K-th Symbol in Grammar

K-th Symbol in Grammar

1. What is this problem about?

The K-th Symbol in Grammar interview question is a recursive sequence problem. You start with a single '0' in row 1. Each subsequent row is generated by replacing each '0' with '01' and each '1' with '10'. You need to find the $k^{th}$ symbol (1-indexed) in the $n^{th}$ row. Because the rows double in size, row 30 would have $2^{29}$ characters, making direct generation impossible.

2. Why is this asked in interviews?

Companies like Goldman Sachs, Microsoft, and Google ask the K-th Symbol coding problem to test a candidate's ability to find mathematical patterns and use Recursion. It evaluations whether you can identify the relationship between a symbol and its "parent" in the previous row. It's a classic test of logarithmic problem-solving.

3. Algorithmic pattern used

This problem follows the Recursive Parent-Child Mapping pattern.

1. The Insight: Row $n$ is formed by two halves. The first half is identical to row $n-1$. The second half is the bitwise inverse of row $n-1$.
2. Recursive Step:
   • If $k$ is in the first half ($k \leq 2^{n-2}$): The symbol is the same as the $k^{th}$ symbol in row $n-1$.
   • If $k$ is in the second half ($k > 2^{n-2}$): The symbol is the inverse of the $(k - ext{half\_size})^{th}$ symbol in row $n-1$.
3. Bitwise Alternative: The $k^{th}$ symbol is actually just the parity of the number of set bits in $k-1$ (i.e., countSetBits(k-1) % 2).

4. Example explanation

$n = 3, k = 3$

• Row 1: 0
• Row 2: 01
• Row 3: 0110

1. $k=3$ is in the second half of Row 3 (which has 4 elements).
2. $k$ relative to first half: $3 - 2 = 1$.
3. Symbol at Row 3, index 3 is the inverse of Row 2, index 1.
4. Row 2, index 1 is '0'. Inverse is '1'. Result: 1.

5. Common mistakes candidates make

• Brute Force: Trying to build the strings. $2^{30}$ characters will cause a memory overflow.
• Off-by-one: Confusion between 0-indexed and 1-indexed $k$.
• Recursive Depth: Forgetting the base case (Row 1 is always 0).

6. Interview preparation tip

Whenever a sequence doubles in every step and you need the $k^{th}$ element, look for a Binary Tree relationship or a property of the binary representation of $k$. This is a vital Bit Manipulation interview pattern.