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🟦 Math & Geometry: Rotate Image

📝 Problem Description

Given an \(n \times n\) 2D matrix representing an image, rotate the image by 90 degrees (clockwise) in-place.

Real-World Application

Essential for image processing and graphics manipulation where pixel arrays must be transformed in memory without extra storage to handle large images efficiently.

🛠️ Constraints & Edge Cases

  • \(n \times n\) matrix.
  • In-place modification.
  • Edge Cases: \(1 \times 1\) matrix.

🧠 Approach & Intuition

The Aha! Moment

Rotating 90 degrees clockwise is equivalent to: 1. Transposing the matrix (swap \(M[i][j]\) with \(M[j][i]\)). 2. Reversing each row.

🐢 Brute Force (Naive)

Create a new matrix and copy elements to their new rotated positions. Requires \(\mathcal{O}(N^2)\) space.

🐇 Optimal Approach

  1. Transpose: For each \(i < j\), swap M[i][j] with M[j][i].
  2. Reverse: For each row, reverse the order of elements.

🧩 Visual Tracing

graph LR
    A[Original] -->|Transpose| B[Matrix]
    B -->|Reverse Rows| C[Rotated]
    style B stroke:#f66,stroke-width:2px

💻 Solution Implementation

(Implementation details need to be added...)

⏱️ Complexity Analysis

  • Time Complexity: \(\mathcal{O}(N^2)\) as we touch every cell.
  • Space Complexity: \(\mathcal{O}(1)\) since it's done in-place.

🎤 Interview Toolkit

  • Harder Variant: Rotate 90 degrees counter-clockwise.
  • Alternative Data Structures: Recursive layer-by-layer rotation is also possible.
  • [Spiral Matrix](#) — 2D Array traversal.
  • [Set Matrix Zeroes](#) — Matrix manipulation.