Solution Explanation for 1702. Maximum Binary String After Change

This problem involves transforming a binary string using two operations to obtain the lexicographically largest possible string. The key insight is that the optimal strategy doesn't require exploring all possible operation sequences. We can achieve the maximum string with a greedy approach.

Core Idea:

The operations allow us to effectively move '1's to the right and change consecutive '0's into '1's (except for potentially the last one). The optimal solution will have as many '1's as possible at the beginning, followed by at most one '0', and then the remaining '1's.

Algorithm:

1. Find the first '0': Locate the index of the first occurrence of '0' in the input string. If no '0' exists, the string is already maximal, and we return it.

2. Count trailing '0's: Count the number of '0's after the first '0' found in step 1.

3. Construct the maximal string: Create a new string with:

   • k leading '1's, where k is the index of the first '0' plus the count of trailing '0's.
   • A single '0' at position k.
   • The remaining characters are filled with '1's.

Time and Space Complexity:

• Time Complexity: O(n), where n is the length of the binary string. We iterate through the string at most twice (once to find the first '0' and once to count trailing '0's). The string construction is also linear in n.

• Space Complexity: O(n) in the worst case. This is due to the creation of the new string in the solution, which has the same length as the input string. In some implementations, the space usage can be improved by modifying the string in place, reducing the space complexity to O(1).

Code Examples (Python):

The Python solution demonstrates the efficiency of this approach:

class Solution:
    def maximumBinaryString(self, binary: str) -> str:
        first_zero_index = binary.find('0')
        if first_zero_index == -1:  # No '0's, already maximal
            return binary
 
        trailing_zeros = binary[first_zero_index + 1:].count('0')
        k = first_zero_index + trailing_zeros
 
        return '1' * k + '0' + '1' * (len(binary) - k - 1)
 

This code directly implements the algorithm described above. It finds the first zero, counts trailing zeros, and constructs the maximal string efficiently. Other languages follow a similar approach, although specific string manipulation functions might differ. The core logic and time/space complexity remain consistent across different programming languages.