From 1982b395fab555e0a129a8a9b7959a86f28ec3e4 Mon Sep 17 00:00:00 2001
From: Sree Vidya <sreevidyapatamsetti@gmail.com>
Date: Sun, 21 Jul 2024 20:35:18 +0530
Subject: [PATCH] Create Find-Kth-Bit-in-Nth-Binary-String.md

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+---
+id: find-kth-bit-in-nth-binary-string
+title: Find Kth Bit in Nth Binary String
+sidebar_label: 1545 - Find Kth Bit in Nth Binary String
+tags: [Binary String, Kth Bit, Inversion, C++]
+description: Given two positive integers n and k, return the kth bit in the nth binary string generated by specific rules.
+---
+
+## Find Kth Bit in Nth Binary String
+
+### Problem Statement
+Given two positive integers `n` and `k`, the binary string `S_n` is formed as follows:
+
+- `S1 = "0"`
+- `S_i = S_(i - 1) + "1" + reverse(invert(S_(i - 1)))` for `i > 1`
+
+Where `+` denotes the concatenation operation, `reverse(x)` returns the reversed string `x`, and `invert(x)` inverts all the bits in `x` (0 changes to 1 and 1 changes to 0).
+
+For example, the first four strings in the above sequence are:
+
+- `S1 = "0"`
+- `S2 = "011"`
+- `S3 = "0111001"`
+- `S4 = "011100110110001"`
+
+Return the `k`th bit in `S_n`. It is guaranteed that `k` is valid for the given `n`.
+
+### Example 1:
+**Input:** `n = 3, k = 1`  
+**Output:** `"0"`  
+**Explanation:** `S3` is `"0111001"`. The 1st bit is `"0"`.
+
+### Example 2:
+**Input:** `n = 4, k = 11`  
+**Output:** `"1"`  
+**Explanation:** `S4` is `"011100110110001"`. The 11th bit is `"1"`.
+
+### Constraints
+
+- `1 &lt;= n &lt;= 20`
+- `1 &lt;= k &lt;= 2^n - 1`
+### Intuition
+
+The goal of the solution is to find the `k`th bit in the `n`th binary string `S_n` generated by specific rules. Here's the step-by-step intuition behind the algorithm:
+
+1. **Base Case**: The base case `S1` is initialized as "0".
+2. **Building Subsequent Strings**: For each subsequent `S_i` where `i > 1`, the string is constructed by concatenating `S_(i-1)`, "1", and the inverted and reversed `S_(i-1)`.
+3. **String Inversion and Reversal**: For constructing the inverted and reversed part, iterate through `S_(i-1)` from end to start, inverting each bit (0 becomes 1 and 1 becomes 0).
+4. **Store Intermediate Strings**: Use a hash map (`mp`) to store each `S_i` to avoid recomputation and facilitate quick access.
+5. **Access the k-th Bit**: After constructing `S_n`, access the `k-1` index of the string to get the `k`th bit.
+
+### Time Complexity
+
+The time complexity of the algorithm is $O(2^n)$, where $n$ is the input integer:
+
+- **Constructing Strings**: Each `S_i` is twice the length of `S_(i-1)`, leading to exponential growth. The length of `S_n` is $2^n - 1$.
+- **Inversion and Reversal**: Inverting and reversing the previous string takes linear time with respect to its length, leading to exponential time overall due to the doubling size at each step.
+
+### Space Complexity
+
+The space complexity of the algorithm is $O(2^n)$ in the worst case:
+
+- **Storage in Hash Map**: The hash map stores each intermediate string `S_i`. The total storage needed is the sum of the lengths of all strings up to `S_n`, which is dominated by the length of `S_n`.
+- **Final String Storage**: The final string `S_n` has length $2^n - 1$, contributing to the space complexity.
+
+Overall, both the time and space complexity are exponential in `n`.
+
+### Code
+#### C++
+```cpp
+class Solution {
+public:
+    char findKthBit(int n, int k) {
+        
+        unordered_map<int,string>mp;
+        mp[1]="0";
+        for(int i=2;i<=n;i++)
+        {
+            
+            int j=mp[i-1].length()-1;
+            string s1="";
+            while(j>=0){
+            if(mp[i-1][j--]=='0')
+            s1+='1';
+            else s1+='0';
+            }
+            
+            mp[i]+=mp[i-1]+"1"+s1;
+        }
+        
+        string ans=mp[n];
+        return ans[k-1];
+    }
+};
+```