• Practice Backtracking
  • Interview Problems on Backtracking
  • MCQs on Backtracking
  • Tutorial on Backtracking
  • Backtracking vs Recursion
  • Backtracking vs Branch & Bound
  • Print Permutations
  • Subset Sum Problem
  • N-Queen Problem
  • Knight's Tour
  • Sudoku Solver
  • Rat in Maze
  • Hamiltonian Cycle
  • Graph Coloring

Related Articles

  • Solve Coding Problems
  • Backtracking Algorithms
  • Introduction to Backtracking - Data Structure and Algorithm Tutorials
  • Difference between Backtracking and Branch-N-Bound technique
  • What is the difference between Backtracking and Recursion?

Standard problems on backtracking

  • The Knight's tour problem
  • Rat in a Maze
  • N Queen Problem
  • Subset Sum Problem using Backtracking
  • M-Coloring Problem
  • Algorithm to Solve Sudoku | Sudoku Solver
  • Magnet Puzzle
  • Remove Invalid Parentheses
  • A backtracking approach to generate n bit Gray Codes
  • Permutations of given String

Easy Problems on Backtracking

  • Print all subsets of a given Set or Array
  • Check if a given string is sum-string
  • Count all possible Paths between two Vertices
  • Find all distinct subsets of a given set using BitMasking Approach
  • Find if there is a path of more than k length from a source
  • Print all paths from a given source to a destination
  • Print all possible strings that can be made by placing spaces

Medium prblems on Backtracking

  • 8 queen problem
  • Combinational Sum
  • Warnsdorff's algorithm for Knight’s tour problem
  • Find paths from corner cell to middle cell in maze
  • Find Maximum number possible by doing at-most K swaps
  • Rat in a Maze with multiple steps or jump allowed
  • N Queen in O(n) space

Hard problems on Backtracking

  • Power Set in Lexicographic order
  • Word Break Problem using Backtracking
  • Partition of a set into K subsets with equal sum
  • Longest Possible Route in a Matrix with Hurdles
  • Find shortest safe route in a path with landmines
  • Printing all solutions in N-Queen Problem
  • Print all longest common sub-sequences in lexicographical order

Warnsdorff’s algorithm for Knight’s tour problem

Problem : A knight is placed on the first block of an empty board and, moving according to the rules of chess, must visit each square exactly once. 

Following is an example path followed by Knight to cover all the cells. The below grid represents a chessboard with 8 x 8 cells. Numbers in cells indicate move number of Knight.   

knight-tour-problem

We have discussed Backtracking Algorithm for solution of Knight’s tour . In this post Warnsdorff’s heuristic is discussed.  Warnsdorff’s Rule:  

  • We can start from any initial position of the knight on the board.
  • We always move to an adjacent, unvisited square with minimal degree (minimum number of unvisited adjacent).

This algorithm may also more generally be applied to any graph. 

Some definitions:   

  • A position Q is accessible from a position P if P can move to Q by a single Knight’s move, and Q has not yet been visited.
  • The accessibility of a position P is the number of positions accessible from P.

Algorithm:   

  • Set P to be a random initial position on the board
  • Mark the board at P with the move number “1”
  • let S be the set of positions accessible from P.
  • Set P to be the position in S with minimum accessibility
  • Mark the board at P with the current move number
  • Return the marked board — each square will be marked with the move number on which it is visited.

Below is implementation of above algorithm.  

Output:  

Time complexity: O(N^3) Space complexity: O(N^2)

The Hamiltonian path problem is NP-hard in general. In practice, Warnsdorff’s heuristic successfully finds a solution in linear time.

Do you know?   “On an 8 × 8 board, there are exactly 26,534,728,821,064 directed closed tours (i.e. two tours along the same path that travel in opposite directions are counted separately, as are rotations and reflections). The number of undirected closed tours is half this number, since every tour can be traced in reverse!”

If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to [email protected]. See your article appearing on the GeeksforGeeks main page and help other Geeks.  

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  • Introduction
  • Analyze Your Algorithm
  • Growth of a Function
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  • Recurrences
  • Let's Iterate
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  • Knight's Tour Problem
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Knight's Tour Problem | Backtracking

In the previous example of backtracking , you have seen that backtracking gave us a $O(n!)$ running time. Generally, backtracking is used when we need to check all the possibilities to find a solution and hence it is expensive. For the problems like N-Queen and Knight's tour, there are approaches which take lesser time than backtracking, but for a small size input like 4x4 chessboard, we can ignore the running time and the backtracking leads us to the solution.

Knight's tour is a problem in which we are provided with a NxN chessboard and a knight.

chessboard and knight

For a person who is not familiar with chess, the knight moves two squares horizontally and one square vertically, or two squares vertically and one square horizontally as shown in the picture given below.

movement of a knight

Thus if a knight is at (3, 3), it can move to the (1, 2) ,(1, 4), (2, 1), (4, 1), (5, 2), (5, 4), (2, 5) and (4, 5) cells.

Thus, one complete movement of a knight looks like the letter "L", which is 2 cells long.

movement of knight to form L

According to the problem, we have to make the knight cover all the cells of the board and it can move to a cell only once.

There can be two ways of finishing the knight move - the first in which the knight is one knight's move away from the cell from where it began, so it can go to the position from where it started and form a loop, this is called closed tour ; the second in which the knight finishes anywhere else, this is called open tour .

Approach to Knight's Tour Problem

Similar to the N-Queens problem, we start by moving the knight and if the knight reaches to a cell from where there is no further cell available to move and we have not reached to the solution yet (not all cells are covered), then we backtrack and change our decision and choose a different path.

So from a cell, we choose a move of the knight from all the moves available, and then recursively check if this will lead us to the solution or not. If not, then we choose a different path.

valid cells for movement of knight

As you can see from the picture above, there is a maximum of 8 different moves which a knight can take from a cell. So if a knight is at the cell (i, j), then the moves it can take are - (i+2, j+1), (i+1, j+2), (i-2,j+1), (i-1, j+2), (i-1, j-2), (i-2, j-1), (i+1, j-2) and (i+2, j-1).

tracking movement of knight

We keep the track of the moves in a matrix. This matrix stores the step number in which we visited a cell. For example, if we visit a cell in the second step, it will have a value of 2.

last move is NxN

This matrix also helps us to know whether we have covered all the cells or not. If the last visited cell has a value of N*N, it means we have covered all the cells.

Thus, our approach includes starting from a cell and then choosing a move from all the available moves. Then we check if this move will lead us to the solution or not. If not, we choose a different move. Also, we store all the steps in which we are moving in a matrix.

Now, we know the basic approach we are going to follow. So, let's develop the code for the same.

Code for Knight's Tour

Let's start by making a function to check if a move to the cell (i, j) is valid or not - IS-VALID(i, j, sol) . As mentioned above, 'sol' is the matrix in which we are storing the steps we have taken.

A move is valid if it is inside the chessboard (i.e., i and j are between 1 to N) and if the cell is not already occupied (i.e., sol[i][j] == -1 ). We will make the value of all the unoccupied cells equal to -1.

As stated above, there is a maximum of 8 possible moves from a cell (i, j). Thus, we will make 2 arrays so that we can use them to check for the possible moves.

