1886. Determine Whether Matrix Can Be Obtained By Rotation

Problem Description

The problem provides two n x n binary matrices: mat and target. A binary matrix is a matrix where each element is either 0 or 1. The goal is to determine whether it is possible to make the matrix mat identical to the matrix target by rotating mat in 90-degree increments. The task is to check all possible rotations of mat and see if any rotation matches the target. If at least one rotation results in mat being the same as target, the function should return true. Otherwise, if none of the rotations yield the target matrix, the function should return false. Note that it is possible to rotate mat up to three times to achieve this since a fourth rotation would bring the matrix back to its original orientation.

Intuition

To solve this problem, we need to simulate the rotation of the matrix and compare the result with the target matrix after each rotation. A 90-degree rotation of a matrix can be achieved by reversing the matrix along its horizontal axis (i.e., flipping it upside down) and then taking the transpose. The transpose of a matrix is obtained by switching the rows with the columns, which means element [i][j] becomes [j][i].

The intuition behind the solution is to apply this transformation to mat up to four times (since rotating four times would return the matrix to its original state), each time checking if the resulting matrix is equal to target. In Python, this rotation can be conveniently performed using a combination of list comprehensions and the zip function, which groups the elements of the rows (after reversing mat) into columns, effectively transposing them.

The steps for a 90-degree rotation are:

1. Flip the matrix upside down (mat[::-1])
2. Transpose the matrix (achieved by zip(*mat) after the flip)
3. Convert the zipped elements back into lists ([list(col) for col in zipped])
4. Compare with target (if mat == target)

The function findRotation continuously applies this rotation method up to four times or until mat matches target. If a match is found before the fourth rotation, it returns true. If no match is found after all possible rotations, it returns false.

Solution Approach

The solution approach can be explained as follows:

1. Create a Rotation Function: The solution defines an inline function within the loop that performs a 90-degree clockwise rotation on mat. This is done by flipping the matrix vertically first (mat[::-1]) and then transposing it, which in Python can be effortlessly implemented with the zip function. The zip(*mat[::-1]) statement pairs row elements with column indices, effectively rotating the matrix.

2. Loop Through Rotations: It initiates a loop that will run four times, each iteration representing a 90-degree rotation. The loop is used because four rotations will bring the matrix back to its initial position. Hence, beyond this, additional rotations would only repeat previous states.

3. Transform and Compare: Inside the loop, the matrix mat is rotated using the rotation function from step 1. After each rotation, mat is compared to target (if mat == target). If at any point the matrices match, the function immediately returns true because we have found a valid rotation that turns mat into target.

4. Handling Non-Matching Cases: If the loop completes without finding a matching rotation (i.e., all four rotations do not result in a matrix equal to target), the function returns false. This implies that there is no sequence of 90-degree rotations that can transform mat into target.

The solution elegantly leverages Python's advanced list comprehension and the zip function to perform matrix rotations cleanly and concisely. The decision to check for equality only four times is based on the mathematical fact that any square matrix will return to its original orientation after four 90-degree rotations. The process is highly efficient, avoiding unnecessary calculations or rotations.

This solution has a time complexity of O(n^2) for each rotation due to the matrix traversal, where n is the number of rows/columns in the matrix. Since a fixed number of rotations (at most 4) are performed, the overall time complexity remains O(n^2). The space complexity is O(n^2) as well, which arises from the storage needed for the rotated matrix at each step.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have the following 3x3 matrices mat and target:

mat:

11 2 3
24 5 6
37 8 9

target:

17 4 1
28 5 2
39 6 3

First Rotation (0-degree, the original mat):

We compare mat with target. Clearly, they are not identical, so we proceed to rotate mat.

Second Rotation (90-degree clockwise rotation):

1. Flip mat upside down:

17 8 9
24 5 6
31 2 3

2. Transpose the flipped matrix:

17 4 1
28 5 2
39 6 3

3. After rotation, we compare the matrix with target. Now, mat and target are identical.

Since mat matches target after the second rotation, the function would return true at this point, indicating that it is possible to make the matrix mat identical to target by rotating mat in 90-degree increments.

If mat had not matched target, we would have continued to the third and fourth rotations to check all possibilities before determining that mat cannot be matched with target through rotations.

Solution Implementation

Time and Space Complexity

The given Python function findRotation checks whether one matrix is a rotation of another. It does this by rotating mat up to four times (0 degrees, 90 degrees, 180 degrees, and 270 degrees rotations) and comparing it to target after each rotation.

The time complexity of the function is determined by these major factors:

1. The number of rotations - which is constant, at 4.
2. The cost of each rotation - which includes reversing the rows of mat (O(n)) and then zipping and list conversion (O(n^2)).
3. The comparison of mat and target - which is O(n^2) where n is the dimension size of the matrix mat.

So for each rotation, the total time cost is O(n) + O(n^2) = O(n^2), since zip(*mat[::-1]) essentially involves looking at all n^2 elements of the matrix. And since we rotate up to 4 times, the total time complexity is 4 * O(n^2), which simplifies to O(n^2).

The space complexity is determined by the extra space needed to store the rotated matrix:

1. The reversed matrix mat[::-1] does not use extra space as it is a shallow copy that references the same rows of mat.
2. However, [list(col) for col in zip(*mat[::-1])] creates a new list of lists for each rotation. This list of lists contains n lists of n integers, so it uses O(n^2) space.

Therefore, the space complexity of the function is O(n^2) as it needs space to store a copy of the matrix each time it is rotated.

In summary:

• Time complexity: O(n^2)
• Space complexity: O(n^2)

Learn more about how to find time and space complexity quickly using problem constraints.