779. K-th Symbol in Grammar

Problem Statement

We build a table of n rows (1-indexed). We start by writing 0 in the 1st row. Now in every subsequent row, we look at the previous row and replace each occurrence of 0 with 01, and each occurrence of 1 with 10.

For example, for n = 3, the 1st row is 0, the 2nd row is 01, and the 3rd row is 0110. Given two integer n and k, return the kth (1-indexed) symbol in the nth row of a table of n rows.

Test Cases

Example 1:

Input: n = 1, k = 1 Output: 0 Explanation: row 1: 0

Example 2: Input: n = 2, k = 1 Output: 0 Explanation: row 1: 0 row 2: 01

Example 3: Input: n = 2, k = 2 Output: 1 Explanation: row 1: 0 row 2: 01

Solution

Approach Our approach is to leverage the recursive pattern observed. The length of the nth row is double the length of the (n-1)th row. If k is in the first half of the nth row, the symbol at position k in the nth row is the same as the symbol at position k in the (n-1)th row. If k is in the second half of the nth row, the symbol at position k in the nth row is the opposite of the symbol at position (k - length of (n-1)th row) in the (n-1)th row.

Complexity

Time complexity: The time complexity is O(n) as there is a single recursive call for every level of n. Space complexity: The space complexity is also O(n) due to the stack space used by the recursive calls.