Thus if we are on a cell (i, j), we can iterate over these arrays to find the possible move i.e., (i+2, j+1), (i+1, j+2), etc.

Now, we will make a function to find the solution. This function will take the solution matrix, the cell where currently the knight is (initially, it will be at (1, 1)), the step count of that cell and the two arrays for the move as mentioned above - KNIGHT-TOUR(sol, i, j, step_count, x_move, y_move) .

We will start by checking if the solution is found. If the solution is found ( step_count == N*N ), then we will just return true.

KNIGHT-TOUR(sol, i, j, step_count, x_move, y_move)   if step_count == N*N     return TRUE

Our next task is to move to the next possible knight's move and check if this will lead us to the solution. If not, then we will select the different move and if none of the moves are leading us to the solution, then we will return false.

As mentioned above, to find the possible moves, we will iterate over the x_move and the y_move arrays.

KNIGHT-TOUR(sol, i, j, step_count, x_move, y_move)   ...   for k in 1 to 8     next_i = i+x_move[k]     next_j = j+y_move[k]

Now, we have to check if the cell (i+x_move[k], j+y_move[k]) is valid or not. If it is valid then we will move to that cell - sol[i+x_move[k]][j+y_move[k]] = step_count and check if this path is leading us to the solution ot not - if KNIGHT-TOUR(sol, i+x_move[k], j+y_move[k], step_count+1, x_move, y_move) .

KNIGHT-TOUR(sol, i, j, step_count, x_move, y_move)   ...   for k in 1 to 8     ...     if IS-VALID(i+x_move[k], j+y_move[k])       sol[i+x_move[k]][j+y_move[k]] = step_count       if KNIGHT-TOUR(sol, i+x_move[k], j+y_move[k], step_count+1, x_move, y_move)         return TRUE

If the move (i+x_move[k], j+y_move[k]) is leading us to the solution - if KNIGHT-TOUR(sol, i+x_move[k], j+y_move[k], step_count+1, x_move, y_move) , then we are returning true.

If this move is not leading us to the solution, then we will choose a different move (loop will iterate to a different move). Also, we will again make the cell (i+x_move[k], j+y_move[k]) of solution matrix -1 as we have checked this path and it is not leading us to the solution, so leave it and thus it is backtracking.

KNIGHT-TOUR(sol, i, j, step_count, x_move, y_move)   ...   for k in 1 to 8     ...       sol[i+x_move[k], j+y_move[k]] = -1

At last, if none the possible move returns us false, then we will just return false.

We have to start the KNIGHT-TOUR function by passing the solution, x_move and y_move matrices. So, let's do this.

As stated earlier, we will initialize the solution matrix by making all its element -1.

for i in 1 to N   for j in 1 to N     sol[i][j] = -1

The next task is to make x_move and y_move arrays.

x_move = [2, 1, -1, -2, -2, -1, 1, 2] y_move = [1, 2, 2, 1, -1, -2, -2, -1]

We will start the tour of the knight from the cell (1, 1) as its first step. So, we will make the value of the cell(1, 1) 0 and then call KNIGHT-TOUR(sol, 1, 1, 1, x_move, y_move) .

Analysis of Code

We are not going into the full time complexity of the algorithm. Instead, we are going to see how bad the algorithm is.

There are N*N i.e., N 2 cells in the board and we have a maximum of 8 choices to make from a cell, so we can write the worst case running time as $O(8^{N^2})$.

But we don't have 8 choices for each cell. For example, from the first cell, we only have two choices.

movement of cells from 1

Even considering this, our running time will be reduced by a factor and will become $O(k^{N^2})$ instead of $O(8^{N^2})$. This is also indeed an extremely bad running time.

So, this chapter was to make you familiar with the backtracking algorithm and not about the optimization. You can look for more examples on the backtracking algorithm with the backtracking tag of the BlogsDope .

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The Knight's Tour Problem (using Backtracking Algorithm)

  • Jun 30, 2023
  • 6 Minute Read
  • By Vedanti Kshirsagar

The Knight's Tour Problem (using Backtracking Algorithm)

Ever wondered how a computer playing as a chess player improves its algorithm to defeat you in the game? In this article, we will learn about the Knight's Tour Problem, the various ways to solve it along with their time and space complexities. So, let's get started!

What is Knight's Tour Problem?

Just for a reminder, the knight is a piece in chess that usually looks like a horse and moves in an L-shaped pattern. This means it will first move two squares in one direction and then one square in a perpendicular direction.

The Knight's Tour problem is about finding a sequence of moves for the knight on a chessboard such that it visits every square on the board exactly one time. It is a type of Hamiltonian path problem in graph theory, where the squares represent the vertices and the knight's moves represent the edges of the graph.

This problem has fascinated many mathematicians for centuries, but the solutions they found were very difficult. The simple solution will be to find every conceivable configuration and selecting the best one is one technique to solve this problem. But that will take a load of time.

One popular solution to solve the Knight's tour problem is Warnsdorff's rule, which involves choosing the next move that leads to a square with the fewest available moves. There is also a backtracking approach. 

 But first moving to all that, let's take a quick understanding of the Hamiltonian path problem.

Hamiltonian Path Problem

The Hamiltonian path problem is a well-known problem in graph theory that asks whether a given graph contains a path that visits every vertex exactly once.

A path that visits every vertex exactly once is called a Hamiltonian path, and a graph that contains a Hamiltonian path is called a Hamiltonian graph. 

Let's take an example of the Hamiltonian path problem. Suppose we have a graph with five vertices and the following edges:

hamiltonian path problem example

This graph can be represented as:

hamiltonian path problem example 2

Knight's Tour Backtracking Algorithm

The backtracking algorithm works by exploring all possible moves for the knight, starting from a given square, and backtracking to try different moves if it reaches a dead end.

Here's the basic outline of the backtracking algorithm to solve the Knight's tour problem:

  • Choose a starting square for the knight on the chessboard.
  • Mark the starting square as visited.
  • For each valid move from the current square, make the move and recursively repeat the process for the new square.
  • If all squares on the chessboard have been visited, we have found a solution. Otherwise, undo the last move and try a different move.
  • If all moves have been tried from the current square and we have not found a solution, backtrack to the previous square and try a different move from there.
  • If we have backtracked to the starting square and tried all possible moves without finding a solution, there is no solution to the problem.

We have given the full C++ program for Backtracking Algorithm to solve Knight's Tour Problem below:

Check the image below before we explain the code:

Knight Tour Backtracking Algorithm

In this implementation, we first define a function validMoves  which takes the current row and column of the knight as input and returns a vector of pairs representing all the valid moves that the knight can make from that position.

The solve function is a recursive function that takes the current row and column of the knight, as well as the current move number, as input. We mark the current square as visited by setting board[row][column] it to the current move number, and then we recursively try all possible moves from the current position.

If we reach the last move, then we found a solution and return true. If no valid move is found from the current position, we backtrack by setting the current square to 0 and returning false.

In the main function, we start the solve function at a specified starting position and then output the number of moves it took to find a solution and print the final chessboard with the solution.

Time & Space Complexity for Backtracking

The backtracking algorithm used to solve the Knight's Tour problem has an exponential time complexity. The number of possible paths for the knight grows very quickly as the size of the chessboard increases, which means that the time taken to explore all possible paths grows exponentially.

