Given a number N, the task is to find the sum of Hamming difference of consecutive numbers from 0 to N.
Hamming Distance between two integers is the number of bits which are different at the same position in both numbers.
Difference between (0, 1) = 1, (1, 2) = 2,
(2, 3) = 1, (3, 4) = 3, (4, 5) = 1.
So the total sum is 1 + 2 + 1 + 3 + 1 = 8
Naive and Logarithmic Approach: Refer to Sum of bit differences for numbers from 0 to N for the naive approach and a logarithmic approach to calculate sum based on checking whether N as powers of 2 or not.
Approach: In this article, an approach based on observing the number of changes occurring in all the bits starting from the LSB to the MSB is discussed.
Follow the steps below to solve the problem:
- It can be observed that the Least Significant Bit will change N times. The 2nd Least significant bit will change floor(N/2) times (i.e (N – 1)/2 if N is odd and N/2 if N is even). Hence, the ith bit will change floor(N/2i) times.
- Hence, to solve this problem we can simply store the sum
N + floor(N/2) + … floor(N/2i)
- until the term does not become zero.
Below is the implementation of the above approach:
Time Complexity: O(logN)
Auxiliary Space: O(1)
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
- Generate permutation of 1 to N such that absolute difference of consecutive numbers give K distinct integers
- Print numbers such that no two consecutive numbers are co-prime and every three consecutive numbers are co-prime
- Permutation of numbers such that sum of two consecutive numbers is a perfect square
- 1 to n bit numbers with no consecutive 1s in binary representation.
- Expressing factorial n as sum of consecutive numbers
- Express a number as sum of consecutive numbers
- Fibbinary Numbers (No consecutive 1s in binary)
- 1 to n bit numbers with no consecutive 1s in binary representation
- Fibbinary Numbers (No consecutive 1s in binary) - O(1) Approach
- Prove that atleast one of three consecutive even numbers is divisible by 6
- Check three or more consecutive identical characters or numbers
- Find the number of consecutive zero at the end after multiplying n numbers
- Check if a number can be expressed as a sum of consecutive numbers
- Count of N digit Numbers whose sum of every K consecutive digits is equal | Set 2
- Count of N digit Numbers whose sum of every K consecutive digits is equal
- Find the prime numbers which can written as sum of most consecutive primes
- Count ways to express a number as sum of consecutive numbers
- Count of N-digit numbers in base K with no two consecutive zeroes
- Generate a list of n consecutive composite numbers (An interesting method)
- Absolute Difference of all pairwise consecutive elements in an array
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.