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This approach involves directly simulating the sequence construction up to the desired nth binary string and retrieving the k-th bit from it. Since the problem ensures n and k are constrained, this method remains efficient enough.
Time Complexity: O(2^n), as each step fully constructs the sequence of size approximately 2^n.
Space Complexity: O(2^n), owing to storing the full sequence.
1def invertChar(c):
2 return '1' if c == '0' else '0'
3
4def invertAndReverse(s):
5 return ''.join(invertChar(c) for c in s)[::-1]
6
7def findKthBit(n, k):
8 S = "0"
9 for i in range(2, n + 1):
10 S = S + "1" + invertAndReverse(S)
11 return S[k - 1]
12
13n = 4
14k = 11
15print("Output:", findKthBit(n, k))The Python solution iteratively builds the binary string by continually appending the transformed version of the previously constructed string. The transformation involves reversing and inverting the bits following the pattern given in the problem definition.
This approach leverages the recursive nature and mathematical properties of the sequence to find the k-th bit without constructing the entire string. By recognizing the symmetry and structure, we use recursive calculations to directly determine the desired bit.
Time Complexity: O(n), limited by the recursion depth of n.
Space Complexity: O(n), due to recursive call stack.
1
public class SolutionRecursive {
public static char FindKthBitRecursively(int n, int k) {
if (n == 1) return '0';
int length = (1 << n) - 1; // S_n length
int mid = (length + 1) / 2;
if (k == mid) return '1';
if (k < mid) return FindKthBitRecursively(n - 1, k);
else {
return FindKthBitRecursively(n - 1, length - k + 1) == '0' ? '1' : '0';
}
}
public static void Main() {
int n = 4, k = 11;
Console.WriteLine("Output: " + FindKthBitRecursively(n, k));
}
}This C# technique solely utilizes recursive manipulation concepts outlined within the reverse, contract, and determine pattern. Attains resultant k-bit via stepwise recursive deviations without constructing each sequence in entirety.