The exact time complexity of the Knight's Tour Backtracking algorithm is O(8^(n^2)), where n is the size of the chessboard. This is because each move has a maximum of 8 possible directions, and we have to explore all possible moves until we find a solution.

The space complexity of the backtracking algorithm is O(n^2), where n is the size of the chessboard. So, we can say that the backtracking algorithm is efficient for smaller chessboards.

Warnsdorff's Rule

Warndorff's rule is a heuristic greedy algorithm used to solve this problem. It tries to move the knight to the square with the fewest valid moves, hoping that this will lead to a solution.

Here's an overview of how Warndorff's rule algorithm works:

  • Start with a random square on the chessboard.
  • From the current square, consider all possible moves and count the number of valid moves from each adjacent square.
  • Move to the square with the lowest number of valid moves. In case of a tie, move to the square with the lowest number of valid moves from its adjacent squares.
  • Repeat steps 2 and 3 until all squares on the chessboard have been visited.

Here is the pseudocode for Warndorff's rule algorithm:

The time complexity of Warndorff's rule algorithm is O(n^2 log n), where n is the size of the chessboard. This is because we have to visit each square once, and for each square, we have to compute the number of valid moves from each adjacent square. The space complexity of the algorithm is O(n^2) since we need to store the chessboard and the current position of the knight.

Overall, The Knight's Tour problem is related to chess, and solving it can help chess players improve their strategy and become better at the game. In the real world, you can also use it to design efficient algorithms for finding the shortest path between two points in a network.

Now we know The Knight's Tour Problem and its solutions using Backtracking and Warnsdorff's Rule. It has several applications in various fields such as Game theory, Network Routing etc, making it an important problem to study in computer science and mathematics.

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2664 - The Knight’s Tour

Welcome to subscribe on youtube, 2664. the knight’s tour, description.

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 .

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
  • RenderScript
  • class Solution { private int [][] g ; private int m ; private int n ; private boolean ok ; public int [][] tourOfKnight ( int m , int n , int r , int c ) { this . m = m ; this . n = n ; this . g = new int [ m ][ n ]; for ( var row : g ) { Arrays . fill ( row , - 1 ); } g [ r ][ c ] = 0 ; dfs ( r , c ); return g ; } private void dfs ( int i , int j ) { if ( g [ i ][ j ] == m * n - 1 ) { ok = true ; return ; } int [] dirs = {- 2 , - 1 , 2 , 1 , - 2 , 1 , 2 , - 1 , - 2 }; for ( int k = 0 ; k < 8 ; ++ k ) { int x = i + dirs [ k ], y = j + dirs [ k + 1 ]; if ( x >= 0 && x < m && y >= 0 && y < n && g [ x ][ y ] == - 1 ) { g [ x ][ y ] = g [ i ][ j ] + 1 ; dfs ( x , y ); if ( ok ) { return ; } g [ x ][ y ] = - 1 ; } } } }
  • class Solution { public: vector < vector < int >> tourOfKnight ( int m , int n , int r , int c ) { vector < vector < int >> g ( m , vector < int > ( n , - 1 )); g [ r ][ c ] = 0 ; int dirs [ 9 ] = { - 2 , - 1 , 2 , 1 , - 2 , 1 , 2 , - 1 , - 2 }; bool ok = false ; function < void ( int , int ) > dfs = [ & ]( int i , int j ) { if ( g [ i ][ j ] == m * n - 1 ) { ok = true ; return ; } for ( int k = 0 ; k < 8 ; ++ k ) { int x = i + dirs [ k ], y = j + dirs [ k + 1 ]; if ( x >= 0 && x < m && y >= 0 && y < n && g [ x ][ y ] == - 1 ) { g [ x ][ y ] = g [ i ][ j ] + 1 ; dfs ( x , y ); if ( ok ) { return ; } g [ x ][ y ] = - 1 ; } } }; dfs ( r , c ); return g ; } };
  • class Solution : def tourOfKnight ( self , m : int , n : int , r : int , c : int ) -> List [ List [ int ]]: def dfs ( i : int , j : int ): nonlocal ok if g [ i ][ j ] == m * n - 1 : ok = True return for a , b in pairwise (( - 2 , - 1 , 2 , 1 , - 2 , 1 , 2 , - 1 , - 2 )): x , y = i + a , j + b if 0 <= x < m and 0 <= y < n and g [ x ][ y ] == - 1 : g [ x ][ y ] = g [ i ][ j ] + 1 dfs ( x , y ) if ok : return g [ x ][ y ] = - 1 g = [[ - 1 ] * n for _ in range ( m )] g [ r ][ c ] = 0 ok = False dfs ( r , c ) return g
  • func tourOfKnight ( m int , n int , r int , c int ) [][] int { g := make ([][] int , m ) for i := range g { g [ i ] = make ([] int , n ) for j := range g [ i ] { g [ i ][ j ] = - 1 } } g [ r ][ c ] = 0 ok := false var dfs func ( i , j int ) dfs = func ( i , j int ) { if g [ i ][ j ] == m * n - 1 { ok = true return } dirs := [] int { - 2 , - 1 , 2 , 1 , - 2 , 1 , 2 , - 1 , - 2 } for k := 0 ; k < 8 ; k ++ { x , y := i + dirs [ k ], j + dirs [ k + 1 ] if x >= 0 && x < m && y >= 0 && y < n && g [ x ][ y ] == - 1 { g [ x ][ y ] = g [ i ][ j ] + 1 dfs ( x , y ) if ok { return } g [ x ][ y ] = - 1 } } } dfs ( r , c ) return g }
  • function tourOfKnight ( m : number , n : number , r : number , c : number ): number [][] { const g : number [][] = new Array ( m ). fill ( 0 ). map (() => new Array ( n ). fill ( - 1 )); const dirs = [ - 2 , - 1 , 2 , 1 , - 2 , 1 , 2 , - 1 , - 2 ]; let ok = false ; const dfs = ( i : number , j : number ) => { if ( g [ i ][ j ] === m * n - 1 ) { ok = true ; return ; } for ( let k = 0 ; k < 8 ; ++ k ) { const [ x , y ] = [ i + dirs [ k ], j + dirs [ k + 1 ]]; if ( x >= 0 && x < m && y >= 0 && y < n && g [ x ][ y ] === - 1 ) { g [ x ][ y ] = g [ i ][ j ] + 1 ; dfs ( x , y ); if ( ok ) { return ; } g [ x ][ y ] = - 1 ; } } }; g [ r ][ c ] = 0 ; dfs ( r , c ); return g ; }
  • impl Solution { pub fn tour_of_knight ( m : i32 , n : i32 , r : i32 , c : i32 ) -> Vec < Vec < i32 >> { let mut g : Vec < Vec < i32 >> = vec! [ vec! [ - 1 ; n as usize ]; m as usize ]; g [ r as usize ][ c as usize ] = 0 ; let dirs : [ i32 ; 9 ] = [ - 2 , - 1 , 2 , 1 , - 2 , 1 , 2 , - 1 , - 2 ]; let mut ok = false ; fn dfs ( i : usize , j : usize , g : & mut Vec < Vec < i32 >> , m : i32 , n : i32 , dirs : & [ i32 ; 9 ], ok : & mut bool ) { if g [ i ][ j ] == m * n - 1 { * ok = true ; return ; } for k in 0 .. 8 { let x = (( i as i32 ) + dirs [ k ]) as usize ; let y = (( j as i32 ) + dirs [ k + 1 ]) as usize ; if x < ( m as usize ) && y < ( n as usize ) && g [ x ][ y ] == - 1 { g [ x ][ y ] = g [ i ][ j ] + 1 ; dfs ( x , y , g , m , n , dirs , ok ); if * ok { return ; } g [ x ][ y ] = - 1 ; } } } dfs ( r as usize , c as usize , & mut g , m , n , & dirs , & mut ok ); g } }

All Problems

All solutions.

