The Knight’s Tour

Let’s break down the solution to the given problem. We’re supposed to find an order of movements for a knight on an m x n chessboard so that every cell is visited exactly once.

The knight’s legal moves can be represented as a combination of horizontal and vertical jumps. We can represent these moves as an array of 8 possible (dx, dy) pairs.

We’ll also maintain a 2D array, board, to keep track of the order in which cells are visited. We’ll run a Depth-First Search (DFS) starting from the given (r, c) position to explore all possible paths.

Here’s the code implementation:

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class Solution:
    def tourOfKnight(self, m: int, n: int, r: int, c: int) -> List[List[int]]:
        moves = [(2, 1), (1, 2), (2, -1), (-1, 2), (-2, 1), (1, -2), (-2, -1), (-1, -2)]  # 8 possible knight moves
        board = [[-1 for _ in range(n)] for _ in range(m)]  # Initialize board with -1

        def isValid(x, y):
            return 0 <= x < m and 0 <= y < n

        def dfs(x, y, count):
            if count == m * n:  # All cells are visited
                return True

            for dx, dy in moves:
                nx, ny = x + dx, y + dy
                if isValid(nx, ny) and board[nx][ny] == -1:
                    board[nx][ny] = count
                    if dfs(nx, ny, count + 1):  # Continue DFS
                        return True
                    board[nx][ny] = -1  # Backtrack
            return False

        board[r][c] = 0  # Starting cell
        dfs(r, c, 1)  # Start DFS from initial position

        return board

Explanation:

  • We use a depth-first search to explore all possible moves from the current position.
  • isValid function checks whether a given cell is within the board boundaries.
  • In the dfs function, we try all possible knight moves, update the board, and continue the search. If a path is found that visits all cells, we return the result.
  • If a move leads to a dead-end, we undo it (backtrack) and continue exploring other possibilities.
  • The constraints of m, n <= 5 ensure that the running time of this solution is manageable.

Example:

Using the function with the given example m = 3, n = 4, r = 0, c = 0 will give us a valid order of movements to visit the entire board.

Identifying Problem Isomorphism

“The Knight’s Tour” can be mapped to “Word Ladder II”. Both involve exploring all possible paths in a certain structure until reaching a particular destination.

In “The Knight’s Tour”, the structure is a chessboard, and the paths are all the legal moves of the knight. The problem is to find a sequence of moves that allow the knight to visit every square once.

In “Word Ladder II”, the structure is a word ladder, and the paths are the sequences of word transformations. The problem involves finding all shortest transformation sequences from a given ‘start’ word to a ’target’ word.

The reasoning is based on the search nature of both problems. In both cases, it’s a matter of exploring all potential “moves” (whether they’re actual moves on a chessboard or word transformations) to find a particular sequence or reach a certain goal.

“Word Ladder II” is more complex due to the additional challenge of dealing with linguistic rules and constraints.

10 Prerequisite LeetCode Problems

For “2664. The Knight’s Tour”, the following is a good preparation:

  1. “1197. Minimum Knight Moves” - This problem requires finding the minimum number of knight moves to reach a certain cell. It is a good practice for path finding and handling knight movements.

  2. “200. Number of Islands” - This problem involves depth-first search (DFS) or breadth-first search (BFS) in a grid, which is similar to the knight’s movements in the chessboard.

  3. “994. Rotting Oranges” - This problem is about BFS traversal in a grid, which could help in understanding the knight’s tour problem.

  4. “130. Surrounded Regions” - This problem requires you to understand how to traverse a grid using DFS or BFS which could be relevant for the knight’s tour problem.

  5. “286. Walls and Gates” - This problem requires you to find the shortest path in a grid, which is similar to knight’s movement.

  6. “490. The Maze” - This problem is about finding a path in a maze which would be useful for understanding grid traversal.

  7. “994. Rotting Oranges” - This problem requires a breadth-first search on a grid, similar to what is required in the knight’s tour problem.

  8. “752. Open the Lock” - Although this problem does not involve movement on a grid, it does involve finding a path within a constrained set of moves, similar to the knight’s tour problem.

  9. “787. Cheapest Flights Within K Stops” - This problem involves path finding with a limited number of steps, which is similar to the knight’s tour problem.

  10. “1091. Shortest Path in Binary Matrix” - This problem involves finding the shortest path in a grid, which is relevant to the knight’s tour problem.

These are about traversing a grid or finding paths, which is a core part of solving the Knight’s Tour problem.

Problem Classification

Problem Statement:Given two positive integers m and n which are the height and width of a 0-indexed 2D-array board, a pair of positive integers (r, c) which is the starting position of the knight on the board.

Your task is to find an order of movements for the knight, in a manner that every cell of the board gets visited exactly once (the starting cell is considered visited and you shouldn’t visit it again).

Return the array board in which the cells’ values show the order of visiting the cell starting from 0 (the initial place of the knight).

Note that a knight can move from cell (r1, c1) to cell (r2, c2) if 0 <= r2 <= m - 1 and 0 <= c2 <= n - 1 and min(abs(r1 - r2), abs(c1 - c2)) = 1 and max(abs(r1 - r2), abs(c1 - c2)) = 2.

Example 1:

Input: m = 1, n = 1, r = 0, c = 0 Output: [[0]] Explanation: There is only 1 cell and the knight is initially on it so there is only a 0 inside the 1x1 grid.

