How Many 0s (Recursive)

Unit 7 Session 2 (Click for link to problem statements)

Problem Highlights

• 💡 Difficulty: Medium
• ⏰ Time to complete: 20 mins
• 🛠️ Topics: Binary Search, Recursion, Arrays

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 behave if the entire array contains only 1s?
  • A: The function should return 0 since there are no 0s present.

HAPPY CASE
Input: [0, 0, 0, 0, 1, 1, 1]
Output: 4
Explanation: There are four 0's before the first 1 appears.

EDGE CASE
Input: [1, 1, 1, 1, 1]
Output: 0
Explanation: No zeros are present in the array.

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 problem leverages binary search within a recursive strategy:

• Adapting binary search to identify the transition from 0 to 1, which allows efficient counting of zeros.

3: P-lan

Plan the solution with appropriate visualizations and pseudocode.

General Idea: Utilize a recursive binary search method to locate the first occurrence of 1 and then compute the number of 0s based on its position.

1) Establish a recursive function to perform binary search and find the first 1's index.
2) Base Case: If the search space is exhausted, handle cases based on the position and the element found.
3) Recursive Steps:
   - Calculate the mid-point.
   - If mid-point is 1 and it's the first element or the previous is 0, return its index.
   - Adjust the search range based on whether the mid-point element is 0 or 1.
4) Use the index returned to calculate the number of 0s.

⚠️ Common Mistakes

• Incorrect handling of arrays containing only 1s or only 0s.
• Not accurately identifying the first 1's position could lead to incorrect count of zeros.

4: I-mplement

Implement the code to solve the algorithm.

def find_first_one(lst, low, high):
    """ Recursively find the index of the first '1' in the list using binary search. """
    if low > high:
        return high + 1  # This handles the case when there are no '1's in the list.

    mid = (low + high) // 2
    if lst[mid] == 1:
        # Check if it's the first '1' or if the element before it is '0'
        if mid == 0 or lst[mid - 1] == 0:
            return mid
        else:
            return find_first_one(lst, low, mid - 1)
    else:
        return find_first_one(lst, mid + 1, high)

def count_zeros(lst):
    """ Uses the recursive binary search to find the first '1' and calculate the number of '0's. """
    if not lst:
        return 0
    n = len(lst)
    # If thelist has elements, find the first '1'
    first_one_index = find_first_one(lst, 0, n - 1)
    return first_one_index  # The index itself represents the number of '0's since it's zero-indexed

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 an array [0, 0, 0, 0, 1, 1, 1] to ensure it correctly identifies 4 zeros.
• Check with an array of all ones [1, 1, 1, 1, 1] to confirm that it returns 0.

6: E-valuate

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

• Time Complexity: O(log n) for the binary search.
• Space Complexity: O(log n) for the recursion stack.