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2596. Check Knight Tour Configuration ¶

  • Time: $O(8n^2) = O(n^2)$
  • Space: $O(1)$

Check Knight Tour Configuration

10 prerequisite leetcode problems.

For the problem “2596. Check Knight Tour Configuration”, the following are a good preparation:

“200. Number of Islands” - This problem introduces the concept of traversing a 2D grid, which is a crucial aspect of the Knight Tour problem.

“79. Word Search” - This problem requires visiting every cell in a grid to construct possible words, somewhat similar to the knight’s tour where you need to check all cells.

“130. Surrounded Regions” - The problem will give a good understanding of grid traversal and connectivity, relevant to the knight’s moves.

“286. Walls and Gates” - Teaches handling of special cases in a grid (walls and gates), similar to the constraints a knight’s move may have.

“994. Rotting Oranges” - This problem uses a similar approach to grid exploration using breadth-first search (BFS), a potential approach for the knight’s tour problem.

“752. Open the Lock” - This problem also involves step-by-step transformations akin to a knight moving step by step on a grid.

“127. Word Ladder” - This problem introduces the concept of transformation sequences which is similar to the sequence of the knight’s moves.

“207. Course Schedule” - This problem uses depth-first search (DFS) and helps understand the notion of prerequisites, similar to the ordered steps in the knight’s tour.

“542. 01 Matrix” - Another grid-based problem that requires calculating minimum distances, related to the knight making optimal moves.

“490. The Maze” - This problem involves navigating a grid with obstacles, similar to the knight’s movements where certain moves are not possible.

These cover grid traversal, breadth-first search, depth-first search, and dealing with sequential steps - which are important for solving the target problem.

Problem Classification

This is a problem involving validating the moves of a knight on a chessboard. It falls under the domain of games and combinatorics.

The key ‘What’ components are:

  • A chessboard modeled as an n x n grid
  • A sequence of knight moves from cell 0 to n*n-1
  • Moves follow the valid L-shape knight pattern
  • Determine if the sequence is a valid tour covering all cells

Based on these aspects, we can categorize this problem as:

  • Game theory - Modeling legal moves and tours for a chess piece
  • Combinatorics - Generating and validating permutations and combinations
  • Graph theory - Grid can be modeled as graph, knight moves as edges
  • Validation - Checking move sequence adheres to constraints

So in summary, this is a validation problem focused on verifying if a given sequence of knight moves on a chessboard grid graphs a valid tour covering all cells. It involves concepts from game theory, combinatorics and graph theory. The core challenge is modeling and validating the move constraints.

Clarification Questions

Here are some potential clarification questions that could be asked about this problem:

Can the knight move off the board or is it contained within the nxn grid?

Is the sequence of moves required to follow the exact order given in the grid?

What should happen if there are repeated cell values in the grid input?

Are diagonal knight moves allowed or only the standard L-shaped moves?

Does the start position have to be the top-left cell or can it start elsewhere?

Can we assume the grid dimensions and values will be within the stated constraints?

Should we validate the move distances/geometry or just cell coverage?

Do we need to reconstruct the path if it is valid, or just return a boolean?

Is performance important? What constraints exist for large grids?

Can we mutate the input grid or should it remain unchanged?

How should invalid values outside the 0-n*n range be handled?

Asking questions like these would help clarify ambiguous requirements, reveal hidden assumptions, and highlight border cases. This enables creating a more robust and optimized solution within problem constraints.

Identifying Problem Isomorphism

An isomorphism establishes a mapping between two systems in a way that preserves the underlying relationships and structure. For this knight’s tour problem, we can draw an isomorphism between:

The chessboard grid and a graph data structure.

The cells on the grid map to vertices/nodes in the graph.

The valid knight moves on the grid become edges between nodes in the graph.

A sequence of knight moves translates to a path traversing edges in the graph.

A complete knight’s tour on the grid corresponds to a Hamiltonian path in the graph, touching each node exactly once.

So the key mapping is:

  • Grid cell -> Graph vertex
  • Knight move -> Graph edge
  • Knight path -> Graph path
  • Full grid tour -> Hamiltonian graph path

This allows us to leverage graph algorithms and properties to model and solve the problem by essentially transforming the grid system into an isomorphic graph representation.

Some benefits are:

  • Established graph traversal algorithms like DFS, BFS etc. can be applied
  • Graph theory concepts like connectivity, cycles, Hamiltonian paths directly map
  • Generic graph validation approaches translate to validating tours

So in summary, the isomorphic mapping between grid and graph systems allows bringing graph concepts and algorithms to bear on the knight tour problem. It transforms the problem into more familiar graph terms.

Problem Analysis and Key Insights

Here are some key insights gained from analyzing the problem statement:

The grid of cell values represents a sequence of knight moves rather than board state. This informs the validation approach.

Knight move rules lend themselves to modeling as graph edges for traversal algorithms.

Values being distinct integers from 0 to n*n indicates each cell is visited once. This gives a clear coverage criterion.

Order of visitation matters, not just coverage, due to sequence in grid. Path construction is key.

Small fixed board size allows brute force approaches without optimizations for pruning.

Knowing the start position is fixed at top left simplifies initialization logic.

Only standard L-shaped knight moves are possible based on constraints given.

Problem is focused on validation rather than actual path construction.

Can likely mutate and annotate grid rather than needing separate data structures.

These insights around the input representation, movable modeling, and validation-focused problem statement help narrow the scope and simplify the solution design.

Problem Boundary

Based on the problem statement and analysis, the scope of this problem is:

Input space - A 2D grid/matrix of size n x n, with unique integer values from 0 to n*n-1.

Output - A boolean indicating if the grid represents a valid knight’s tour.

Rules - Move sequence must use standard L-shaped knight moves. All cells must be visited exactly once in order.

Objective - Validate the move sequence, not construct or output the actual path taken.

Non-goals - Finding the optimal tour, reconstructing the path, handling boards bigger than 7x7, or alternate move rules.

So in summary, the scope is limited to:

Validating the knight move sequence encoded in the grid input

Checking coverage and connectivity constraints are satisfied

Returning a boolean for overall sequence validity

The focus is on move sequence validation given the grid encoding. Finding the actual optimal tour or path construction details are out of scope.

Here are some ways we can establish boundaries for this problem:

  • Grid is fixed size n x n, where n is 3 to 7.
  • Cell values are distinct integers from 0 to n*n-1.
  • No other inputs are provided.

