Spiral Matrix Traversal in DSA Python - Time & Space Complexity

Time Complexity: Spiral Matrix Traversal

O(n * m)

Understanding Time Complexity

We want to understand how the time needed to traverse a matrix in a spiral way grows as the matrix size increases.

Specifically, how does the number of steps change when the matrix gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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

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

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

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

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

    return result

This code collects all elements of a 2D matrix in a spiral order starting from the top-left corner.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Visiting each element of the matrix exactly once.
• How many times: Each element is visited once during the spiral traversal loops.

How Execution Grows With Input

As the matrix size grows, the number of elements to visit grows too, so the steps increase proportionally.

Input Size (n x n)	Approx. Operations
10 x 10	100
100 x 100	10,000
1000 x 1000	1,000,000

Pattern observation: The operations grow roughly with the total number of elements, so if the matrix size doubles in each dimension, the operations increase by about four times.

Final Time Complexity

Time Complexity: O(n * m)

This means the time to complete the spiral traversal grows linearly with the total number of elements in the matrix.

Common Mistake

[X] Wrong: "The spiral traversal has nested loops that multiply to a higher time complexity like O(n² * m²)."

[OK] Correct: Although there are nested loops, each element is visited only once, so the total steps are just the number of elements, not multiplied repeatedly.

Interview Connect

Understanding how to analyze traversal of 2D structures like matrices is a key skill that shows you can reason about loops and data size clearly.

Self-Check

"What if the matrix was not rectangular but jagged (rows of different lengths)? How would the time complexity change?"