**π‘ Problem Formulation:** In binary operations, determining if the bits in a specific range between two numbers are complements of each other is an intriguing puzzle. This problem involves taking two integers `A`

and `B`

, and a range from bit position `l`

to `r`

, and ensuring that for every bit `i`

in this range, `A[i]`

is the complement of `B[i]`

. For instance, if `A = 13`

(binary 1101) and `B = 2`

(binary 0010), and our range is from bit 1 to 2 (1-indexed), the function should return `True`

because the bits in that range are complements.

## Method 1: Basic Bit Manipulation

This method employs fundamental bit manipulation techniquesβmasking and bitwise operationsβto verify if bits of two numbers within a specified range are complements. We create a bit mask corresponding to the desired range and apply it to both numbers to isolate the range of bits. Then by performing the AND operation on their complements, we can determine if they’re complements over the entire range.

Here’s an example:

def are_complementary(A, B, l, r): mask = ((1 << (r - l + 1)) - 1) << (l - 1) # Create the mask for the specified range return (mask & A) == (~mask & B) # Check if bits are complements # Example usage: result = are_complementary(13, 2, 1, 2) print(result)

Output:

True

This code snippet defines a function `are_complementary`

that accepts two numbers and the range as parameters. It creates a mask for extracting the range of bits from `l`

to `r`

, then checks if the corresponding bits of `A`

and `B`

are complements within that range. If they are, it returns `True`

; otherwise, it returns `False`

.

## Method 2: Using XOR Operation

The XOR operation is a powerful tool in validating the complementarity of bits because if two bits are complements, their XOR equals 1. This method is about using XOR operation on the masked bits of both numbers within the desired range. If the result equals the mask, then the bits in the given range are complements.

Here’s an example:

def are_complementary_xor(A, B, l, r): mask = ((1 << (r - l + 1)) - 1) << (l - 1) return ((A ^ B) & mask) == mask # Example usage: result = are_complementary_xor(13, 2, 1, 2) print(result)

Output:

True

The function `are_complementary_xor`

performs a bitwise XOR between `A`

and `B`

, then uses the mask to focus on the range of interest. The resulting bits are compared with the mask. If they match, the bits in the range are complements.

## Method 3: String Manipulation

In this approach, the integers are first converted to their binary string representations. We then slice the strings according to the range and compare character by character. The strength of this method lies in its simplicity and ease of understanding for those more comfortable with string operations over bit manipulation.

Here’s an example:

def are_complementary_string(A, B, l, r): A_bin = bin(A)[2:][-r:-l+1] if l > 1 else bin(A)[2:][-r:] B_bin = bin(B)[2:][-r:-l+1] if l > 1 else bin(B)[2:][-r:] return all(a == '0' if b == '1' else a == '1' for a, b in zip(A_bin, B_bin)) # Example usage: result = are_complementary_string(13, 2, 1, 2) print(result)

Output:

True

This function `are_complementary_string`

starts by converting both numbers to binary strings, then slices the strings according to the bit range. It iterates over both slices using `zip`

to check if each pair of characters are logical complements.

## Method 4: Using Bit Length for Masking

For this method, we calculate the bit length of the larger number to apply the mask appropriately, ensuring its length matches the number’s bit length. We then proceed as in the previous methods to check for bit complementarity using bit manipulation.

Here’s an example:

def are_complementary_bit_length(A, B, l, r): max_len = max(A, B).bit_length() mask = ((1 << (r - l + 1)) - 1) << (l - 1) mask = mask | ((1 << max_len) - 1) ^ ((1 << r) - 1) return (A & mask) == (~B & mask) # Example usage: result = are_complementary_bit_length(13, 2, 1, 2) print(result)

Output:

True

The `are_complementary_bit_length`

function calculates the mask adjusting to the maximum bit length between `A`

and `B`

. The masking and complementation check is then performed, making sure to consider only the relevant bits.

## Bonus One-Liner Method 5: Using Built-in Functions

Utilizing Python’s powerful one-liners, one can condense the complement check into a single, albeit dense, line of code. By leveraging bitwise operations and slicing, this method achieves the same result without explicitly defining separate functions or loops.

Here’s an example:

result = lambda A, B, l, r: ((1 << (r - l + 1)) - 1) << (l - 1) & (A ^ B) == (((1 << (r - l + 1)) - 1) << (l - 1)) print(result(13, 2, 1, 2))

Output:

True

This one-liner defines an anonymous function that performs the complement check using a combination of bit shifts, XOR, and comparisons, offering a quick and elegant solution.

## Summary/Discussion

**Method 1: Basic Bit Manipulation.**Clear and easy to understand. Requires knowledge of bitwise operations. May not be the most efficient for large ranges of bits.

**Method 2: Using XOR Operation.**Straightforward with logical simplicity. Efficient for executing a single check, and no looping is required.

**Method 3: String Manipulation.**Intuitive for those more comfortable with string processing. Not the most efficient due to string operations and possible slicing overhead.

**Method 4: Using Bit Length for Masking.**Adaptive for numbers with different bit lengths. A bit more complex due to additional steps for mask adjustment. Provides accurate masking for various input sizes.

**Bonus One-Liner Method 5:**Compact and clever. May sacrifice readability for brevity. Excellent for quick, one-off checks without wrapping in a function.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.