Processing:

  • Only standard L-shaped knight moves allowed.
  • Sequence order in grid must be followed.
  • All cells must be visited exactly once.
  • Only output is a boolean indicating overall validity.
  • No need to reconstruct or return actual path or tour.
  • No externally stored state apart from input grid.
  • Can modify grid in-place to track visited, invalid moves etc.

Performance:

  • No specific time or space complexity constraints given.

So in summary, the fixed-sized input grid, limited move options, boolean-only output, and modifiable input-only state establish clear boundaries for validation-focused processing using standard knight moves and coverage criteria.

Distilling the Problem to Its Core Elements

This problem is based on the fundamental concepts of combinatorics and constraint validation.

At its core, it involves determining if a sequence of moves for a game piece follows certain rules and covers the full range of spaces. I would describe it simply as:

“Given a sequence of chess knight moves, check if all squares are visited exactly once following the allowed L-shaped knight moves.”

The core problem is validating the move sequence, not finding or constructing it. We can simplify it to:

“Does the provided sequence represent a valid knight’s tour on a chessboard?”

The key components are:

  • The grid encoding the move sequence
  • Rules for valid knight moves
  • Coverage criteria requiring visiting all cells
  • Checking adherence to rules and coverage

The minimum operations are:

  • Parse the grid input
  • Check each move’s validity
  • Track coverage
  • Return true if fully covered per rules, false otherwise

So in essence, this problem focuses on validating a pre-defined move sequence adheres to constraints, rather than constructing the actual sequence. The rest are details built on top of this core validation need.

Visual Model of the Problem

Here are some ways we could visualize this problem statement:

Show an empty chessboard overlayed with a sample sequence of knight moves step-by-step using arrows or highlights.

Animate a knight traversing the board following a provided sequence, coloring visited squares.

Illustrate valid vs invalid moves on a chessboard diagram to make the constraints more clear.

Use a flow chart to demonstrate validating each move and tracking coverage, returning true/false result.

Provide small example grids and animate the knight’s path, coloring valid and invalid moves differently.

Show counter examples of paths that don’t visit all squares or follow invalid moves.

Visualize the chessboard as a graph with knight moves as edges between nodes. Highlight resulting path.

Leverage chessboard/graph libraries to animate sample traversals and paths.

The goal is to leverage visuals like animations, examples and diagrams to make the constraints and coverage requirements clear by demonstration. This engages intuition rather than just mathematical descriptions.

Problem Restatement

Here’s how I would paraphrase the problem statement in my own words:

We are given an n x n grid representing a sequence of chess knight moves on a board, where each cell contains a unique number from 0 to n*n-1 indicating the order each cell was visited.

Our goal is to determine whether this sequence follows the rules for valid knight moves and covers the entire board. Specifically, we need to check that each move goes 2 cells vertically/horizontally and 1 cell horizontally/vertically in an L-shape based on the previous position, and all grid cells are visited exactly once.

The key aspects are:

The grid encodes a move sequence rather than just board state

We need to check if each move follows the valid L-shaped knight pattern

The full n x n board must be covered by the sequence

The specific start position and order does not matter as long as rules are followed

In summary, we need to validate that the provided sequence in the grid represents a legal knight’s tour that fully covers the board using only permitted moves. The focus is on move validity and coverage rather than constructing the actual tour.

Does this help summarize the essence of the problem? Please let me know if I’m missing anything or need to clarify any part of the paraphrasing.

Abstract Representation of the Problem

Here is one way we could formulate an abstract representation of this problem:

Let’s define:

S = A finite set of states {s1, s2,…,sn} representing board positions

s0 = The initial start state

G = A sequence of transitions between states (g1, g2,…,gn)

T(si, sj) = The function defining allowed transitions from state si to sj

Then the problem can be modeled as:

  • A finite state space S
  • Initial state s0
  • State transition sequence G
  • Transition function T
  • If G represents a valid path starting from s0, such that all states are visited exactly once following allowed transitions T.
  • True if G is a valid path per the above criteria
  • False otherwise

This frames the problem more generically in terms of states, transitions between states, and validating if a sequence of transitions represents a valid path covering all states.

The chess-specific concepts like knight moves and grid cells are captured through the abstraction of state transitions and coverage.

Does this help provide an abstract perspective? Please let me know if you would modify or clarify this abstraction in any way.

Terminology

Here are some key terms and concepts relevant to this problem:

Knight’s tour - A sequence of knight moves that visits every cell on a chessboard exactly once. Validating a tour is the core problem.

Graph - Abstract data structure of vertices and edges. The grid can be modeled as a graph.

Hamiltonian path - A path through a graph that visits each vertex exactly once. Mapping to knight’s tour.

DFS/BFS - Depth/breadth-first search algorithms used to traverse graphs and trees. Useful for exploring possible tours.

Recursion - Technique where a function calls itself. Helpful for incrementally building solutions.

Backtracking - Abandoning partial solutions that violate constraints. Allows pruning invalid tours early.

Permutation - Ordered sequence that covers a set of elements. A tour is a permutation over board cells.

Heuristic function - Guides search direction based on domain knowledge. Can help steer towards valid tours.

The key concepts are modeling the grid as a graph to leverage graph theory and algorithms like DFS as well as combinatorics approaches like generating and testing permutations. Recursion and backtracking help iterate through possibilities efficiently.

Problem Simplification and Explanation

Here’s one way I can explain this problem in simpler terms:

Key Concepts:

Grid - The chessboard grid containing the sequence of moves

Moves - Each cell represents a knight move from the previous position

Sequence - The order of cell values indicates the full move sequence

Rules - Moves must follow valid knight L-shape patterns

Coverage - All cells must be visited exactly once

Imagine the grid is like a map showing the route a knight took across a chessboard. Each cell is a stop on the route in order.

We want to check that the full map is covered by the route. Also that each step on the route follows the rule that knights can only move in L-shapes.

It’s like verifying if someone completed a puzzle correctly by checking if they visited every tile on the board and followed the movement rules between tiles.

We don’t need to know the order they put it together - just that their route hits all tiles legally.

In summary, we validate coverage and adherence to constraints rather than reconstructing the actual sequence. The key is verifying the rules were followed to create the end result.

Constraints

Here are some characteristics of the problem that we can leverage:

Small fixed grid size of n x n where n <= 7 allows brute force approaches without complex pruning.

Grid encoding move sequence means we validate rather than construct solution. Simplifies logic.

Distinct integer values from 0 to n*n-1 enables direct mapping to cell coordinates.

Exactly one visit per cell provides a clear coverage criterion to check.

Known starting point at top left simplifies initialization.

Fixed L-shaped move pattern makes validity prediction and checking straightforward.

No specific constraints on efficiency given, so optimize for code simplicity rather than performance.

Can directly annotate and modify grid to track state rather than separate data structures.

Output is binary, so can terminate on first invalid move found rather than exhaustively validating.

Key aspects like small fixed size, encoded input, and binary output allow simplifying the implementation. We can focus on correctness first rather than efficiency and leverage the problem structure for state tracking.

Here are some key insights gained by analyzing the constraints:

Small fixed grid size allows brute force approaches without needing complex optimizations.

Encoding sequence in input simplifies logic since we validate rather than construct.

Unique values enable direct mapping to coordinates for tracking.

Clear coverage criteria makes validation straightforward.

Known start position simplifies initialization.

