Question

Apply the Backtracking algorithm for the \(n\) Queens 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 8-Queens problem can be solved using a Backtracking algorithm. Starting with the first Queen, place each queen in a safe square, and back up (or 'backtrack') if no safe square can be found for a queen, moving the previous queen to another place, until a solution is found.

Step-by-step solution

Initialization

Start by placing the first Queen (Queen 1) at the first square of the board (first row, first column). This will be the root of our tree.

Backtracking Process: Queen 2

Place Queen 2 in a safe square in the second row, which doesn't place it in the path of Queen 1. If there's no safe spot, back up to Queen 1 and move it to the next square in the first row.

Backtracking Process: Queen 3

Place Queen 3 in a safe square in the third row, avoiding the paths of Queen 1 and 2. If there's no safe spot, back up to Queen 2 and move it to the next square in the second row. If no safe spot is found there, back up further to Queen 1.

Repetition of Backtracking Process

Repeat the process of placing a Queen and backtracking if no safe square is found, until all 8 Queens are placed.

Finding the First Solution

Once 8 Queens are placed and no more queens are left to be placed, the first solution to the problem is found. Document the position of the queens.

Pruned State Space Tree

The pruned state space tree can be drawn to represent the process in this algorithm. The root of the tree corresponds to the initial state where Queen 1 is placed. The branches indicate the possible positions for the next queens and pruned branches (those cut off) will highlight the sequences that would not lead to a solution. In this tree, a solution is represented by a leaf node which has all 8 queens placed safely.

Key concepts

n Queens Problem

The n Queens Problem is a fascinating challenge in combinatorial optimization. It involves placing \(n\) queens on an \(n\times n\) chessboard in such a way that no two queens threaten each other. A queen attacks any square in the same row, column, or diagonal. This means that, in a valid configuration, no two queens can share the same row, column, or diagonal on the board.

While the problem was originally framed with 8 queens on an 8x8 board, it can be generalized to any number \(n\). The complexity increases with \(n\) as the number of possible configurations grows significantly. The primary goal is to find all valid arrangements if they exist, or determine that no such arrangement is possible for a given \(n\). This problem has been thoroughly studied in computer science, often serving as a classic example of constraint satisfaction problems and is commonly solved using backtracking algorithms.

State Space Tree

A State Space Tree is an abstract representation used to systematically explore all possible states of a problem. In the context of the n Queens Problem, each node in the State Space Tree represents a board configuration with a certain number of queens placed.

The root of this tree starts with an empty board, and each level corresponds to placing an additional queen on the board.

• The branches from each node explore the various positions a new queen can occupy without being in a threatening position.
• The depth of the tree will be \(n\), which is the number of queens to be placed.
• Leaves of the tree that successfully place all \(n\) queens denote valid solutions.

This visualization helps in understanding the progression of the solution and highlights the decision points during the backtracking process.

Algorithm 5.1

Algorithm 5.1 is a specific approach based on the Backtracking algorithm to solve the n Queens Problem. This algorithm leverages a systematic trial and error method to place queens one by one in different rows, ensuring no two queens threaten each other.

The process involves:

• Starting at the first row and attempting to place a queen in each column one by one.
• If a queen can be placed without conflicts, proceed to the next row (next recursive call).
• If no valid position is available, back up to the previous row and move the previous queen to the next possible position, a process known as backtracking.

The algorithm efficiently cuts off branches early if the current configuration blocks further queens from being placed, avoiding unnecessary computation and accelerating the discovery of a solution.

Pruned State Space Tree

The Pruned State Space Tree is an optimized version of the State Space Tree, showing only the pathways explored by the Backtracking algorithm where solutions could potentially exist. This pruning process eliminates certain branches early when it is determined they cannot lead to a complete solution, therefore "pruning" the exploration.

The pruned tree focuses on realistic paths, emphasizing the power of backtracking in reducing computational overhead. It uses:

• Truncated branches to represent dead-end paths which do not lead to a valid placement.
• Full branches reaching leaf nodes that solve the n Queens Problem, indicating successful configurations.

Visualizing this pruned tree is essential to understanding both the efficiency gains achieved by pruning and the logical flow of the backtracking process in reaching a valid solution.