The **local beam search** algorithm keeps track of k states rather than just one. It begins with k randomly generated states. At each step, all the successors of all k states are generated. If any one is a goal, the algorithm halts. Otherwise, it selects the k best successors from the complete list and repeats

At first sight, a local beam search with k states might seem to be nothing more than running k random restarts in parallel instead of in sequence. In fact, the two algorithms are quite different. In a random-restart search, each search process runs independently of the others.

In a local beam search, useful information is passed among the parallel search threads. In effect, the states that generate the best successors say to the others, “Come over here, the grass is greener!” The algorithm quickly abandons unfruitful searches and moves its resources to where the most progress is being made

In its simplest form, local beam search can suffer from a lack of diversity among the k states—they can quickly become concentrated in a small region of the state space, making the search little more than an expensive version of hill climbing.

A variant called stochastic beam search, analogous to stochastic hill climbing, helps alleviate this problem.

Instead of choosing the best k from the the pool of candidate successors, stochastic beam search chooses k successors at random, with the probability of choosing a given successor being an increasing function of its value. Stochastic beam search bears some resemblance to the process of natural selection, whereby the “successors” (offspring) of a “state” (organism) populate the next generation according to its “value” (fitness).

In the field of artificial intelligence, search algorithms play a vital role in problem-solving and decision-making processes. Local beam search is one such powerful algorithm that enables AI systems to efficiently explore and find optimal solutions within a given search space. In this article, we will delve into the concept of local beam search, its working principles, and provide illustrative examples to enhance your understanding.

**Understanding Local Beam Search:**

Local beam search is a heuristic search algorithm that is widely used in various AI applications, including problem-solving, optimization, and planning. It belongs to the family of local search algorithms, which focus on improving the current solution iteratively rather than exhaustively exploring the entire search space.

Working Principles:

**Initialization:**

The local beam search algorithm begins by initializing a set of randomly generated candidate solutions, called the “beam.” The beam represents a population of potential solutions to the problem at hand.**Evaluation:**

Each candidate solution in the beam is evaluated using a scoring function or evaluation metric. This function measures the quality or fitness of a solution based on problem-specific criteria. The evaluation helps in ranking the solutions within the beam.**Expansion:**

In the expansion phase, new candidate solutions are generated by applying operators or transformations to the existing solutions in the beam. These operators aim to produce neighboring solutions that may potentially be better than the current ones.**Selection:**

After the expansion phase, the newly generated solutions, along with the original solutions, are evaluated using the scoring function. A predetermined number of the best solutions are selected to form the new beam for the next iteration. This selection process ensures that the beam retains the most promising solutions.**Termination:**

The algorithm iterates through the expansion and selection steps until a termination condition is met. This condition can be a maximum number of iterations, a predefined threshold for the quality of solutions, or any other stopping criterion based on the specific problem.

**Example: Traveling Salesman Problem (TSP):**

To illustrate the application of local beam search, let’s consider the classic Traveling Salesman Problem. In this problem, a salesman needs to find the shortest possible route that visits a set of cities and returns to the starting city.

Suppose we have five cities: A, B, C, D, and E. The local beam search algorithm starts with an initial beam of randomly generated tours, each representing a potential solution. The tours are evaluated based on their total distance traveled. Then, through the expansion phase, new tours are created by swapping the order of two cities in a given tour.

Let’s say the initial beam contains the following tours:

- A -> B -> C -> D -> E -> A
- A -> C -> B -> E -> D -> A
- A -> D -> E -> C -> B -> A

After evaluation, the best two tours are selected based on the total distance traveled. Let’s assume the first two tours are chosen for the next iteration:

- A -> B -> C -> D -> E -> A
- A -> C -> B -> E -> D -> A

Through the expansion phase, new tours are generated by swapping cities within the selected tours. The evaluation is performed again, and the best solutions are selected to form the new beam. This process continues until the termination condition is met or the optimal solution is found.

Here’s an example implementation of the Local Beam Search algorithm in Java:

```
import java.util.ArrayList;
import java.util.Collections;
import java.util.List;
import java.util.Random;
public class LocalBeamSearch {
private static final int BEAM_WIDTH = 5; // Number of candidate solutions in the beam
private static final int MAX_ITERATIONS = 100; // Maximum number of iterations
public static void main(String[] args) {
// Define the problem-specific data or representation
// Example: TSP cities
List<City> cities = new ArrayList<>();
cities.add(new City("A", 0, 0));
cities.add(new City("B", 1, 3));
cities.add(new City("C", 4, 1));
cities.add(new City("D", 3, 5));
cities.add(new City("E", 6, 2));
// Initialize the beam with random tours
List<List<City>> beam = new ArrayList<>();
Random random = new Random();
for (int i = 0; i < BEAM_WIDTH; i++) {
List<City> tour = new ArrayList<>(cities);
Collections.shuffle(tour);
beam.add(tour);
}
// Perform local beam search
int iteration = 0;
while (iteration < MAX_ITERATIONS) {
// Evaluate the tours in the beam
List<City> bestTour = null;
double bestDistance = Double.MAX_VALUE;
for (List<City> tour : beam) {
double distance = calculateTotalDistance(tour);
if (distance < bestDistance) {
bestDistance = distance;
bestTour = tour;
}
}
// Termination condition: If a satisfactory solution is found
if (bestDistance == 0.0) {
break;
}
// Generate new tours through expansion
List<List<City>> newBeam = new ArrayList<>();
for (List<City> tour : beam) {
for (int i = 0; i < cities.size() - 1; i++) {
for (int j = i + 1; j < cities.size(); j++) {
List<City> newTour = new ArrayList<>(tour);
Collections.swap(newTour, i, j);
newBeam.add(newTour);
}
}
}
// Select the best tours for the next iteration
Collections.sort(newBeam, (tour1, tour2) -> Double.compare(calculateTotalDistance(tour1), calculateTotalDistance(tour2)));
beam = newBeam.subList(0, BEAM_WIDTH);
iteration++;
}
// Print the best tour found
System.out.println("Best Tour: " + beam.get(0));
System.out.println("Total Distance: " + calculateTotalDistance(beam.get(0)));
}
// Calculate the total distance traveled in a tour
private static double calculateTotalDistance(List<City> tour) {
double totalDistance = 0.0;
for (int i = 0; i < tour.size() - 1; i++) {
City currentCity = tour.get(i);
City nextCity = tour.get(i + 1);
totalDistance += calculateDistance(currentCity, nextCity);
}
return totalDistance;
}
// Calculate the Euclidean distance between two cities
private static double calculateDistance(City city1, City city2) {
int xDiff = city1.getX() - city2.getX();
int yDiff = city1.getY() - city2.getY();
return Math.sqrt(xDiff * xDiff + yDiff * yDiff);
}
// City class representing the coordinates of a city
private static class City {
private String name;
private int x;
private int y;
public City(String name, int x, int y) {
this.name = name;
this.x = x;
this.y = y;
}
public int getX() {
return x;
}
public int getY() {
return y;
}
@Override
public String toString() {
return name;
}
}
}
```

**Conclusion:**

Local beam search is a valuable algorithm in the domain of artificial intelligence, allowing systems to efficiently search for optimal solutions within a given search space. Its ability to explore the solution space iteratively, focusing on promising areas, makes it a popular choice for various AI applications. By understanding the working principles and exploring examples like the Traveling Salesman Problem, we gain insights into the potential of local