#3307

Hard

LeetCode Problem Workspace

Find the K-th Character in String Game II

Find the K-th character in a string game using bit manipulation and recursion, optimizing performance for large k values.

math bit manipulation recursion

Problem Statement

Alice starts with a string word = "a". A sequence of operations is given where each operation modifies this string. Each operation either doubles the string or adds a character to it, depending on whether the operation is 0 or 1. Alice must perform all operations in sequence.

Given an integer k, you must determine which character is at the k-th position of the resulting string after performing all operations. The constraints on k and the number of operations make a brute-force solution infeasible, and efficient simulation using bit manipulation is required.

Examples

Example 1

Input: k = 5, operations = [0,0,0]

Output: "a"

Initially, word == "a" . Alice performs the three operations as follows:

Example 2

Input: k = 10, operations = [0,1,0,1]

Output: "b"

Initially, word == "a" . Alice performs the four operations as follows:

Constraints

1 <= k <= 1014
1 <= operations.length <= 100
operations[i] is either 0 or 1.
The input is generated such that word has at least k characters after all operations.

Solution Approach

Simulate Operations Efficiently

Rather than building the entire string, track its length after each operation. Use bit manipulation to track how the string evolves and how the k-th character is affected without constructing the full string.

Use Recursive Backtracking

Perform a recursive approach where the solution traces backward through the operations to determine the exact character at position k. This avoids the need to simulate the entire string and optimizes performance.

Mathematical Insight on String Doubling

Recognize that operations 0 and 1 cause the string to grow exponentially. Efficiently calculate the new string size after each operation using the properties of doubling, reducing the need for direct string construction.

Complexity Analysis

Metric	Value
Time	O(\log k)
Space	O(1)

The time complexity is O(log k) due to the recursive approach that traces the operations. Each operation is efficiently processed without building the string. The space complexity is O(1), as no additional space is required aside from the current state and recursive stack.

What Interviewers Usually Probe

Candidate demonstrates an understanding of recursion and bit manipulation.
Candidate uses mathematical insight to reduce the problem size.
Candidate optimizes space complexity by not constructing the full string.

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the role of bit manipulation and simulating the string directly.
Incorrectly handling the recursive backtracking step, leading to errors in the character's position.
Overcomplicating the problem by trying to build the full string rather than focusing on the positions.

Follow-up variants

The operations array could have a different length, changing the string evolution process.
The string could have more complex transformations, such as alternating operations of 0 and 1.
Considerations on handling very large k values may differ, affecting performance and approach.

FAQ

How do I optimize the search for the k-th character in "Find the K-th Character in String Game II"?

By using recursion and bit manipulation to track the length and evolution of the string without constructing it entirely. This minimizes time complexity.

What is the primary pattern in "Find the K-th Character in String Game II"?

The problem uses a combination of mathematical insight and bit manipulation to efficiently trace the k-th character through multiple string operations.

Can this problem be solved with brute force?

No, due to the constraints on k and the number of operations, a brute-force approach would be too slow and inefficient.

How do I handle large k values in "Find the K-th Character in String Game II"?

By simulating the string transformations mathematically and recursively, without building the entire string, making it scalable for large k values.

What is the time complexity of "Find the K-th Character in String Game II"?

The time complexity is O(log k), as the solution involves recursively simulating the string’s evolution without constructing it.

terminal

Solution

Solution 1: Recurrence

Since the length of the string doubles after each operation, if we perform $i$ operations, the length of the string will be $2^i$.

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class Solution:
    def kthCharacter(self, k: int, operations: List[int]) -> str:
        n, i = 1, 0
        while n < k:
            n *= 2
            i += 1
        d = 0
        while n > 1:
            if k > n // 2:
                k -= n // 2
                d += operations[i - 1]
            n //= 2
            i -= 1
        return chr(d % 26 + ord("a"))

Continue Practicing

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