Print X in Python with O(n) Time Complexity

Given a positive integer n print matrix containing an X as in the example below

Example:

Input: n = 5
Output:
x---x
-x-x-
--x--
-x-x-
x---x
Explanation:
Each line contains exactly n = 5 characters and two 'x's.
Each diagonal contains 'x'

Understanding the Problem

The core challenge of this problem is to generate a matrix of size n x n where the diagonals are marked with 'x' and the rest of the elements are '-'. This problem is significant in understanding how to manipulate and generate patterns in matrices, which is a common task in computer graphics, game development, and algorithm design.

Potential pitfalls include incorrectly indexing the matrix or not properly updating the indices for the 'x' characters, leading to incorrect patterns.

Approach

To solve this problem, we can follow these steps:

1. Initialize two variables, firstX and secondX, to keep track of the positions of 'x' in each row.
2. Iterate through each row and update the positions of 'x' accordingly.
3. For each row, construct a string with 'x' at the positions indicated by firstX and secondX, and '-' elsewhere.
4. Print the constructed string for each row.

Algorithm

Here is a step-by-step breakdown of the algorithm:

1. Initialize firstX to 0 and secondX to n-1.
2. For each row from 0 to n-1:
   • Initialize an empty string row.
   • For each column from 0 to n-1:
     • If the column index is equal to firstX or secondX, append 'x' to row.
     • Otherwise, append '-' to row.
   • Print the row.
   • Increment firstX and decrement secondX.

Code Implementation

def print_x(n):
    # Initialize the positions of 'x'
    firstX = 0
    secondX = n - 1
    
    # Iterate through each row
    for i in range(n):
        row = ""
        # Construct the row string
        for j in range(n):
            if j == firstX or j == secondX:
                row += "x"
            else:
                row += "-"
        # Print the row
        print(row)
        # Update the positions of 'x'
        firstX += 1
        secondX -= 1

# Example usage
n = 5
print_x(n)

Complexity Analysis

The time complexity of this approach is O(n^2) because we have a nested loop where both the outer and inner loops run n times. The space complexity is O(1) as we are only using a few extra variables and not storing the entire matrix in memory.

Edge Cases

Some potential edge cases include:

• n = 1: The output should be a single 'x'.
• n = 2: The output should be:

  x-
  -x

These edge cases are handled by the algorithm as it correctly updates the positions of 'x' and constructs the rows accordingly.

Testing

To test the solution comprehensively, we can use a variety of test cases:

• Simple cases like n = 1 and n = 2.
• Odd and even values of n.
• Large values of n to ensure the algorithm handles them efficiently.

We can use Python's built-in unittest framework to automate the testing process.

Thinking and Problem-Solving Tips

When approaching such problems, it's important to:

• Break down the problem into smaller, manageable parts.
• Think about how to construct the desired output step by step.
• Consider edge cases and how the algorithm handles them.
• Practice similar problems to improve problem-solving skills.

Conclusion

In this blog post, we discussed how to generate a matrix containing an 'X' pattern given a positive integer n. We covered the problem definition, approach, algorithm, code implementation, complexity analysis, edge cases, and testing. Understanding and solving such problems is crucial for developing strong problem-solving skills and algorithmic thinking.

Additional Resources

For further reading and practice, consider the following resources:

• LeetCode - A platform for practicing coding problems.
• GeeksforGeeks - A comprehensive resource for learning algorithms and data structures.
• Python's unittest Documentation - Official documentation for Python's built-in testing framework.