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🟦 Math & Geometry: Spiral Matrix

📝 Problem Description

Given an \(m \times n\) matrix, return all elements of the matrix in spiral order.

Real-World Application

Spiral traversal is used in computer vision for searching patterns, and in data storage indexing where spatially correlated data needs to be retrieved in a sequence.

🛠️ Constraints & Edge Cases

  • \(m \times n\) matrix.
  • Edge Cases: Matrix with one row or one column.

🧠 Approach & Intuition

The Aha! Moment

Use four boundaries (top, bottom, left, right) to represent the current layer of the spiral, and contract these boundaries as you visit elements.

🐢 Brute Force (Naive)

Creating a visited matrix is \(\mathcal{O}(MN)\) space.

🐇 Optimal Approach

  1. Initialize top=0, bottom=m-1, left=0, right=n-1.
  2. While boundaries do not cross:
    • Traverse top row (increment top).
    • Traverse right column (decrement right).
    • Traverse bottom row if top <= bottom (decrement bottom).
    • Traverse left column if left <= right (increment left).

🧩 Visual Tracing

graph TD
    A[Start: 0,0] --> B[Right]
    B --> C[Down]
    C --> D[Left]
    D --> E[Up]
    E --> F[Next Inner Loop]

💻 Solution Implementation

(Implementation details need to be added...)

⏱️ Complexity Analysis

  • Time Complexity: \(\mathcal{O}(MN)\) as we visit each cell once.
  • Space Complexity: \(\mathcal{O}(1)\) (excluding output storage).

🎤 Interview Toolkit

  • Harder Variant: Generate a spiral matrix given a size.
  • Alternative Data Structures: Using recursion to process layers.
  • [Rotate Image](#) — Matrix manipulation.
  • [Set Matrix Zeroes](#) — Matrix manipulation.