Decoding the 779th K-th Symbol in Grammar

In the world of computer science and mathematics, there exist intriguing problems that seem simple on the surface but hide complex patterns and symmetries within. The 779th K-th Symbol in Grammar is one such problem that can be solved using a variety of methods. This article delves into the details of this problem and explores three different solutions to decipher the K-th symbol.

Understanding the Problem

The problem statement is straightforward. We are tasked with creating a table of rows, each containing a sequence of 0s and 1s. Starting with the first row as 0, in each subsequent row, we replace each 0 with 01 and each 1 with ten from the previous row. The objective is to find the K-th symbol in the N-th row.

For example, when N = 3, the first row is 0, the second row is 01, and the third row is 0110.

Solution 1: Recursion

The first solution provided uses a recursive approach. It defines a function 'getPrevRowValue' that calculates the K-th symbol by working its way backward through the rows. This method employs recursive calls to determine the previous row's value and then calculates the K-th symbol based on the pattern of 0s and 1s.

class Solution {

    fun kthGrammar(n: Int, k: Int): Int {

        return getPrevRowValue(n, k - 1)

    }



    fun getPrevRowValue(prevRow: Int, k: Int): Int {

        if (prevRow <= 1) return 0



        val prevValue = getPrevRowValue(prevRow - 1, k / 2)

        val productedValue = if (prevValue == 0) 1 else 10



        if (k % 2 == 0) {

            return (productedValue / 10)

        } else {

            return productedValue % 10

        }

    }

}

Solution 2: Bit Manipulation

The second solution leverages bitwise operations to efficiently find the K-th symbol. It iteratively divides the row into two halves and inverts the symbol whenever the value of K crosses the midpoint of the current row. This approach is based on the observation that each row is essentially divided into two halves, and the symbols on one half are inverted compared to the other half.

class Solution {

    fun kthGrammar(n: Int, k: Int): Int {

        var mid = 1.shl(n - 1) // 2^(n-1)

        var i = k

        var base = true // true - 0, false - 1



        while (mid > 1) {

            mid = mid shr 1   // mid / 2

            if (mid < i) {

                i -= mid       // shift i to the left side

                base = !base   // inversion

            }

        }



        return if (base) 0 else 1

    }

}

Solution 3: Recursion With Inversion

The third solution also employs recursion but simplifies the problem by considering the inversion of the symbols. It recursively calls the function with modified values, essentially flipping 0 to 1 and 1 to 0 when needed. This approach reduces the problem to its core and avoids calculating the entire row explicitly.

class Solution {

fun kthGrammar(n: Int, k: Int): Int = if (n == 1) 0 else 

    (if (kthGrammar(n - 1, (k + 1) / 2) == 0) k.inv() else k)

}

Conclusion

The 779th K-th Symbol in Grammar is a captivating problem that demonstrates the power of recursive thinking and mathematical patterns in computer science. These three solutions provide different ways to decode the K-th symbol efficiently, catering to various preferences and coding styles. When tackling this problem, choosing the right approach depends on the specific requirements and constraints of your project.

Whether you opt for the recursive approach, utilize bitwise operations, or work with symbol inversion, it is crucial to understand the problem's underlying principles. This problem is a prime example of how seemingly simple questions can lead to elegant and optimized solutions in the world of algorithms and computer science.