Fixed move pattern allows predicting and checking validity easily.

No performance constraints allows simpler solutions focused on correctness.

Annotating input grid avoids separate data structures.

Binary output allows short-circuiting on first failure.

Overall, these constraints allow us to simplify the implementation and avoid over-engineering for performance. We can use the encoded input directly rather than translations. Simple exhaustive searches would suffice.

The problem becomes more about properly modeling rules and constraints rather than complex optimizations. This allows focusing on correctness and simplicity.

Case Analysis

Here are some additional test cases covering different aspects:

  • Minimal valid tour

Grid: [[0,1,2], [3,4,5], [6,7,8]]

Output: True

Analysis: Small valid example, tests core logic.

  • Invalid move

Grid: [[0,1,2], [4,3,5], [7,6,8]]

Output: False

Analysis: Check invalid non-L-shape moves.

  • Out of sequence

Grid: [[0,1,2], [4,3,5], [6,7,8]]

Analysis: Validate order matters.

  • Missing cell

Grid: [[0,1], [3,2]]

Analysis: Ensure full coverage.

Edge cases:

  • Minimal grid size e.g. 3x3
  • Starting position other than top-left
  • Boundary and corner moves

The key aspects tested are validity, order, and coverage. The edge cases explore bare minimum and boundary configurations.

Here are some ideas for visualizing these test cases:

Animate the knight’s tour on a chessboard, highlighting each move on the grid

Use arrows and labels to indicate invalid moves that violate constraints

Color code cells visited out of sequence to identify order violations

Cross out or gray out missing cells not covered by the tour

Show traversal on smallest possible 3x3 grid as well as largest 7x7 grid

Illustrate moves along edges and corners which are boundary cases

Visualize tours that follow edges and perimeter vs spanning entire board

Show counter examples demonstrating incorrect behavior

Keep animations clear and concise, focusing on highlighting invalid moves

Use consistent color scheme throughout examples for clarity

Leverage chessboard visualization libraries to animate tours

The goal is to vividly convey concepts like invalid moves, coverage, sequence and boundaries through targeted animations and highlight. Counter examples are particularly useful for identifying pitfalls. Short animations work best for comprehension.

Here are some key insights gained from looking at these different test cases:

Small valid examples are crucial to verify core validation logic works as expected. They serve as a minimal baseline.

Input sequences that violate constraints, like invalid moves or missing coverage, are useful for testing edge cases and error handling.

Order violations highlight that visitation sequence matters just as much as coverage.

Testing a variety of grid sizes reveals assumptions around input size boundaries.

Evaluating moves along edges and corners checks assumptions about knight movement from those positions.

Animations and visuals make comprehending the sequential process much more intuitive.

Counter examples showing incorrect behavior are invaluable for clarifying the problem statement.

A wide range of test cases builds confidence in the solution’s robustness.

Overall, these cases illustrate how comprehensive input coverage combined with visuals for illustration make validating complex sequential processes easier and more foolproof. The tests improve understanding of assumptions and constraints.

Identification of Applicable Theoretical Concepts

Here are some mathematical and algorithmic concepts that could help solve this problem:

Graph theory - Model grid as graph, knight moves as edges to leverage graph algorithms

Hamiltonian path algorithms - Well-studied approaches like backtracking or branch-and-bound for finding paths hitting all nodes

DFS/BFS - Traversal algorithms that can systematically explore possible tours

Permutation generation - Generate and test all permutations of moves that cover board

Pruning strategies - Eliminate branches unlikely to lead to valid solutions to reduce search space

Heuristics - Priority functions to guide search direction, like staying close to unvisited nodes

Bitmasking - Compactly encode visited cells using bit vectors for fast lookup

Matrix transformations - Rotate/flip board without changing validity to generalize logic

Concepts from discrete math like graph theory, combinatorics, algorithms, and data structures provide useful techniques for modeling the problem, exploring the search space efficiently, and optimization. The constraints allow balancing simplicity and efficiency.

Simple Explanation

Here is how I would explain this problem in simple non-technical terms:

Imagine a chess board with numbers written in each square from 0 to n*n-1, where n is the size of the board. These numbers show the order a knight chess piece moved to visit each square.

Our goal is to check that the full board is covered exactly once by the knight’s path, and that each step follows the valid L-shape knight moves.

Knights can only move 2 squares vertically or horizontally, then 1 square horizontally or vertically. So we need to verify each pair of numbers is a valid knight step apart.

It’s like verifying someone fully colored a map using the correct rules. We don’t need to know the exact path they took - just that their steps were valid and covered the whole map.

We can think of each number as a stop on a knight’s journey across the board. We just validate that the journey was complete and followed the movement rules at each step.

The key is to incrementally check each move rather than reconstructing the full path. We focus on verifying adherence to constraints rather than the actual sequencing.

Problem Breakdown and Solution Methodology

Here is a step-by-step explanation of how I would approach solving this problem:

  • Encode grid as graph

Map each cell to a node, valid knight moves to edges

Allows leveraging graph algorithms

  • Validate each move incrementally

Check current and next cell form a valid knight move edge

Update tracking structures (e.g. visited set) after each valid move

Mark each visited node to check entire grid covered

Can annotate grid directly rather than separate data

  • Return overall validity

If completed with full coverage per constraints, return true

If invalid move, return false

This builds up validation step-by-step rather than reconstructing overall tour.

If move constraints were more complex, I’d encapsulate move validation in a separate function. For L-shaped moves, it is straightforward.

Example on 3x3 grid:

Map grid to graph

Check 0->1 is valid move

Check 1->2 is valid

Update tracking structures after each

Repeat until full coverage or invalid move

Return true if fully covered through valid moves

The incremental validation of each transition is key. I could visualize this using simple animated examples with invalid moves highlighted.

Inference of Problem-Solving Approach from the Problem Statement

Here are some key terms and how they guide my approach:

Grid - Indicates 2D matrix structure for the input board

Sequence - Sequential ordering of cell values represents move order, informs validation approach

Knight moves - Specific L-shape move constraints enable direct encoding of valid transitions

Graph - Knight moves lend themselves to graph representation for using graph algorithms

Hamiltonian path - Full coverage goal maps to finding a Hamiltonian path in the graph

Coverage - Visiting all cells maps to covering all nodes in the graph, guides tracking

Validation - Checking if sequence follows rules rather than constructing it informs solution

Incremental - Step-by-step validation of each transition rather than full reconstruction

The core terms of grid, sequence, incremental validation, knight moves, graphs, and coverage lead towards transforming the grid to a graph model and using incremental validation algorithms like BFS/DFS rather than direct construction. The constraints allow simplifying the interpretation and approach.

We can visualize some key properties and aspects of this problem using tables and diagrams:

Grid Encoding:

| 0 | 1 | 2 | | 3 | 4 | 5 | | 6 | 7 | 8 |

  • Show grid structure with cell values representing move sequence

Knight Moves:

0 –> 2 –> 4 –> 6 | | | 1 –> 3 –> 5 –> 7

  • Illustrate possible valid knight moves on grid

Graph Model:

(0) — (2) — (4) — (6) | | | (1) — (3) — (5) — (7)

  • Map grid to graph with knight moves as edges

(0) —> (2) — (4) — (6) | | | (1) —> (3) — (5) — (7)

  • Annotate visited nodes during traversal

These visualizations capture the core aspects - encoding sequence in grid, constraints dictating valid moves, graph abstraction, and tracking state during validation.

