LeetCode, 𝐃𝐚𝐲-22/365-DSA-Coding 𝐉𝐨𝐮𝐫𝐧𝐞𝐲…Google Prep…

Today I Learned This Problem: 48. Rotate Image

Problem Overview

The task is to rotate an n×nn \times nn×n matrix 90 degrees clockwise in place. This means we must modify the matrix directly without using extra space.

The steps involve two transformations:

1. Transpose the Matrix: Swap elements symmetrically across the main diagonal.
2. Reverse Each Row: Flip the elements in each row horizontally.

This problem is an exercise in:

• Matrix Manipulation: Modifying 2D arrays efficiently.
• In-place Transformations: Using no extra space for intermediate structures.

Challenges Involved

Understanding Matrix Indices:

• Identifying which elements to swap during the transpose step.
• Carefully handling indices to avoid overlapping swaps.

Reversing Rows Efficiently:

• Reversing rows without creating a copy of the row.

In-Place Modifications:

• Completing the rotation without allocating additional memory.

Approach to Solve the Problem

Step 1: Transpose the Matrix

• Swap elements symmetrically along the main diagonal.
• For a matrix element at position (i,j)(i, j)(i,j), swap it with the element at (j,i)(j, i)(j,i) where j>ij > ij>i to avoid duplicate swaps.

Step 2: Reverse Each Row

• Reverse the elements in each row by swapping the first element with the last, the second with the second-to-last, and so on.

Python Code

class Solution:
    def rotate(self, matrix: List[List[int]]) -> None:
        for row in range(len(matrix)):
            for col in range(row + 1, len(matrix[0])):
                matrix[row][col], matrix[col][row] = matrix[col][row], matrix[row][col]
        
        for i in range(len(matrix)):
            j = 0
            while j < len(matrix) // 2:
                matrix[i][j], matrix[i][len(matrix) - 1 - j] = matrix[i][len(matrix) - 1 - j], matrix[i][j]
                j += 1

Explanation

Step 1: Transpose

• Transpose swaps elements across the main diagonal.
• Example for a 3x3 matrix:

Input Matrix:
1 2 3
4 5 6
7 8 9

After Transpose:
1 4 7
2 5 8
3 6 9

Step 2: Reverse Rows

• Flip each row horizontally.
• Continuing from the transposed matrix:

Transposed Matrix:
1 4 7
2 5 8
3 6 9

After Reversing Rows:
7 4 1
8 5 2
9 6 3

Optimization Insights

In-Place Transformations:

• All operations modify the matrix directly, using O(1)O(1)O(1) extra space.

Efficient Index Management:

• The nested loops for transposing and reversing are easy to follow, ensuring clarity and correctness.

Time Complexity:

• Both transposing and reversing involve O(n2)O(n²)O(n2) operations for an n×nn \times nn×n matrix, which is optimal for this problem.

Learnings

Matrix Manipulation:

• Problems involving 2D arrays often require understanding row-column relationships and careful index handling.

In-Place Operations:

• Efficient algorithms prioritize space-saving transformations without extra allocations.

Iterative Swapping:

• Many problems involving arrays or matrices can be reduced to iterative swap operations.

Real-Life Applications

Image Processing:

• Rotating images (or sub-images) is a common operation in graphics programming and computer vision.

Grid-Based Problems:

• Transformations like rotation or mirroring are foundational in grid-based simulations, games, and optimizations.

Matrix Transformations in AI:

• Many AI and ML algorithms rely on efficient matrix manipulations, such as rotations in convolutional neural networks (CNNs).

To Sum Up

The Rotate Image problem demonstrates how in-place matrix transformations can be efficiently implemented using simple operations like transposition and row reversal. By understanding indices and managing space, this problem lays the foundation for solving more advanced grid and matrix challenges.

Let me know if you have alternative approaches or insights to share!