#2596

Medium

auto_awesomeArray plus Depth-First Search

LeetCode Problem Workspace

Check Knight Tour Configuration

Validate a knight's movement configuration on an n x n chessboard to check if it forms a valid Knight's Tour.

array depth first search breadth first search matrix simulation

Problem Statement

You are given an n x n chessboard, with a knight starting at the top-left corner of the grid. The knight should visit every cell exactly once in a valid Knight's Tour configuration. You are provided with an n x n integer matrix, where grid[row][col] indicates the order in which the knight visits that cell. The challenge is to determine whether the configuration represents a valid sequence of knight moves.

A valid knight move in chess consists of an L-shaped movement: two squares in one direction and one square in a perpendicular direction. The task is to verify if the knight follows this valid movement pattern at each step across the board. Return true if the configuration is valid, and false otherwise.

Examples

Example 1

Input: grid = [[0,11,16,5,20],[17,4,19,10,15],[12,1,8,21,6],[3,18,23,14,9],[24,13,2,7,22]]

Output: true

The above diagram represents the grid. It can be shown that it is a valid configuration.

Example 2

Input: grid = [[0,3,6],[5,8,1],[2,7,4]]

Output: false

The above diagram represents the grid. The 8th move of the knight is not valid considering its position after the 7th move.

Constraints

n == grid.length == grid[i].length
3 <= n <= 7
0 <= grid[row][col] < n * n
All integers in grid are unique.

Solution Approach

Depth-First Search for Valid Move Sequence

The main approach is to simulate the knight's movements using Depth-First Search (DFS). Starting from the top-left corner, we must check if the knight can reach each subsequent position in the sequence by making a valid L-shaped move. Each step needs to be validated against the knight's movement pattern.

Array Representation and Position Validation

Since the grid provides a specific order of moves, we can represent the grid as an array. For each move, check if the current cell corresponds to a valid knight's move from the previous cell. The knight’s movement is defined by the array index positions, and each step must follow a valid knight move pattern.

Efficient Boundaries and Uniqueness Check

To ensure the knight stays within the bounds of the board and does not revisit any cell, we must check if the position lies within the grid boundaries. Additionally, verify that all integers in the grid are unique and represent distinct moves to prevent invalid cycles or revisiting cells.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the final implementation. The DFS approach requires checking each move sequentially, which has a time complexity of O(n^2), where n is the grid size. Space complexity is O(n^2) due to storing the grid and tracking visited cells.

What Interviewers Usually Probe

Candidates may struggle with efficiently verifying valid moves in a DFS traversal.
Be aware of performance concerns for larger grid sizes (n=7).
Look for clear understanding of knight’s L-shaped movement and boundary validation.

Common Pitfalls or Variants

Common pitfalls

Overlooking the knight's movement pattern can lead to incorrect configurations.
Failing to track visited cells or revisiting a cell can result in an invalid configuration.
Misunderstanding the bounds of the grid could cause out-of-bound errors or incorrect results.

Follow-up variants

Change the board size to test efficiency across different n values.
Modify the starting position of the knight to test other valid configurations.
Add a constraint where the knight moves in a more complex pattern.

FAQ

What is the primary approach for solving the Check Knight Tour Configuration problem?

The primary approach involves Depth-First Search (DFS) to validate the sequence of knight's moves and ensure each move follows the valid L-shaped movement pattern.

How can I verify the knight's moves efficiently in this problem?

You can efficiently verify the knight's moves by checking the difference in indices between two consecutive moves and ensuring they follow the valid L-shaped pattern.

What are the key challenges in solving the Knight's Tour configuration?

Key challenges include ensuring the knight's moves are valid at each step and managing the complexity of larger grid sizes (up to 7x7).

Why is boundary checking crucial in this problem?

Boundary checking ensures the knight does not move off the board, preventing invalid configurations and out-of-bound errors during the move validation process.

What role does the array representation of the grid play in solving this problem?

The array representation of the grid allows you to track the knight's exact movement order and easily validate if each move is valid based on the knight's L-shaped movement pattern.

terminal

Solution

Solution 1: Simulation

We first use an array $\textit{pos}$ to record the coordinates of each cell visited by the knight, then traverse the $\textit{pos}$ array and check if the coordinate difference between two adjacent cells is $(1, 2)$ or $(2, 1)$. If not, return $\textit{false}$.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

class Solution:
    def checkValidGrid(self, grid: List[List[int]]) -> bool:
        if grid[0][0]:
            return False
        n = len(grid)
        pos = [None] * (n * n)
        for i in range(n):
            for j in range(n):
                pos[grid[i][j]] = (i, j)
        for (x1, y1), (x2, y2) in pairwise(pos):
            dx, dy = abs(x1 - x2), abs(y1 - y2)
            ok = (dx == 1 and dy == 2) or (dx == 2 and dy == 1)
            if not ok:
                return False
        return True

Continue Practicing

#2556 disconnect path in a binary matrix by at most one flip #2658 maximum number of fish in a grid #2328 number of increasing paths in a grid