Example 2:

Input: m = 3, n = 4, r = 0, c = 0 Output: [[0,3,6,9],[11,8,1,4],[2,5,10,7]] Explanation: By the following order of movements we can visit the entire board. (0,0)->(1,2)->(2,0)->(0,1)->(1,3)->(2,1)->(0,2)->(2,3)->(1,1)->(0,3)->(2,2)->(1,0)

Constraints:

1 <= m, n <= 5 0 <= r <= m - 1 0 <= c <= n - 1 The inputs will be generated such that there exists at least one possible order of movements with the given condition

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Clarification Questions

What are the clarification questions we can ask about this problem?

Identifying Problem Isomorphism

Can you help me with finding the isomorphism for this problem?

Which problem does it map to on Leetcode for problem?

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Distilling the Problem to Its Core Elements

Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the edge cases?

How to visualize these cases?

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Stepwise Refinement

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Is invariant and loop invariant the same for this problem?

Identify Recursion Invariant

Is there an invariant during recursion in this problem?

Is invariant and invariant during recursion the same for this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

Establishing Preconditions and Postconditions

  1. Parameters:

    • What are the inputs to the method?
    • What types are these parameters?
    • What do these parameters represent in the context of the problem?

  2. Preconditions:

    • Before this method is called, what must be true about the state of the program or the values of the parameters?
    • Are there any constraints on the input parameters?
    • Is there a specific state that the program or some part of it must be in?

  3. Method Functionality:

    • What is this method expected to do?
    • How does it interact with the inputs and the current state of the program?

  4. Postconditions:

    • After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
    • What does the return value represent or indicate?
    • What side effects, if any, does the method have?

  5. Error Handling:

    • How does the method respond if the preconditions are not met?
    • Does it throw an exception, return a special value, or do something else?

Problem Decomposition

  1. Problem Understanding:

    • Can you explain the problem in your own words? What are the key components and requirements?

  2. Initial Breakdown:

    • Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?

  3. Subproblem Refinement:

    • For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?

  4. Task Identification:

    • Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?

  5. Task Abstraction:

    • For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?

  6. Method Naming:

    • Can you give each task a simple, descriptive name that makes its purpose clear?

  7. Subproblem Interactions:

    • How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

  1. Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.

  2. Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?

  3. If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.

  4. If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?

  5. Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.

  6. Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the code for the solution of this problem.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

  1. Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.

  2. Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.

  3. Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

  1. Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.

  2. In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.

  3. Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Q&A

Similar Problems

Can you suggest 10 problems from LeetCode that require similar problem-solving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem. Do not include the original problem. The response text is of the following format. First provide this as the first sentence: Here are 10 problems that use similar underlying concepts:

excerpt: This covers backtracking algorithm to solve knight’s tour problem. tags: backtracking

In the knight’s tour problem, the goal is to make a knight visit every position on a chessboard without visiting any square twice. A tour is closed if the final position is one move away from the starting position and the knight could immediately start the tour again. A tour that is not closed is said to be open.

A knight moves two squares horizontally or vertically and then one square perpendicularly from its current position.

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# Move the knight to position [row, col]. Then recursively try
# to make other moves. Return true if we find a valid solution
def knights_tour(row, col, move_number, moves_taken)
  # Move the knight to this position
  moves_taken = moves_taken + 1
  move_number[row, col] = moves_taken
  
  # Check if we have made all the required moves
  if moves_taken == 64 
    return true
  end
  
  # Build arrays to determine where legal moves are 
  # with respect to this position
  rows = [-2, -2, -1, 1, 2, 2, 1, -1]
  cols = [-1, 1, 2, 2, 1, -1, -2, -2]
  
  for i in (0..7)
    r = row + rows[i]
    c = col + cols[i]
    if r >= 0 && r < num_rows && 
       c >= 0 && c < num_cols &&
       move_number[r, c] == 0
       
       # This move is legal and available. Make this move
       # and then recursively try other assignments
       if knights_tour(r, c, move_number, moves_taken)
         return true
       end
     end
  end
  
  # This move did not work. Undo it.
  move_number[row, col] = 0
  
  # If we get here, we did not find a valid solution
  return false
end

The method takes the row and column where the knight should move next as the the parameters. It also takes an move_number array that gives the number of the move when the knight visited each square and the number of moves made so far.

The knight’s move to the current square is recorded and the number of moves made is incremented. If the number of moves made is 64, the knight has finished a tour of the board, so the method returns true to indicate success.

If the tour is not finished, the method initializes two arrays to represent the moves that are possible from the current square. For example, the first entries in the arrays are -2 and -1, indicating that the knight can move from square (row, col) to (row - 2, col - 1) if that square is on the board.

Next the method loops through all the possible moves from the position (row, col). If a move is on the board and has not already been visited in the test tour, the method makes the move and recursively calls itself to see if that leads to a full solution.

Any position the knight can reach that has not yet been visited gives a new test solution. There may be some cases where you can easily conclude that a test solution won’t work, such as if the board has an unvisited square that is more than one move away from any other unvisited square, but recognizing that situation is difficult.

The fact that it is hard to eliminate test solutions early on means the algorithm often follows a test solution for a long while before discovering that the solution is infeasible. It is difficult to solve the knight’s tour problem for a normal 8 x 8 chessboard.