SBS (serial backward search)

Serial Backward Search (SBS) is a search algorithm used for finding solutions or patterns in combinatorial optimization problems. It is particularly applicable to problems where the solution can be represented as a sequence of elements, such as permutations or strings. SBS is a variation of the Backtracking algorithm that systematically explores the search space by examining all possible solutions in a systematic manner.

Here's how SBS works:

1. Problem representation: The problem is defined, and a representation is chosen to encode the potential solutions. For example, if we are searching for a permutation of a set of elements, the solution can be represented as an array where the position of each element represents its position in the permutation.
2. Objective function: An objective function is defined to evaluate the quality or fitness of a solution. The objective function assigns a score or value to each potential solution based on problem-specific criteria. The goal is usually to maximize or minimize this objective function.
3. Search space: The search space is the set of all possible solutions to the problem. In the case of permutations, the search space consists of all possible orderings of the elements.
4. Initialization: The SBS algorithm starts with an initial solution, which can be an empty solution or a randomly generated one. This initial solution serves as the starting point for the search.
5. Backward search: SBS performs a backward search by iteratively modifying the current solution to explore the search space. It systematically moves backward from the current solution to the previous one, evaluating each potential candidate along the way.
6. Candidate generation: At each step of the backward search, SBS generates a set of candidate solutions by making a single modification to the current solution. This modification can involve swapping two elements, inserting an element, or removing an element, depending on the problem's constraints.
7. Solution evaluation: Each candidate solution generated in the previous step is evaluated using the objective function. The objective function determines the quality of the solution and assigns it a score or value.
8. Pruning: SBS uses pruning techniques to eliminate unpromising candidate solutions early in the search process. Pruning involves checking if a candidate solution can potentially lead to a better solution than the current best solution found so far. If not, the candidate solution is discarded, reducing the search space.
9. Solution update: If a candidate solution is deemed better than the current best solution, it becomes the new best solution.
10. Termination condition: The backward search continues until a termination condition is met. This condition can be a fixed number of iterations, a predefined time limit, or a satisfactory solution that meets a specific criterion.
11. Output: Once the termination condition is reached, the algorithm outputs the best solution found during the search.

SBS is a systematic and exhaustive search algorithm, which means it guarantees finding an optimal solution if given enough time and resources. However, due to the combinatorial nature of many optimization problems, the search space can become very large, making it computationally expensive to explore every possible solution. Therefore, SBS is often used in practice for smaller problem instances or combined with other techniques, such as heuristics or pruning strategies, to reduce the search space and improve efficiency.