feat(bit-manipulation, count-bits): count number of bits

Author: BrianLusina

bit_manipulation/number_of_1_bits/README.md

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+# Number of 1 Bits
+
+Write a function that takes a 32-bit binary representation of an unsigned integer n and returns the count of its 1 bits.
+
+The binary representation of an unsigned integer is a sequence of 0s and 1s that represents the integer's value using
+base-2 notation. An example of the 32-bit binary representation of an unsigned integer 13 is 00000000000000000000000000001101.
+
+## Constraint
+
+- The input must be a 32-bit binary representation of an unsigned integer.
+
+> Note: The input is passed to the function as an unsigned integer variable. In this lesson, we have displayed it as a
+> string of binary numbers solely for visualization purposes.
+
+
+## Solution
+
+The first attempt to solve this would be to iterate over each bit in the 32-bit representation and keep increasing the
+count, representing the number of 1 bits, every time we encounter the digit 1. This is the correct way to solve this
+problem, and it even takes constant time and space complexity. However, this approach requires us to perform 32
+iterations regardless of the number of 1 bits, which makes it a less efficient approach, specifically for integers with
+a small number of 1 bits. For example, there are only three 1 bits in the unsigned integer 13, but we'll have to iterate
+the entire 32 times to get to this number.
+To avoid any unnecessary iterations, we can use bitwise manipulation to efficiently count the number of 1 bits in the
+given unsigned integer. Here, we'll iterate through the bits of the binary representation that are actually 1s. This
+means it only performs iterations equal to the number of 1 bits in the binary representation of the integer. For example,
+if the given integer is 12 with the binary representation of 00000000000000000000000000001100, the loop will iterate only
+twice, which is highly efficient.
+
+First, we'll initialize a counter, count, to keep track of the number of 1 bits in the 32-bit binary representation of
+an unsigned integer. Then, we'll start a loop that continues as long as the integer n is not zero, i.e., there is no 1 bit
+left in the integer. This loop will iterate through each bit of the binary representation of n, starting from the least
+significant bit. In each iteration:
+
+> The least significant bit is the rightmost bit in the binary representation of a number.
+
+- Check whether the least significant bit of n is equal to 1 using the bitwise AND operation.
+  - If the last bit is 1, increment the count by 1.
+  - Otherwise, continue with the loop.
+- Once we are done checking the current bit, right-shift n by one position using the right bitwise shift operator. This
+  operation effectively shifts all bits one place to the right, discarding the last bit and making the next bit the new
+  least significant bit.
+
+Eventually, count will have the total number of 1 bits, so we'll return it as the final result.
+
+Let’s look at the following illustration to get a better understanding of the solution:
+
+![Solution 1](./images/solutions/number_of_one_bits_solution_1.png)
+![Solution 2](./images/solutions/number_of_one_bits_solution_2.png)
+![Solution 3](./images/solutions/number_of_one_bits_solution_3.png)
+![Solution 4](./images/solutions/number_of_one_bits_solution_4.png)
+![Solution 5](./images/solutions/number_of_one_bits_solution_5.png)
+![Solution 6](./images/solutions/number_of_one_bits_solution_6.png)
+![Solution 7](./images/solutions/number_of_one_bits_solution_7.png)
+![Solution 8](./images/solutions/number_of_one_bits_solution_8.png)
+![Solution 9](./images/solutions/number_of_one_bits_solution_9.png)
+![Solution 10](./images/solutions/number_of_one_bits_solution_10.png)
+![Solution 11](./images/solutions/number_of_one_bits_solution_11.png)
+![Solution 12](./images/solutions/number_of_one_bits_solution_12.png)
+![Solution 13](./images/solutions/number_of_one_bits_solution_13.png)
+![Solution 14](./images/solutions/number_of_one_bits_solution_14.png)
+![Solution 15](./images/solutions/number_of_one_bits_solution_15.png)
+
+### Time Complexity
+
+In the worst case, the number of iterations in the loop can go up to 32, which is a fixed number. Therefore, the overall
+time complexity for this solution is O(1).
+
+### Space Complexity
+
+The space complexity of this solution is O(1).

bit_manipulation/number_of_1_bits/__init__.py

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+def count_bits(n: int) -> int:
+  count = 0
+
+  while n:
+    # Check the least significant bit by using AND
+    if n & 1:
+        count += 1
+    # Right-shift the number to move to the next bit
+    n >>= 1
+
+  return count