# State Space Tree for 4 Queen Problem

## What is 4 Queen Problem?

The 4 Queen Problem asks you to arrange four queens on a 4x4 chessboard so that none of them pose a threat to the other queens. Finding a solution that allows all four queens to be put on the board without creating any conflicts is the goal of this puzzle.

We must make sure that no two queens on the chessboard share the same row, column, or diagonal to find the solution to this puzzle. This indicates that there should only be one queen in each row and column, and there shouldn't be any queens on the same diagonal. You may get a setup where all four queens live in harmony by placing the queens in this way.

In the 4 Queen problem, we have a 4x4 chessboard, and the task is to place           four queens on the board in such a way that no two queens threaten each other.

Solution

Let's consider one possible solution:

Initial State: An empty chessboard.

### Selecting the first queen

We must first place the first queen on the chessboard.

• Any square in the first row can be chosen as the location for the first queen. Assume we decide to start with the square at location A1.
• As a result of this decision, the first state in our State Space Tree is created, signifying the position of the first queen.

### Placing the second queen

In order to place the second queen, we must first move to the second row and locate a square that is not under danger from the first queen.

• Let's assume that the second queen is placed in the square at B3.
• Our State Space Tree's second state, which is connected to the first state, is formed by this decision.

### Placing the third queen

Under order to place the third queen, we must go on to the third row and choose a square that is not under danger from the first or second queen.

• Let's say we decide to place the third queen on square C2.
• Our State Space Tree's third state, which is connected to the second state, is created by this choice.

### Placing the fourth queen

Under order to place the fourth queen, we get to the fourth row; we must locate a square that is not in danger from the queens that have already been set up.

• Let's assume that the fourth queen will be placed in the square at D4. In our State Space Tree, this option connects to the third state to form the fourth state.
• Moving on to the fourth row, we have to find a square that is not threatened by the previously placed queens.
• Let's say we choose the square at D4 as the position for the fourth queen.
• This choice becomes the fourth state in our State Space Tree, connected to the third state.

This method has allowed us to put all four queens successfully and conflict-free on the chessboard. Our State Space Tree's final state represents a workable solution to the 4 Queen Problem.

The State Space Tree enables us to explore different paths and combinations to find all possible answers or to determine the best solution for the problem.

### Importance of the State Space Tree in problem-solving

The State Space Tree is a key idea in problem solving as it offers a systematic                depiction of all potential states and transitions of an issue. By decomposing the problem-solving process into a tree-like structure, it facilitates the visualisation of the process and makes it easier to explore alternative paths and identify the best.

## Understanding the State Space Tree

### Definition and Purpose of the State Space Tree

A graphical representation of the state space of a problem is the State Space Tree, commonly referred to as the Search Tree. It includes of nodes that stand in for various states and edges that connect them, illuminating potential state transitions. With the help of the State Space Tree, we may assess several approaches, monitor our progress, and ultimately solve the issue by methodically exploring the full state space.

### Representation of Problem States in the Tree

Each node in the State Space Tree represents a distinct problem state. A state in the 4 Queen Problem relates to a specific configuration of queens on the chessboard. The empty chessboard that is the initial state is represented by the tree's root node. Applying legitimate moves or operations to the parent node after that produces child nodes.

To identify the best solution to challenging issues like the 4 Queen Problem, problem solvers can walk through numerous paths, monitor progress, and carefully explore the complete state space by knowing and using the State Space Tree.

## Placing 4 Queens on a 4x4 Chessboard

Explanation of the Chessboard and Queen Placement Constraints:

### What is Chessboard?

The chessboard is a square grid consisting of 16 squares arranged in a 4x4 configuration. Each square represents a possible position for placing a queen.

• The problem involves placing 4 queens on a 4x4 chessboard.
• The chessboard consists of 16 squares arranged in a 4x4 grid.

### Queen Placement Constraints

To prevent the queens from endangering one another, we must follow the restrictions listed below when positioning them on the chessboard.

