This problem can be solved by recognizing the recursive pattern in the grammar table and applying bit manipulation to track flips efficiently. Each row doubles the previous row with complementary flips, allowing calculation of the K-th symbol without generating the full row. Understanding how to map K in row N to a position in row N-1 is key for constant space solutions.
Problem Statement
We define a binary grammar sequence starting with row 1 as '0'. Each subsequent row is generated by replacing every 0 with 01 and every 1 with 10 from the previous row. Given integers n and k, your task is to find the k-th symbol in the n-th row of this sequence without constructing the entire row.
For example, row 1 is 0, row 2 is 01, row 3 is 0110, and so on. Given n = 4 and k = 5, you need to compute the value at that position efficiently using the recursive and bit manipulation pattern, staying within the constraints 1 <= n <= 30 and 1 <= k <= 2^(n-1).
Examples
Example 1
Input: n = 1, k = 1
Output: 0
row 1: 0
Example 2
Input: n = 2, k = 1
Output: 0
row 1: 0 row 2: 01
Example 3
Input: n = 2, k = 2
Output: 1
row 1: 0 row 2: 01
Constraints
- 1 <= n <= 30
- 1 <= k <= 2n - 1
Solution Approach
Recursive Mapping Approach
Track the K-th symbol by mapping it to the parent position in row N-1. If K is even, it corresponds to the flipped value of the previous row's K/2 symbol; if K is odd, it is the same as the (K+1)/2 symbol. Apply recursion until reaching row 1.
Bit Manipulation Shortcut
Count the number of 1s in the binary representation of K-1. If this count is even, the symbol is 0; if odd, it is 1. This leverages the fact that each flip corresponds to a 1 in the path from root to position, allowing O(log k) time with O(1) space.
Iterative Parent Tracking
Start from row N and iteratively move to the corresponding parent in row N-1 using division and modulo operations. Track flips along the path to determine the final symbol without recursion, maintaining constant space.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(\log k) |
| Space | O(1) |
Time complexity is O(log k) because each step halves the position until reaching the first row. Space complexity is O(1) if using iterative bit tracking, as no full row is stored.
What Interviewers Usually Probe
- Expect candidates to notice the recursive generation of the grammar sequence quickly.
- Check if the solution avoids full row construction and leverages math or bit manipulation.
- Look for clarity in mapping K from row N to row N-1 to handle flips correctly.
Common Pitfalls or Variants
Common pitfalls
- Attempting to build the entire row, which causes memory overflow for larger N.
- Miscounting indices due to 1-based versus 0-based numbering.
- Failing to correctly apply flips when K is an even index in the recursion path.
Follow-up variants
- Compute multiple K-th symbols in the same row efficiently using batch bitwise operations.
- Extend the pattern to ternary or quaternary grammar sequences, adjusting the flip logic.
- Query a range of symbols in row N without generating the full sequence.
How GhostInterview Helps
- GhostInterview identifies the core recursive and bit manipulation pattern for K-th symbol computation.
- It guides the mapping from K in row N to its parent in row N-1 to prevent full row expansion.
- The solver highlights iterative bit-tracking techniques that reduce space complexity while preserving correctness.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the most efficient way to find the K-th symbol in the grammar sequence?
Use bit manipulation by counting 1s in the binary representation of K-1 or recursively map K to its parent in row N-1.
Why is recursion commonly used in this problem?
Recursion naturally reflects the sequence's generation pattern where each row is built from the previous row's symbols.
Can I calculate the symbol without constructing the entire row?
Yes, by tracking flips via bit counting or parent mapping, you can determine the K-th symbol directly.
What common mistakes should I avoid?
Avoid generating the full row, misapplying 1-based indexing, or forgetting flips on even positions.
How does the K-th Symbol in Grammar problem illustrate math plus bit manipulation patterns?
It uses binary representation to track symbol flips, combining math insight with bitwise operations to compute the answer efficiently.
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