Question

Apply the Backtracking Algorithm for the \(n\) -Quecns Problem (Algorithm 5.1) to the problem instance in which \(n=8\), and show the actions step by step. Draw the pruned state space tree produced by this algorithm up to the point where the first solution is found.

Short answer

The backtracking algorithm starts by placing a queen in a cell in the first row. It then goes row by row, looking for a safe place to put the next queen. If it can't find a place to put a queen in a row, it goes back to the previous row (backtracks) and moves the queen there. This process continues until all 8 queens have been placed on the board without threatening each other. The pruned state space tree visualizes this process by showing each step and backtracking when no safe cell is found.

Step-by-step solution

Initialize the Chessboard

First, an 8x8 chessboard needs to be initialized. Each cell in the board is set to 0, indicating that no queen is placed.

Determine the Safe Zone

Next, determine if a queen can be placed at a given cell in the chessboard. To do this, check the row, column, and two diagonals for a queen (value=1). If a queen is not present, it is safe to place a queen in the current cell.

Start Placing Queens

Begin by placing the first queen in the safe cell of the first row. Now, move to the next row and apply the backtracking algorithm. Find a safe cell and place a queen there. If no safe cell is found, backtrack to the previous row and change the queen's position.

Continue and Repeat

Continue to the next row and repeat the backtracking process. If no safe cell is found in any row, go back to previous row and change the position of queen.

Reach the Final Solution

Continue this process until all the queens are placed on the board safely without threatening each other according to the chess rules. This is the final solution of the 8-queens problem.

Draw the Pruned State Space Tree

In the pruned state space tree, a node represents an intermediate status of the solution. The tree is generated level by level where each level adds another queen to a newer row. Starting from the root of the tree, branches are followed until a solution is found.

Key concepts

N-Queens Problem

The N-Queens Problem is a classic example of a backtracking algorithm used to solve constraint satisfaction problems. The challenge is to place \(n\) queens on an \(n \times n\) chessboard such that no two queens threaten each other. This means that no two queens can share the same row, column, or diagonal. This is because a queen can move any number of squares along a row, column, or diagonal on a chessboard. Therefore, finding a proper arrangement of queens requires checking multiple constraints in a systematic manner. The N-Queens Problem is a popular topic of study in computer science because it exemplifies the effectiveness and efficiency of backtracking when designed well. Backtracking involves exploring possible solutions and undoing decisions if they don’t lead to a solution, making it an excellent method for exploring large solution spaces like in the N-Queens Problem.

State Space Tree

In the context of the N-Queens Problem, a state space tree is a conceptual tool used to illustrate the decision-making process of the backtracking algorithm. Each node in the tree represents a partial solution, which is a configuration of queens placed on a part of the chessboard.

• The root node represents an empty board.
• Each level adds a queen to the board, with branches showing different options for queen placement.
• A leaf node represents a complete arrangement of queens that satisfies all constraints, or a path that leads to a dead-end or conflict.

Visualizing the state space tree helps in understanding how the algorithm explores different configurations before arriving at a solution. It prunes parts of the tree—meaning it cuts off branches—that cannot possibly lead to a valid solution, which helps in reducing the number of possibilities the algorithm needs to explore.

Chessboard Initialization

The very first step in solving the N-Queens Problem is initializing the chessboard. In this step, an \(n \times n\) board is prepared where every cell is empty and ready to accommodate queens.
This initialization can be visualized as filling up the entire board with zeros. Here:

• Each zero represents an empty cell.
• As queens are placed on the board, these zeros are changed to ones, indicating occupied cells.

The initialized board is essential as it provides a clean starting point from which to begin solving the problem. This sets the stage for placing queens while checking for safe zones, as it is important that each queen is placed optimally without conflicting with the others.

Safe Zone Check

A safe zone check is pivotal in the N-Queens Problem to ensure that placing a queen on a given cell doesn't lead to conflicts. When trying to place a queen in a cell on the chessboard, a comprehensive check is performed to ascertain that:

• No other queen exists in the same row.
• No other queen shares the same column.
• No other queen occupies the same diagonal, both major and minor.

These checks are crucial because queens can attack each other if they are on the same line of movement. The algorithm looks at horizontal, vertical, and diagonal lines originating from the cell being considered for placement. If all these checks return negative, meaning no threats are detected, the cell is deemed a 'safe zone' for placing the queen. If at any during these checks a conflict is identified, the algorithm proceeds to try the next possible cell or backtracks if no further possibilities exist in the current step.