• A queen cannot be put in the same row as another queen. There should be no more than one queen each row.
• A queen cannot be put in the same column as another queen. At most one queen should be present in each column.
• No two queens may be positioned on the same diagonal (point c). The chessboard's top-left to bottom-right and top-right to bottom-left diagonals are examples of diagonals.

### Placing the First Queen

Here are four possible locations for the first queen on the first row of the chessboard. The problem's starting point is created by this decision.

### Placing the Second Queen

The Second Queen can only be placed in one of the remaining three locations in the Second Row in order to meet the no same row and no same diagonal criteria. From the initial state, this produces child states..

### Placing the Third Queen

Considering the no same row and no same diagonal restrictions, the third queen can be put in one of the final two locations in the third row to satisfy the constraints. This results in more kid states.

### Placing the Fourth Queen

After all other queens have been placed, the fourth queen is put in the last open spot in the fourth row.

### Goal State

When all four queens are put on the chessboard without endangering one another, that is, when no two queens are placed in the same row, column, or diagonal, the goal state has been achieved.

### Goal of the Problem

To solve the puzzle, one must locate all four queens on the chessboard in a position that prevents them from endangering one another. No two queens should share the same row, column, or diagonal, according to the definition of "endangering" in this context.

The following are the specific prerequisites for reaching the goal:

• Non-attacking Queens: It is best to arrange the queens so that no two of them are in the same row, column, or diagonal. By doing this, they can be prevented from attacking one another in accordance with chess regulations.
• No Queen Conflicts: To prevent queen conflicts, each queen must be positioned in a different row. When two queens are arranged in a row, they are in the same line of attack and could potentially pose a threat to one another.
• No Queen Conflicts in the Columns: Each queen must be put in a different column. Two queens might attack one another if they were in the same column and on the same vertical line.
• No Diagonal Conflicts: Each queen must be positioned so that no two of her diagonals are shared. If two queens were on the same diagonal, which might be either ascending or descending, they could fight one another.

## Constructing the State Space Tree

1. Define the problem: Placing 4 Queens on a 4x4 Chessboard.

• Clarify the objective of placing 4 queens without threatening each other.

2. Set the initial state: Empty Chessboard.

• Begin with an empty 4x4 chessboard with no queens placed.

3. Generate child states:

Queen Placement at Level 1:

Place a queen in each of the 4 columns of the first row. Each placement creates a child state.

Queen Placement at Level 2:

Put a queen in each of the final three columns of the second row to create a child state for each Level 1 child state. By branching from Level 1 states, these placements establish new child states

Queen Placement at Level 3:

Place a queen in each of the final two columns of the third row to create a child state for each Level 2 child state. By branching from Level 2 states, these placements establish new child states.

Queen Placement at Level 4:

Place a queen in the final column of the fourth row for each child state at Level 3 to create child states. From Level 3 states, these placements new child states.

## Visualization of the State Space Tree

• Visualise the many paths and combinations by graphically representing the state space tree.
• The tree's levels correspond to rows, and each node symbolises a particular positioning of a queen on the chessboard.

## Analyzing the State Space Tree

1.  Establish the goal state(s):

Search for leaf nodes (end states) where four queens are positioned without interfering with one another.

These aim states adhere to the limitations of the issue.

2. Examine Different Paths in the Tree:

Travel through the state space tree to examine several paths that represent various queen placing sequences.

Assess each route to see if it leads to the desired state or a dead end.

3. Use Backtracking:

• When a path comes to a dead end or breaks the limits, go back to the prior state and investigate additional paths.
• The examination of every conceivable combination can be done without repetition by using backtracking.

4. Use Pruning Techniques:

• Pruning helps to optimise the search process by preventing wasteful exploration of incorrect or unpromising paths.
• It can be used to remove specific paths early based on limitations or heuristics.

