First Bad Version

Clarify the problem:

• The problem requires finding the first bad version in a series of versions.
• We need to implement a function that takes an integer n as input, representing the number of versions, and returns an integer indicating the first bad version.

Analyze the problem:

• Input: An integer n representing the number of versions.
• Output: An integer representing the first bad version.
• Constraints:
  • The versions are labeled from 1 to n.
  • There is at least one bad version.

Design an algorithm:

• We can solve this problem using a binary search algorithm.
• We start with the left and right boundaries set to 1 and n, respectively.
• While the left boundary is less than the right boundary:
  • Find the middle version between the left and right boundaries.
  • If the middle version is a bad version, we update the right boundary to be the middle version.
  • If the middle version is not a bad version, we update the left boundary to be the middle version plus one.
• Once the left and right boundaries converge, we have found the first bad version.

Explain your approach:

• We will implement a function called firstBadVersion that takes an integer n as input.
• We initialize the left and right boundaries as 1 and n, respectively.
• While the left boundary is less than the right boundary:
  • Find the middle version using the formula mid = left + (right - left) // 2 to avoid integer overflow.
  • If the middle version is a bad version, we update the right boundary to be the middle version.
  • If the middle version is not a bad version, we update the left boundary to be the middle version plus one.
• Once the left and right boundaries converge, we return the left boundary as the first bad version.

Write clean and readable code:

python

def firstBadVersion(n):
    left = 1
    right = n

    while left < right:
        mid = left + (right - left) // 2
        if isBadVersion(mid):
            right = mid
        else:
            left = mid + 1

    return left

Test your code:

• To test the code, we need to define the isBadVersion function that checks if a given version is bad or not. For testing purposes, we can assume a specific version to be bad.

python

# Test case:
# Suppose version 4 is the first bad version.
def isBadVersion(version):
    return version >= 4

n = 5
print(firstBadVersion(n))  # Output: 4

Optimize if necessary:

• The binary search algorithm used in the solution has a time complexity of O(log n), as we halve the search space in each iteration.
• The space complexity is O(1) since we use a constant amount of extra space.

Handle error cases:

• We assume that the input n is a positive integer since the problem statement mentions that there is at least one bad version. If n is not a positive integer, the behavior of the function is not defined.

Discuss complexity analysis:

• Let n be the input representing the number of versions.
• The time complexity of the solution is O(log n) because we use a binary search algorithm.
• The space complexity is O(1) because we use a constant amount of extra space to store variables.