LeetCode 779. K-th Symbol in Grammar (Solution)

Introduction

In the vast universe of algorithmic problems, every once in a while, we stumble upon a puzzle that’s as intriguing as it is deceptive. One such riddle is LeetCode’s Problem 779: “K-th Symbol in Grammar.” At first glance, it might seem like a mere exercise in array manipulation. But as we dig deeper, a beautiful pattern, reminiscent of the famous fractals in mathematics, begins to emerge.

Intuition

Imagine a table built from rows of binary digits. The first row starts with a single digit: 0. Each subsequent row evolves from the one above, but not in the typical manner you might expect. A 0 transforms into “01”, and a 1 becomes “10”. The task? Given n (the row number) and k (the position in that row), determine the value at that position.

For instance, if n = 3, the sequence evolves as:

n = 1: 0
n = 2: 01
n = 3: 0110

As we analyze the sequence’s growth, a pattern emerges (see the figure below to visualize the pattern):

1. The first half of any row is identical to the previous row.
2. The second half is the inverse of the previous row.

This recursive nature is the key to unlocking our solution. If we know which half of the row our k-th position lies in, we can relate it back to a position in the n-1 row.

Code

1. Initialization:
   flip = False: This variable keeps track of whether the final result should be flipped. If flip is True by the end of the algorithm, the result is 1; otherwise, it’s 0.
   k -= 1: Since the problem is 1-indexed but Python uses 0-indexing, we subtract 1 from k to adjust.
   
2. Traversal:
   The while n != 1: loop ensures that we keep traversing upwards in the binary tree until we reach the root. Remember, the root is at n=1, and it always has a value of 0.
   
3. Logic Inside the Loop:
   if k % 2:: This checks if k is odd. If it’s odd, it means that we’re at the right child of the parent in the binary tree. Recall from the problem statement that 0 transforms to 01 and 1 transforms to 10. The right child always inverts the value of its parent. Therefore, if we’re at a right child, we flip the value.
   k = k//2: This line moves us up the binary tree, effectively halving the index. This is similar to integer division by 2.
   n -= 1: We move one row up in our virtual binary tree.
   
4. Result:
   return 1 if flip else 0: If flip is True, it returns 1; otherwise, it returns 0.

Relating to the Binary Tree Pattern:

The transformation pattern (0 -> 01 and 1 -> 10) essentially forms a binary tree. The left child of any node is the same as the parent, while the right child is the inverse.

With this in mind, the code is traversing from a leaf node (specified by n and k) up to the root of this binary tree. As it traverses, it checks if it’s currently at a right child (using the k % 2 check) and flips the value accordingly.

In essence, this uses the properties of the binary tree to avoid explicitly constructing the entire sequence, resulting in an efficient solution.

class Solution:
    def kthGrammar(self, n, k):
        flip = False
        k -= 1

        while n != 1:
            if k % 2:
                flip = not(flip)
            k = k//2
            n -= 1

        return 1 if flip else 0

If you are using Python version 3.10 or higher, you can try bit_count().

class Solution:
    def kthGrammar(self, n, k):
        return 1 if (k-1).bit_count()%2 else 0

Conclusion

The “K-th Symbol in Grammar” problem serves as a brilliant testament to the intertwined relationship between sequences and binary structures. On the surface, it may present itself as a mere generation of a sequence, but diving deeper reveals its inherent binary tree structure and the rules governing its growth.

The realization that each row in the sequence mirrors a level in a binary tree was pivotal. And within this tree, the transformation rules of 0 to 01 and 1 to 10 governed the progression. This binary tree structure wasn’t just a visual aid; it was intrinsically linked to the binary representation of numbers.

Our optimal solution, concise yet powerful, utilized the binary nature of numbers to determine the k-th symbol. By observing the set bits in the binary representation of our position, we discerned the number of ‘flips’ from the root, which in turn determined the value at that position. Python’s bit_count() method was the crowning jewel, streamlining this logic into an efficient solution.

In conclusion, the “K-th Symbol in Grammar” problem is more than just a sequence generation challenge. It’s an embodiment of the depth and beauty that lies beneath seemingly straightforward computational problems. It teaches us the importance of pattern recognition, the elegance of binary structures, and the power of efficient problem-solving.