By following these steps, the state space tree is constructed, allowing for systematic exploration and analysis of different queen placement combinations   on the chessboard. The goal is to identify the leaf nodes where 4 queens are placed without threatening each other, while applying backtracking and pruning techniques to efficiently search the state space.

## Algorithmic Approach to Solve the 4 Queen Problem

### A. Depth-First Search (DFS) Algorithm

Algorithmic Approach to Solve the 4 Queen Problem:

1. Definition:

• DFS is an algorithm that explores a path as far as it can go before turning around in a graph or tree structure.

2. Strategy:

• Recursively search the neighbours of a particular node or state starting from the node or state that was visited.
• Depth is prioritised over breadth; therefore it explores one branch in as much depth as possible.

3. Algorithm Steps:

1. Initialize an empty stack and mark the starting node or state as visited.

2. Push the starting node or state onto the stack.

3. While the stack is not empty:

• Pop the top node or state from the stack.
• Process the node or state.
• Explore the unvisited neighbours of the current node or state.
• If a neighbour is unvisited, mark it as visited and push it onto the stack.
• Repeat until all nodes or states are visited or the goal is reached.

4. Key Characteristics:

• The nodes or states are tracked using a stack (implemented with recursion or an explicit data structure).
• It explores a path as thoroughly as possible before turning around, visiting nodes or states in depth first.
• In sometimes, it might be a best or fastest route.

5. Applications:

• Navigating mazes and tracing routes over a graph.
• Coming up with combinations and permutations.
• Recognising connections or cycles in a graph.
• Directed acyclic graph (DAG) topological sorting.

We can examine paths systematically in the state space tree for the 4 Queen issue by using the DPS strategy.

1. Path Systems:

It enables to move through the tree, various queen placement arrangements , going backwards when necessary to locate the ideal goal state.

2. Recursive Algorithm Implementation:

• Use recursion to implement the depth-first search technique.
• To investigate child states and go backwards as necessary, recursive calls are made.

3. Creating Pseudocode to Represent the Algorithm:

• Create pseudocode to represent the algorithm at a high level that is independent of programming language.
• Pseudocode offers a detailed procedure for fixing the issue.

4. Initial State:

• No queens are initially placed on the chessboard; instead, the game starts with an empty board.

5. Create Child States:

• Create child states by putting queens at each level on the chessboard.
• Start with Level 1, then move on to the following levels, installing queens until all four are in place.

6. Restrictions Checking:

• Determine whether the present queen positioning violates any restrictions at each level.
• Queens cannot be placed in the same row, column, or diagonal,

7. Backtracking:

• If a constraint is broken, go back to the level before and try different locations.
• Backtracking enables investigation of other routes without having to use the same setups repeatedly.

8. Goal State(s):

• In the state space tree, locate the goal state(s).
• When four queens are positioned without endangering one another, the goal state(s) are reached.

9. Exploration and Pruning:

Consider alternate queen placement combinations as you explore various state space tree pathways.

• Use pruning strategies to get rid of bad pathways determined by restrictions or heuristics.
• Pruning improves search results by preventing pointless investigation.

10. Solution Retrieval:

• Retrieve the answers from the goal state(s) that adhere to the limitations of the challenge.
• The solutions indicate legitimate arrangements for the four queens on the four by four checkerboard.

The 4 Queen Problem can be successfully solved by using this algorithmic approach, DFS, recursion, and backtracking.  To find combinations that do not pose  to one another on the chessboard is made easier with the help of constraints checking, pruning, and state space tree exploration.

## Solution Example Using the State Space Tree

### Navigating the State Space Tree for the Four Queen Problem

Begin with an empty chessboard as the initial state.

• Create child states by arranging the queens in the first row's various columns.
• Advance to the following level and create kid states using the proper placements from the preceding level.
• Repeat these steps until you reach the fourth row.
• Navigate the state space tree by investigating all potential routes and retracing your steps as necessary.

