Square Root

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Unit 7 Session 1 (Click for link to problem statements)

Problem Highlights

• 💡 Difficulty: Medium
• ⏰ Time to complete: 15 mins
• 🛠️ Topics: Binary Search, Mathematics

1: U-nderstand

Understand what the interviewer is asking for by using test cases and questions about the problem.

• Established a set (2-3) of test cases to verify their own solution later.
• Established a set (1-2) of edge cases to verify their solution handles complexities.
• Have fully understood the problem and have no clarifying questions.
• Have you verified any Time/Space Constraints for this problem?

• Q: How should the function handle very small values, like 0 or 1?
  • A: The function should return the input itself for 0 and 1, as the square root of 0 is 0 and of 1 is 1.

HAPPY CASE
Input: 16
Output: 4
Explanation: The square root of 16 is 4, which is a perfect square.

EDGE CASE
Input: 27
Output: 5
Explanation: The square root of 27 is approximately 5.196, so the floor value is 5.

2: M-atch

Match what this problem looks like to known categories of problems, e.g. Linked List or Dynamic Programming, and strategies or patterns in those categories.

This is a number computation problem that can effectively be solved using binary search to find the floor of the square root:

• Utilizing binary search to narrow down the possible values for the square root.

3: P-lan

Plan the solution with appropriate visualizations and pseudocode.

General Idea: Implement a binary search to find the largest integer whose square is less than or equal to the given number.

1) If `x` is 0 or 1, return `x` as the square root is the number itself.
2) Initialize the search range with `left` = 1 and `right` = x / 2.
3) While `left` is less than or equal to `right`:
   - Calculate the middle value (`mid`).
   - Compute the square of `mid`.
   - If the square is exactly `x`, return `mid`.
   - If the square is less than `x`, adjust `left` to `mid + 1`.
   - If the square is more than `x`, adjust `right` to `mid - 1`.
4) Return `right` as the largest integer whose square is less than `x`.

⚠️ Common Mistakes

• Not adjusting the binary search bounds correctly could lead to infinite loops or incorrect results.

4: I-mplement

Implement the code to solve the algorithm.

def sqrt(x):
    if x < 2:
        return x
    left, right = 1, x // 2
    while left <= right:
        mid = (left + right) // 2
        mid_squared = mid * mid
        if mid_squared == x:
            return mid
        elif mid_squared < x:
            left = mid + 1
        else:
            right = mid - 1
    return right

5: R-eview

Review the code by running specific example(s) and recording values (watchlist) of your code's variables along the way.

• Test the function with input 16 to ensure it returns 4.
• Validate the function with input 27 to check that it returns 5 as the floor of the square root.

6: E-valuate

Evaluate the performance of your algorithm and state any strong/weak or future potential work.

• Time Complexity: O(log n) due to the binary search process, which halves the search range with each iteration.
• Space Complexity: O(1) as it only uses a few variables for computation.