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

Here is one way to break this problem down into more granular steps:

High-level approach:

Validate knight’s tour encoded in grid

Return overall validity result

Refined steps:

  • Check if move is valid knight move
  • Mark current node as visited
  • Increment counter tracking nodes visited
  • Return false if invalid
  • Call validation function on current and next cell
  • If false returned, exit early and return false
  • After full loop, return true if all nodes visited

Independent parts:

  • Graph encoding
  • Move validation logic
  • Coverage tracking

Repeatable patterns:

  • Checking node visited status
  • Validating moves
  • Updating tracking variables
  • Short-circuiting on failure

This breaks the problem into modeling, modular validation, and tracking coverage - then iterates through sequence validating each transition. The validation logic is reusable and combinable.

Solution Approach and Analysis

  • Map each cell to a node, knight moves to edges between nodes
  • Define validation function
  • Checks if current move is valid based on knight rules
  • Updates tracking variables like visited set
  • Loop through sequence
  • Call validation on current and next cell
  • If invalid, return false and exit early
  • After full loop, check coverage
  • If all nodes visited, return true
  • Else false (some unvisited)

This breaks the solution into encoding the input, modular move validation, iterating through sequence validating each move, and finally checking full coverage.

For example on 3x3 grid:

Encode grid

Define isValidMove()

Loop through cells in order:

  • Call isValidMove(0, 1)
  • Call isValidMove(1, 2)

After loop, check all cells visited

Return true if fully covered through valid moves.

Keeping the validation logic modular and reusable is key. We build up validity incrementally with each call. Animated visuals could help illustrate the process.

Identify Invariant

An invariant in this problem is a condition or statement that remains true throughout the execution of the solution. Some potential invariants are:

The input grid encoding the sequence remains unchanged. The grid values stay the same.

The rules defining valid knight moves do not change. The move validity criteria is constant.

Once a cell is marked visited, it remains visited for the duration. Its visited status does not get reversed.

The grid dimensions and total number of cells stays the same.

An invalid move remains invalid later. Move validity does not change with more iterations.

The output boolean indicating overall tour validity starts out empty/undefined, and is only set once after full traversal.

So in essence, key invariants are:

  • Unchanged encoded input grid
  • Consistent, fixed move validity rules
  • Monotonically growing set of visited nodes
  • Preserved grid dimensions and cell count
  • Single final assignment of output value

The traversal cannot modify these core aspects defining the input space and move constraints.

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

Here is how I would think through solving this problem step-by-step:

The problem statement cues:

Grid encoding a sequence tells me to validate rather than construct solution

Knight move rules lend themselves to modeling valid transitions

Full coverage requirement indicates tracking visited status

So the steps would be:

Encode grid as a graph with nodes and knight move edges

Define a function to validate moves:

  • Check if transition is a valid edge
  • Mark nodes visited

Loop through sequence:

After loop, check if all nodes were visited

If fully visited, return true, else false

This models moves as a graph, validates each transition, and finally checks full coverage.

Claude generates buggy code.

Establishing Preconditions and Postconditions

Parameters:

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

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?

Method Functionality:

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

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?

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

Problem Understanding:

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

Initial Breakdown:

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

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?

Task Identification:

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

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?

Method Naming:

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

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

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

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?

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.

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?

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

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.

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

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

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

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

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

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.

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.

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.

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.

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.

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.

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.

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:

Backtracking | Set 1 (The Knight's tour problem) | GeeksforGeeks

knight tour problem leetcode

  • public   class  HorseStep {  
  •      static   final   int [] dx = { - 1 , - 2 , - 2 , - 1 ,  1 ,  2 ,  2 ,  1  };  // x方向的增量   
  •      static   final   int [] dy = {  2 ,  1 , - 1 , - 2 , - 2 , - 1 ,  1 ,  2  };  // y方向的增量   
  •      static   final   int  N =  8 ;  
  •      static   int [][] board =  new   int [N][N];  // 棋盘   
  •   
  •      // 计算结点出口   
  •      int  waysOut( int  x,  int  y) {  
  •          int  tx, ty;  
  •          int  count =  0 ;  
  •          // 结点位置非法或已踏过,返回-1   
  •          if  (x <  0  || y <  0  || x >= N || y >= N || board[x][y] >  0 ) {  
  •              return  - 1 ;  
  •         }  
  •          for  ( int  i =  0 ; i < N; ++i) {  
  •             tx = x + dx[i];  
  •             ty = y + dy[i];  
  •              if  (tx <  0  || ty <  0  || tx >= N || ty >= N) {  
  •                  continue ;  
  •             }  
  •              if  (board[tx][ty] ==  0 ) {  
  •                 count++;  
  •          return  count;  
  •     }  
  •      // 按结点出口数,从小到大排序   
  •      void  sortnode(HorseNode[] hn,  int  n) // 采用简单排序法,因为子结点数最多只有8   
  •     {  
  •          int  i, j, t;  
  •         HorseNode temp;  
  •          for  (i =  0 ; i < n; ++i) {  
  •              for  (t = i, j = i +  1 ; j < n; ++j)  
  •                  if  (hn[j].waysOutNum < hn[t].waysOutNum)  
  •                     t = j;  
  •              if  (t > i) {  
  •                 temp = hn[i];  
  •                 hn[i] = hn[t];  
  •                 hn[t] = temp;  
  •      // 搜索函数,count代表当前第几步   
  •      void  dfs( int  x,  int  y,  int  count) {  
  •          int  i, tx, ty;  
  •         HorseNode[] tExit =  new  HorseNode[N];  // 记录出口结点的出口数   
  •          if  (count > N * N) {  
  •             output();  
  •              return ;  
  •          // 计算[x,y]的出口结点和出口结点的出口数   
  •          for  (i =  0 ; i < N; i++) {  
  •             HorseNode h =  new  HorseNode();  
  •             tExit[i] = h;  
  •             tExit[i].x = tx;  
  •             tExit[i].y = ty;  
  •             tExit[i].waysOutNum = waysOut(tx, ty);  
  •           
  •         sortnode(tExit, N);  
  •          for (i =  0 ; tExit[i].waysOutNum <  0 ; ++i)  
  •             ;  
  •          for (; i < N; ++i){  
  •             tx = tExit[i].x;  
  •             ty = tExit[i].y;  
  •             board[tx][ty] = count;  
  •             dfs(tx, ty, count +  1 );  
  •             board[tx][ty] =  0 ;  
  •       
  •      public   static   void  main(String[] args) {  
  •          int  x =  1 ;  
  •          int  y =  3 ;  
  •         HorseStep test =  new  HorseStep();  
  •         board[x][y] =  1 ;  
  •         test.dfs(x, y,  2 );  
  •      //打印结果   
  •      void  output(){  
  •          for ( int  i = N -  1 ; i >=  0 ; --i){  
  •              for ( int  j =  0 ; j < N; ++j){  
  •                 System.out.printf( "%2d " , board[i][j]);  
  •             System.out.println();  
  •         System.exit( 0 );  
  • }  
  • class  HorseNode {  
  •      int  x;  
  •      int  y;  
  •      int  waysOutNum;  