### Choosing Valid Queen Configurations

1. Look at each leaf node in the state space tree:

• Examine the leaf nodes, which stand in for various queen positions on the chessboard.
• These arrangements represent the final conditions attained after investigating every avenue.

2. Determine the proper queen configurations:

• Determine whether each configuration complies with the limitations of the problem.
• Make certain that no two queens are in the same row, column, or diagonal.

3. No same row and  column:

• The limitation is broken if there are several queens in a row.
• Verify that there is one queen in each column precisely.
• The limitation is broken if there are many queens in a column.

4. No diagonal threats:

• Look over each queen's diagonals, including main and anti-diagonal.
• Be certain that no two queens share the same diagonal.
• Two queens can attack one another if they are on the same diagonal.

5. Determine the total number of valid configurations:

• Record all valid configurations that are encountered.
• The 4-queens problem has a solution in each viable configuration.
• Each valid configuration represents a solution to the 4-queens problem.

### Visual Representation of the Solution

1. Visually illustrate the correct queen placements on the chessboard:

• Produce a visual depiction of the solution to indicate the correct queen placements.
• A leaf node in the state space tree represents each valid configuration.

2. Use a graphic representation to see the solution:

• To represent the location of queens, use a 4x4 chessboard grid.
• Every square on the chessboard symbolises a possible location for a queen.

3.  Mark the queens' locations on the chessboard:

• Determine the queens' locations in the legitimate configurations gleaned from the state space tree.
• Mark the chessboard square that corresponds to the presence of a queen to indicate its presence.

4.  Show the placements:

• Display the chessboard with the noted queen positions for each legal combination.
• To demonstrate the lack of dangers, highlight the rows, columns, or diagonals where the queens are positioned.

5.  Highlight how distinctive each configuration:

• Each legal configuration must show a different configuration of queens on the chessboard.
• Aim for originality in the visual depiction by avoiding duplicating combinations.

## A. Techniques to Reduce Symmetry

• Identify Symmetries: Study the issue and look for rotational and reflecting symmetries in the queen placements and chessboard.
• Identify Symmetries and Define Symmetry Classes: Using the symmetries, identify the various symmetry classes.
• Pick Representative Configurations: Pick one representative configuration from each symmetry class to prevent duplicative investigation.
• Restrict the Search field: By restricting the search field to only the representative configurations, efficiency can be increased.

### B. Pruning and Early Backtracking Techniques

• Identify Constraint-Based Pruning: Early on in the search, identify restrictions or criteria that can be utilised to prune specific paths.
• Use Early Backtracking: Instead of continuing the search down that path when a partial configuration violates a constraint, backtrack right away.
• Prioritise promising pathways: To possibly find a solution more quickly, arrange the examination of pathways based on heuristics, concentrating on more promising combinations to search space by extrapolating new constraints from the incomplete configuration that is already in place
• Domain-Specific Pruning: Based on the specifics of the problem, use domain-specific knowledge or rules to eliminate specific configurations or routes.

### C. Algorithmic Improvements

• Iterative Deepening: Implement an iterative deepening approach to gradually increase the depth of the search, allowing for better resource allocation and potential early termination when a solution is found.
• Dynamic Ordering: Dynamically adjust the ordering of queen placements or child state generation based on heuristics or information gained during the search process.
• Parallelization: Explore parallel algorithms or techniques to distribute the search across multiple processors or threads, reducing the overall search time.
• Memory Optimization: Optimize memory usage by only storing necessary information and discarding irrelevant or redundant data during the search.
• that is necessary and delete any redundant or irrelevant information.

## Conclusion

To explore various queen placement arrangements on the chessboard, the State Space Tree offers a systematic depiction of all potential states and transitions in the 4 Queen Problem. The problem can be solved effectively and optimally by using optimisations including symmetry reduction, early backtracking, and pruning techniques, as well as the depth-first search algorithm and recursive implementation.