
        return 1;
      }
      else
        return -1;
    }
  }

  public GFG(int v)
  {
    for (int i = 0; i < v; i++) {
      adj.add(new ArrayList<>());
    }
  }

  // Function For Implementing Best First Search
  // Gives output path having lowest cost
  static void best_first_search(int source, int target, int v)
  {

    // MIN HEAP priority queue
    PriorityQueue<edge> pq = new PriorityQueue<>();
    boolean visited[] = new boolean[v];
    visited = true;

    // sorting in pq gets done by first value of pair
    pq.add(new edge(source, -1));
    while (!pq.isEmpty()) {
      int x = pq.poll().v;

      // Displaying the path having lowest cost
      System.out.print(x + " ");
      if (target == x) {
        break;
      }
      for (edge adjacentNodeEdge : adj.get(x)) {
        if (!visited[adjacentNodeEdge.v]) {
          visited[adjacentNodeEdge.v] = true;
          pq.add(adjacentNodeEdge);
        }
      }
    }
  }
  void addedge(int u, int v, int cost)
  {
    adj.get(u).add(new edge(v, cost));
    adj.get(v).add(new edge(u, cost));
  }

  // Driver code to test above methods
  public static void main(String args[])
  {

    // No. of Nodes
    int v = 14;

    // The nodes shown in above example(by alphabets) are
    // implemented using integers addedge(x,y,cost);
    GFG graph = new GFG(v);
    graph.addedge(0, 1, 3);
    graph.addedge(0, 2, 6);
    graph.addedge(0, 3, 5);
    graph.addedge(1, 4, 9);
    graph.addedge(1, 5, 8);
    graph.addedge(2, 6, 12);
    graph.addedge(2, 7, 14);
    graph.addedge(3, 8, 7);
    graph.addedge(8, 9, 5);
    graph.addedge(8, 10, 6);
    graph.addedge(9, 11, 1);
    graph.addedge(9, 12, 10);
    graph.addedge(9, 13, 2);

    int source = 0;
    int target = 9;

    // Function call
    best_first_search(source, target, v);
  }
}

// This code is contributed by Prithi_Dey

Python3

from queue import PriorityQueue
v = 14
graph = [[] for i in range(v)]

# Function For Implementing Best First Search
# Gives output path having lowest cost


def best_first_search(actual_Src, target, n):
    visited = [False] * n
    pq = PriorityQueue()
    pq.put((0, actual_Src))
    visited[actual_Src] = True

    while pq.empty() == False:
        u = pq.get()[1]
        # Displaying the path having lowest cost
        print(u, end=" ")
        if u == target:
            break

        for v, c in graph[u]:
            if visited[v] == False:
                visited[v] = True
                pq.put((c, v))
    print()

# Function for adding edges to graph


def addedge(x, y, cost):
    graph[x].append((y, cost))
    graph[y].append((x, cost))


# 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)

source = 0
target = 9
best_first_search(source, target, v)

# This code is contributed by Jyotheeswar Ganne

C#

using System;
using System.Collections.Generic;

using System.Collections;
// C# program to implement Best First Search using priority
// queue


class HelloWorld {

    public static LinkedList<Tuple<int, int>>[] graph;

    // Function for adding edges to graph
    public static void addedge(int x, int y, int cost)
    {
        graph[x].AddLast(new Tuple<int, int>(cost, y));
        graph[y].AddLast(new Tuple<int, int>(cost, x));
    }

    // Function for finding the minimum weight element.
    public static Tuple<int,int> get_min(LinkedList<Tuple<int,int>> pq){
        // Assuming the maximum wt can be of 1e5.
        Tuple<int,int> curr_min = new Tuple<int,int>(100000, 100000);
        foreach(var ele in pq){
            if(ele.Item1 == curr_min.Item1){
                if(ele.Item2 < curr_min.Item2){
                    curr_min = ele;
                }
            }
            else{
                if(ele.Item1 < curr_min.Item1){
                    curr_min = ele;
                }
            }
        }

        return curr_min;
    }

    // Function For Implementing Best First Search
    // Gives output path having lowest cost
    public static void best_first_search(int actual_Src, int target, int n)
    {
        int[] visited = new int[n];
        for(int i = 0; i < n; i++){
            visited[i] = 0;
        }

        // MIN HEAP priority queue
        LinkedList<Tuple<int, int>> pq = new LinkedList<Tuple<int,int>>();

        // sorting in pq gets done by first value of pair
        pq.AddLast(new Tuple<int, int> (0, actual_Src));
        int s = actual_Src;
        visited[s] = 1;
        while (pq.Count > 0) {

            Tuple<int,int> curr_min = get_min(pq);
            int x = curr_min.Item2;
            pq.Remove(curr_min);

            Console.Write(x + " ");
            // Displaying the path having lowest cost
            if (x == target)
                break;

            LinkedList<Tuple<int,int>> list = graph[x];
            foreach(var val in list)
            {
                if (visited[val.Item2] == 0) {
                    visited[val.Item2] = 1;
                    pq.AddLast(new Tuple<int,int>(val.Item1, val.Item2));
                }
            }

        }
    }

    static void Main() {
        // No. of Nodes
        int v = 14;
        graph = new LinkedList<Tuple<int, int>>[v];
        for (int i = 0; i < graph.Length; ++i){
            graph[i] = new LinkedList<Tuple<int, int>>();
        }

        // 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);
    }
}

// The code is contributed by Nidhi goel.

Javascript

<script>
// javascript program to implement Best First Search using priority
// queue


// Function for adding edges to graph
function addedge(x, y, cost)
{
    graph[x].push([cost, y]);
    graph[y].push([cost, x]);
}

// Function For Implementing Best First Search
// Gives output path having lowest cost
function best_first_search(actual_Src, target, n)
{
    let visited = new Array(n).fill(false);

    // MIN HEAP priority queue
    let pq = [];

    // sorting in pq gets done by first value of pair
    pq.push([0, actual_Src]);
    let s = actual_Src;
    visited[s] = true;
    while (pq.length > 0) {
        let x = pq[0][1];
        // Displaying the path having lowest cost
        document.write(x + " ");
        pq.shift();
        if (x == target) break;

        for (let i = 0; i < graph[x].length; i++) {
            if (visited[graph[x][i][1]] == 0) {
                visited[graph[x][i][1]] = true;
                pq.push([graph[x][i][0],graph[x][i][1]]);
                pq.sort((a, b)=>{
                    if(a[0] != b[0]){
                        return a[0]-b[0];
                    }
                    return a[1]-b[1];
                });
            }
        }
    }
}

// Driver code to test above methods

// No. of Nodes
let v = 14;
let graph = new Array(v);
for(let i = 0; i < v; i++){
    graph[i] = new Array(0);
}

// 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);

let source = 0;
let target = 9;

// Function call
best_first_search(source, target, v);

// The code is contributed by Nidhi goel.
</script>

Output

0 1 3 2 8 9 

Analysis :

• The worst-case time complexity for Best First Search is O(n * log n) where n is the number of nodes. In the 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.
• The performance of the algorithm depends on how well the cost or evaluation function is designed.

Special cases of Best first search:

1. Greedy Best first search algorithm
2. A* search algorithm

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