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IMAGES

  1. 2596. Check Knight Tour Configuration

    knight tour problem leetcode

  2. Leetcode 2596: Check Knight Tour Configuration

    knight tour problem leetcode

  3. GitHub

    knight tour problem leetcode

  4. Solving Knight's Tour variation problem in python : r/learnprogramming

    knight tour problem leetcode

  5. The Knight's Tour Problem (using Backtracking Algorithm)

    knight tour problem leetcode

  6. 2596. Check Knight Tour Configuration (Leetcode Medium)

    knight tour problem leetcode

VIDEO

  1. A closed Knight's Tour

  2. The Dark Knight Rises Is A Peak#tdk#thebatman#christianbale#ae#viral#film#viralvideo#sigma#edit

  3. Mã Đi Tuần

  4. [Leetcode 62/63] Unique Paths I/II

  5. #Knight tour Base raid M K 7

  6. Knight Dialer LEETCODE

COMMENTS

  1. Check Knight Tour Configuration

    Can you solve this real interview question? Check Knight Tour Configuration - There is a knight on an n x n chessboard. In a valid configuration, the knight starts at the top-left cell of the board and visits every cell on the board exactly once. You are given an n x n integer matrix grid consisting of distinct integers from the range [0, n * n - 1] where grid[row][col] indicates that the cell ...

  2. The Knight's tour problem

    The Knight's tour problem. Backtracking is an algorithmic paradigm that tries different solutions until finds a solution that "works". Problems that are typically solved using the backtracking technique have the following property in common. These problems can only be solved by trying every possible configuration and each configuration is ...

  3. The Knight's Tour

    The Knight's Tour - Level up your coding skills and quickly land a job. This is the best place to expand your knowledge and get prepared for your next interview. Can you solve this real interview question? The Knight's Tour - Level up your coding skills and quickly land a job.

  4. Warnsdorff's algorithm for Knight's tour problem

    Warnsdorff's algorithm for Knight's tour problem. Problem : A knight is placed on the first block of an empty board and, moving according to the rules of chess, must visit each square exactly once. Following is an example path followed by Knight to cover all the cells. The below grid represents a chessboard with 8 x 8 cells.

  5. Knight's tour problem using backtracking and its analysis

    For the problems like N-Queen and Knight's tour, there are approaches which take lesser time than backtracking, but for a small size input like 4x4 chessboard, we can ignore the running time and the backtracking leads us to the solution. Knight's tour is a problem in which we are provided with a NxN chessboard and a knight.

  6. 2596

    In a valid configuration, the knight starts at the top-left cell of the board and visits every cell on the board exactly once. You are given an n x n integer matrix grid consisting of distinct integers from the range [0, n * n - 1] where grid[row][col] indicates that the cell (row, col) is the grid[row][col] th cell that the knight visited.

  7. 2596. Check Knight Tour Configuration (Leetcode Medium)

    Larry solves and analyzes this Leetcode problem as both an interviewer and an interviewee. This is a live recording of a real engineer solving a problem liv...

  8. 2596. Check Knight Tour Configuration

    Here in this video we have discussed the approach to solve 2596. Check Knight Tour Configuration of Weekly Contest 337📢📢 Our complete Placement Preparation...

  9. The Knight's Tour Problem (using Backtracking Algorithm)

    The backtracking algorithm used to solve the Knight's Tour problem has an exponential time complexity. The number of possible paths for the knight grows very quickly as the size of the chessboard increases, which means that the time taken to explore all possible paths grows exponentially. The exact time complexity of the Knight's Tour ...

  10. Check Knight Tour Configuration

    There is a knight on an n x n chessboard. In a valid configuration, the knight starts at the top-left cell of the board and visits every cell on the board exactly once.. You are given an n x n integer matrix grid consisting of distinct integers from the range [0, n * n - 1] where grid[row][col] indicates that the cell (row, col) is the grid[row][col] th cell that the knight visited.

  11. 2664

    Welcome to Subscribe On Youtube 2664. The Knight's Tour Description 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 ...

  12. Knight Probability in Chessboard

    Knight Probability in Chessboard - Level up your coding skills and quickly land a job. This is the best place to expand your knowledge and get prepared for your next interview. Can you solve this real interview question? Knight Probability in Chessboard - Level up your coding skills and quickly land a job.

  13. Check Knight Tour Configuration

    Problem: https://leetcode.com/contest/weekly-contest-337/problems/check-knight-tour-configuration/Solution: https://github.com/aithalvishwas/Leetcode_solutio...

  14. 2596. Check Knight Tour Configuration

    LeetCode Solutions in C++ 17, Java, and Python. ... Follow @pengyuc_ on Twitter. LeetCode Solutions 2596. Check Knight Tour Configuration Initializing search walkccc/LeetCode LeetCode Solutions walkccc/LeetCode Preface Style Guide Problems Problems 1. Two Sum 2. Add Two Numbers 3. Longest Substring Without Repeating Characters

  15. Knight Tour

    Initially a knight is placed at the cell(0, 0) of this chessboard, Moving according to the rules of chess, the knight must visit each cell exactly once. Find out the order of each cell in which they are visited. Note : 1. There are multiple possible orders in which a knight can visit each cell of the chessboard exactly once.

  16. Check Knight Tour Configuration

    Check Knight Tour Configuration - LeetCode. Can you solve this real interview question? Check Knight Tour Configuration - Level up your coding skills and quickly land a job. This is the best place to expand your knowledge and get prepared for your next interview.

  17. 6322. Check Knight Tour Configuration

    6322. Check Knight Tour Configuration | Weekly Contest 337 | Leetcode SolutionSubscribe : Telegram: https://t.me/free_solution_to_every_contests (@fr...

  18. Leetcode question Check Knight Tour Configuration

    2. 1. without the problem description, we can hardly know if the result is correct or not - and checking if we got the same result as you running that code is not really helpful. Please add the description to the question and including a minimal reproducible example also helps. - user16320675.

  19. Check Knight Tour Configuration

    10 Prerequisite LeetCode Problems. For the problem "2596. Check Knight Tour Configuration", the following are a good preparation: "200. Number of Islands" - This problem introduces the concept of traversing a 2D grid, which is a crucial aspect of the Knight Tour problem.

  20. Knight Dialer

    Can you solve this real interview question? Knight Dialer - The chess knight has a unique movement, it may move two squares vertically and one square horizontally, or two squares horizontally and one square vertically (with both forming the shape of an L). The possible movements of chess knight are shown in this diagram: A chess knight can move as indicated in the chess diagram below: [https ...

  21. Backtracking

    Backtracking works in an incremental way to attack problems. Typically, we start from an empty solution vector and one by one add items (Meaning of item varies from problem to problem. In context of Knight's tour problem, an item is a Knight's move). When we add an item, we check if adding the current item violates the problem constraint ...

  22. Microsoft

    Microsoft - Knight's tour problem - LeetCode Discuss. Level up your coding skills and quickly land a job. This is the best place to expand your knowledge and get prepared for your next interview.

  23. Microsoft

    Microsoft - Knight's tour problem - LeetCode Discuss. Level up your coding skills and quickly land a job. This is the best place to expand your knowledge and get prepared for your next interview.