Hamming Distance

The Hamming distance between two integers can be calculated by finding the number of positions at which the corresponding bits are different. Here’s a simple step-by-step guide to solve this problem:

  1. XOR the two integers x and y. In binary representation, XOR will give us a 1 for each position where the corresponding bits are different.
  2. Convert the result to binary representation.
  3. Count the number of 1s in this binary representation. This count represents the Hamming distance.

Here’s the code:

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class Solution:
    def hammingDistance(self, x: int, y: int) -> int:
        xor_result = x ^ y  # Step 1: XOR the numbers to find different bits
        hamming_distance = 0

        # Step 2 and 3: Convert to binary and count the number of 1s
        while xor_result:
            if xor_result & 1:  # Check if the last bit is 1
                hamming_distance += 1
            xor_result = xor_result >> 1  # Shift right to check the next bit

        return hamming_distance

By performing an XOR operation on the two numbers and counting the number of 1s in the resulting binary number, we calculate the Hamming distance between them. The time complexity of this solution is O(log n), where n is the maximum of x and y.

Identifying Problem Isomorphism

“461. Hamming Distance” involves determining the number of positions at which the corresponding bits are different for two given numbers, a concept related to bitwise manipulation.

A corresponding problem to this is “136. Single Number”. In “Single Number”, you’re asked to find the number in an array that appears only once while others appear twice. The solution uses the property of XOR operation, a type of bitwise manipulation. XOR of all the elements in the array results in the number that appears only once because XOR of a number with itself is 0, and XOR of a number with 0 is the number itself.

While both problems don’t exactly mirror each other, they share a common foundation of understanding and applying bitwise operations. Thus, it forms an approximate mapping.

Understanding these problems can enhance your knowledge of bitwise operations and their potential application in a variety of problem settings.

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def hammingDistance(self, x, y):
        return bin(x^y).count('1')

Problem Classification

This problem can be classified as a Bit Manipulation problem, specifically Bitwise XOR operation, and counting the number of 1 bits (set bits).

What: The problem statement asks to compute the Hamming distance between two integers. Hamming distance is defined as the number of positions at which the corresponding bits are different.

How: One approach to solve this could involve using Bitwise XOR operation between the two numbers which will give a new number with bits set where the two numbers differ. The task then reduces to counting the number of 1 bits in this resultant number.

Given the nature of the problem, it can be classified as a problem in the domain of Bit Manipulation. The task revolves around understanding binary representation of numbers and performing bitwise operations, not the particular details of how these operations are implemented.

Language Agnostic Coding Drills

To understand and build up to this final solution, you would need to understand the following concepts and drill them separately:

  1. Binary Representation of Integers: Understand how to represent integers in binary format. This is crucial because the problem talks about bit differences, so knowing binary representation is a must.

  2. XOR Bitwise Operation: Understand what a bitwise XOR operation is and how it functions. In short, a bitwise XOR operation takes two bit patterns of equal length and performs the logical XOR operation on each pair of corresponding bits. The result in each position is 1 if only the first bit is 1 or only the second bit is 1, but will be 0 if both are 0 or both are 1.

  3. Counting the number of ‘1’ bits in binary representation: Learn how to count the number of ‘1’ bits in a binary number. In the context of the problem, each ‘1’ bit represents a position where the two input numbers differ.

Approach towards the problem:

  1. Convert to Binary and XOR Operation: Given two numbers, you need to first convert them into binary and then perform the XOR operation. The result will be a binary number where each ‘1’ bit indicates a position where the original two numbers differ.

  2. Count ‘1’ bits: Count the number of ‘1’ bits in the resulting binary number. This count will be the Hamming distance, i.e., the number of positions where the two numbers differ.

They involve concepts of binary representation, bitwise operations, and basic counting, which can be performed in almost all modern programming languages.

Targeted Drills in Python

  1. Binary Representation of Integers:

    You can get binary representation of an integer using the bin() function in Python. The function converts an integer to a binary string.

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    def binary_representation(num):
    return bin(num)

print(binary_representation(10))  # Output: '0b1010'

  2. XOR Bitwise Operation:

    In Python, you can use the ^ operator to perform XOR operations. Let’s write a function to perform XOR operation on two numbers.

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    def xor_operation(num1, num2):
    return num1 ^ num2

print(xor_operation(10, 20))  # Output: 30

  3. Counting the number of ‘1’ bits in binary representation:

    We can count the number of ‘1’ bits in a binary string using the count() method in Python.

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    def count_ones(binary_string):
    return binary_string.count('1')

print(count_ones('0b1010'))  # Output: 2

Now, to write an integrated drill that combines all these separate parts into the final solution:

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def hamming_distance(x, y):
    xor_result = xor_operation(x, y)
    binary_representation_result = binary_representation(xor_result)
    return count_ones(binary_representation_result)

print(hamming_distance(10, 20))  # Output: 4

The hamming_distance function uses the three functions we have defined in the drills. First, it performs the XOR operation on the two input numbers. Then, it converts the result into a binary string. Finally, it counts the number of ‘1’ bits in the binary string, which is the Hamming distance between the two input numbers.