Spiral Matrix Traversal

Problem Description

Given a matrix (2D array), the task is to traverse the matrix in a spiral order and return all elements in that order.

Naive Approach

A brute-force approach involves simulating the spiral traversal directly.

1. Initialize boundary indices: top, bottom, left, and right representing the boundaries of the matrix.
2. Iterate while left <= right and top <= bottom.
3. Traverse from left to right along the top row.
4. Traverse from top + 1 to bottom along the right column.
5. Traverse from right - 1 to left along the bottom row.
6. Traverse from bottom - 1 to top + 1 along the left column.
7. Update the boundary indices after each traversal.

Code Example (Python)

def spiral_matrix_naive(matrix):
    result = []
    top, bottom = 0, len(matrix) - 1
    left, right = 0, len(matrix[0]) - 1

    while left <= right and top <= bottom:
        # Traverse top row
        for i in range(left, right + 1):
            result.append(matrix[top][i])
        top += 1

        # Traverse right column
        for i in range(top, bottom + 1):
            result.append(matrix[i][right])
        right -= 1

        if top <= bottom and left <= right:
            # Traverse bottom row
            for i in range(right, left - 1, -1):
                result.append(matrix[bottom][i])
            bottom -= 1

            # Traverse left column
            for i in range(bottom, top - 1, -1):
                result.append(matrix[i][left])
            left += 1

    return result

Time Complexity: O(m * n)

Where 'm' is the number of rows and 'n' is the number of columns in the matrix, as each element is visited once.

Space Complexity: O(1)

Excluding the output array, the algorithm uses constant extra space.

Optimal Approach

The optimal approach is essentially the same as the naive approach because every element of the matrix needs to be visited to produce the spiral order, therefore no optimization can reduce the overall time complexity. The key to the optimal approach is clarity and conciseness in the implementation.

Edge Cases

1. Empty Matrix: If the input matrix is empty (either 0 rows or 0 columns), return an empty list.
2. Single Row/Column Matrix: The algorithm should handle these cases correctly without any special modifications.
3. Non-rectangular Matrix: The problem assumes a rectangular matrix. If the matrix is not rectangular, the algorithm might produce incorrect results.

Code Example (Optimized Python)

def spiral_matrix_optimized(matrix):
    if not matrix:
        return []

    result = []
    top, bottom = 0, len(matrix) - 1
    left, right = 0, len(matrix[0]) - 1

    while top <= bottom and left <= right:
        # Traverse top row
        for i in range(left, right + 1):
            result.append(matrix[top][i])
        top += 1

        # Traverse right column
        for i in range(top, bottom + 1):
            result.append(matrix[i][right])
        right -= 1

        if top <= bottom and left <= right:
            # Traverse bottom row
            for i in range(right, left - 1, -1):
                result.append(matrix[bottom][i])
            bottom -= 1

            # Traverse left column
            for i in range(bottom, top - 1, -1):
                result.append(matrix[i][left])
            left += 1

    return result

Time Complexity: O(m * n)

Where 'm' is the number of rows and 'n' is the number of columns in the matrix.

Space Complexity: O(1)

Excluding the output list, the algorithm uses constant extra space.