Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 5: Problem 2

Write a backtracking algorithm for the \(n\) -Queens Problem that uses a version of procedure expand instead of a version of procedure checknode.

Short Answer

Expert verified
The backtracking algorithm for the n-Queens problem with the expand procedure starts from an empty chessboard and recursively places queens on the board, generating children nodes. It backtracks when it fails to place a queen, returning to the previous node and trying a different placement. The algorithm continues this process until a solution is found or all placements are exhausted.

Step by step solution

01

Creating the initial node

Create an empty chessboard as the initial node for the backtracking algorithm and define the expand procedure. Since the queens are unique (no two are alike), we'll index them and try placing them on the board. Start by placing the first queen (index 0) in every possible space on the board (which would be \(n\) possibilities). Generate \(n\) children nodes, each representing a different placement of the first queen.
02

Recursive call

Let's assume we have a node that represents the placement of k queens on the board. Then the expand procedure will generate the children node by placing the queen \(k + 1\) in all possible spaces on the board where it is not threatened by any other queen already placed on the board. It's important to note that we only need to check the queens that have already been placed. This is the 'expand' procedure which will recursively go down the tree from the root node (empty chessboard), placing queens on the board and generating children nodes for each placement.
03

Handle base cases

There are two base cases to handle here. First, if we have placed all \(n\) queens on the board, then we have found a solution, so return the current board. Second, if we couldn't place a queen in any row during the 'expand' procedure, that means we have hit a dead-end. In that case, we backtrack and remove the queen placed in the previous procedure, marking it as not feasible.
04

Iterative backtracking algorithm

This expands on the third step and outlines how backtracking works. Start from the root node (empty chessboard). Then, while there are untried ways of placing the queens, choose one and update the board accordingly. Apply the expand procedure to generate the children nodes. If we find a solution during this process, return the corresponding board. If not, backtrack to the previous node and try a different placement until a solution is found or all placements have been tested.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the Backtracking Algorithm for the Sum-of-Subsets Problem (Algorithm 5.4) to find all combinations of the following numbers that sum to \(W=52\) \(w_{1}=2 \quad w_{2}=10 \quad w_{3}=13 \quad w_{4}=17 \quad w_{3}=22 \quad w_{6}=42\) Show the actions step by step.

List some of the practical applications that are representable in terms of the \(m-\) Coloring Problem.

Suppose we have a solution to the \(n\) -Queens Problem instance in which \(n=4,\) Can we extend this solution to find a solution to the problem instance in which \(n=5,\) then use the solutions for \(n=4\) and \(n=5\) to construct a solution to the instance in which \(n=6,\) and continue this dynamic programming approach to find a solution to any instance in which \(n>4 ?\) Justify your answer.

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.

Given an \(n \times n \times n\) cube containing \(n^{3}\) cells, we are to place \(n\) queens in the cube so that no two queens challenge each other (so that no two queens are in the same row, column, or diagonal). Can the \(n\) -Queens Algorithm (Algorithm 5.1 ) be extended to solve this problem? If so, write the algorithm and implement it on your system to solve problem instances in which \(n=4\) and \(n=8\)
See all solutions

Recommended explanations on Computer Science Textbooks

Computer Systems

Read Explanation

Big Data

Read Explanation

Computer Network

Read Explanation

Problem Solving Techniques

Read Explanation

Computer Programming

Read Explanation

Data Representation in Computer Science

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free