Approach

To solve the problem of finding the next smallest and next largest integers with the same number of 1 bits in their binary representation, we can follow a structured approach. This involves:

1. Understanding Binary Representation: Recognize how integers are represented in binary and how the number of 1 bits affects their value.

2. Identifying Bit Patterns: Utilize bit manipulation techniques to identify the next smallest and largest integers.

3. Implementing the Algorithm: Develop a systematic method to compute the results efficiently.

Key Points

• Binary Representation: Each positive integer can be represented in binary, consisting of bits that are either 0 or 1. The number of 1 bits is crucial in this problem.

• Next Smallest Integer: To find the next smallest integer with the same number of 1 bits, we need to rearrange the bits in a specific way.

• Next Largest Integer: Similarly, finding the next largest integer involves a different rearrangement of bits.

• Efficiency: Ensure the solution is efficient, ideally operating in linear time relative to the number of bits.

Standard Response

Here is a sample implementation in Python that finds the next smallest and next largest integers with the same number of 1 bits:

def next_smallest_and_largest(n):
 # Function to find the next largest integer
 def next_largest(n):
 c = n
 c0 = c1 = 0
 while (c & 1) == 0 and c != 0:
 c0 += 1
 c >>= 1
 while (c & 1) == 1:
 c1 += 1
 c >>= 1
 if c0 + c1 == 31 or c0 + c1 == 0:
 return -1
 
 pos = c0 + c1
 n |= (1 << pos) # Flip the rightmost non-trailing zero
 n &= ~((1 << pos) - 1) # Clear all bits to the right of pos
 n |= (1 << (c1 - 1)) - 1 # Insert (c1-1) ones on the right.
 return n

 # Function to find the next smallest integer
 def next_smallest(n):
 c = n
 c0 = c1 = 0
 while (c & 1) == 1:
 c1 += 1
 c >>= 1
 if c == 0: # All bits are 1
 return -1
 while ((c & 1) == 0) and (c != 0):
 c0 += 1
 c >>= 1
 p = c0 + c1 # Position of rightmost non-trailing zero
 n &= ((~0) << (p + 1)) # Clear bits from bit p onwards
 mask = (1 << (c1 + 1)) - 1 # Sequence of (c1 + 1) ones
 n |= mask << (c0 - 1) # Position the ones at the correct location
 return n

 smallest = next_smallest(n)
 largest = next_largest(n)
 return smallest, largest

# Example usage
n = 78 # Example input
next_smallest, next_largest = next_smallest_and_largest(n)
print(f"Next smallest: {next_smallest}, Next largest: {next_largest}")

Tips & Variations

Common Mistakes to Avoid

• Ignoring Bit Overflow: Ensure that the number stays within the bounds of a 32-bit integer.

• Incorrect Counting of 1 Bits: Double-check the calculation of the number of 1 bits, as it is crucial for both smallest and largest integers.

Alternative Ways to Answer

• For different programming languages, the logic remains the same, but the syntax will vary. Consider exploring solutions in languages like Java, C++, or JavaScript for broader applicability.

Role-Specific Variations

• Technical Roles: Focus on the efficiency and performance of the algorithm. Discuss time complexity in detail.

• Managerial Roles: Emphasize the importance of problem-solving skills and how this method can be applied in software development projects.

• Creative Roles: Highlight a more conceptual understanding of algorithms and how they can lead to innovative solutions.

Follow-Up Questions

• What is the time complexity of your solution?

• Can you explain how you handled edge cases?

• How would you optimize this solution further?

• Could you adapt this algorithm for larger integers or different numeral systems?

By following this structured approach, job seekers can articulate their thought process clearly and demonstrate