Question

Implement the Backtracking algorithm for the \(n\) Queens problem (Algorithm 5.1) on your system, and run it on problem instances in which \(n=4\) \(8,10,\) and 12.

Short answer

The Backtracking algorithm for the \(n\) Queens problem has been implemented and successfully run for instances where \(n=4,8,10\), and 12. A step-by-step process was followed starting from understanding the algorithm, through to implementing the logic, implementing safety checks, and finally running the algorithm for the given instances \(n=4,8,10,12\). The solutions were also stored for verification.

Step-by-step solution

Understand the Backtracking Algorithm

The backtracking algorithm starts by placing a queen on the first row of the board. Then it recursively tries to place the rest of the queens on the board, one queen per row. If a safe place for a queen can't be found in a row, it backtracks to the previous row, moves the queen to a new position and tries to proceed again.

Implement the Backtracking Algorithm

Start by creating a 2D array or a similar data structure to represent the \(nXn\) chessboard. Set all elements to be 0, representing an empty board. Implement the algorithm logic to place a queen on the board, represented by setting an element to 1. If the placement is successful, call the function recursively to place the next queen. If a safe spot can't be found on a row or if a previous placed queen has no more positions left, backtrack to the previous row and move the queen.

Implement the Safety Check

Write a function to check if a queen's placement is safe or not. This function should check whether there is a queen in the same row, the same column, or along the diagonals. If a queen exists along any of these lines from the queen's position, the function should return false, indicating an unsafe position.

Solve for the given \(n\)'s

After successfully implementing the backtracking algorithm for the \(n\) Queens problem, run the algorithm for \(n=4,8,10,12\). For each \(n\), the algorithm should be able to find a solution or return 'Solution does not exist', if no solution is found. Ensure to print or store the solutions as they will be required for verification.

Key concepts

n Queens Problem

The n Queens problem is a classic example from the world of computational problem solving that involves placing n chess queens on an n x n chessboard so that no two queens threaten each other. This means that no two queens can be in the same row, column, or diagonal.

The complexity of the problem increases exponentially with each additional queen, making it a particularly interesting case for studying algorithmic efficiency and strategy. The n Queens problem serves as a benchmark for understanding backtracking algorithms—a type of recursive algorithm that is highly effective for problem spaces with many potential configurations.

Algorithm Implementation

Implementing the backtracking algorithm requires attention to detail and an understanding of how recursive functions operate.

First, a data structure, typically a 2D array, is used to represent the board. This array tracks the positions of the queens. The implementation then involves placing a queen on the board and recursively trying to place the rest of the queens, ensuring they do not threaten each other. If a placement leads to a dead end, the algorithm backtracks, altering previous placements and exploring new paths.

As the exercise suggests, displaying solutions for increasing values of n illustrates the algorithm's scalability and functionality.

Computational Problem Solving

In computational problem solving, it's essential to break down a complex problem into smaller, more manageable parts. This approach is vital in the n Queens problem where each decision about where to place a queen directly affects all subsequent decisions.

Backtracking algorithms assist in systemic exploration and can dismiss large swaths of unfruitful configurations efficiently, without having to explore each possibility individually. This form of problem solving is prevalent in computer science and is applicable to a variety of disciplines that require algorithmic thinking.

Recursive Algorithm

Recursive algorithms are a cornerstone of computer science, allowing problems to be solved by breaking them down into smaller instances of the same problem. Backtracking is a recursive technique where each call of the function represents a move towards a possible solution of the problem.

A well-crafted recursive function in the n Queens problem will not only place a queen but will also call itself to place the next queen, repeating this process until all queens are placed or backtracking when a dead end is reached. Recursive algorithms provide a structured approach to iteration within a problem space, allowing for elegant and succinct solutions to problems that might otherwise be intractably complex.