Find k-th bit in a binary string created by repeated invert and append operations
You are given an initial string s starting with “0”. The string keeps duplicating as follows. Invert of it is appended to it.
Input : k = 2 Output : 1 Initially s = "0". First Iteration : s = s + s' = "01" Second Iteration : s = s + s' = "0110" The digit at index 2 of s is 1. Input : k = 12 Output : 0
1. Naive Approach
We can build the string s while its length is smaller than or equal to i in the manner mentioned in the problem description and then simply do a lookup of the required index in the string s.
Time Complexity: O(k log k).
Let’s take a look at a few string constructions:
1. s = 0
2. s = 01
3. s = 0110
4. s = 01101001
5. s = 0110100110010110
Let’s consider finding the bit at position 11 in Line 5. The bit value at this position is effectively the complement of the bit at index 3 in Line 4. So we effectively need to find the complement of the bit at index 3.
s = ~(s). However we do not know s either. Let’s move ahead. Now s = ~(s) using the same explanation in Line 3. And s = ~(s). However, s is always 0 from the problem statement.
Plugging these results,
s = ~(~(~0)) = 1 where ‘~’ is the bitwise NOT operator.
Now, k was initially 11 which is 1011 in binary. The next value of k was 3 that is 011. Notice how the first set bit has reduced from the original k. Subsequently, the next value of k is 1. Another set bit has reduced. Finally for k = 0, the last set bit has vanished. So for k = 11 which is 1011 in binary, the number of complements are 3, which is incidentally equal to the number of set bits in 11. Now we see for odd number of inversions the final result is 1. We can derive the same reasoning for even number of inversions yielding a final result as 0.
To count the number of set-bits in an integer, refer to this article – Count set bits in an integer.
Time Complexity: O(log k)
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