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Special Binary String

Number: 763

Difficulty: Hard

Paid? No

Companies: Nvidia, Google, Grammarly, Adobe, Quip (Salesforce), Coursera

Problem Description

Given a special binary string s (which is defined as a binary string that has an equal number of 0's and 1's and every prefix has at least as many 1's as 0's), the task is to perform swaps between consecutive, non-empty special substrings so that the resulting string is lexicographically largest possible.

Key Insights

  • The problem has an inherent recursive structure where each valid special string can be broken down into smaller special strings.
  • The inner structure between a matching pair of 1 and 0 (i.e. a segment "1" + sub-string + "0") is itself a special binary string.
  • Once you recursively transform the inner segments, sorting these segments in descending order helps maximize the lexicographical order of the overall string.
  • The crucial trick is recognizing that by recursively processing the inner parts and then reassembling them after sorting, you capture all possible valid swaps.

Space and Time Complexity

Time Complexity: O(n^2 * log n) in the worst-case scenario due to the recursive partitioning and sorting of segments. Space Complexity: O(n) for recursion depth and storage of sub-segments.

Solution

We solve the problem using a recursive approach. The algorithm works as follows:

  1. Initialize a counter and traverse string s.
  2. For every substring that forms a valid special binary string (when the counter reaches zero), recursively process the inner substring (after removing the leading '1' and trailing '0').
  3. Wrap the processed substring with '1' and '0' and add it to a list of candidate segments.
  4. Sort these segments in descending (lexicographically largest) order.
  5. Concatenate the sorted segments and return the result.

Key data structures used include:

  • A list (or array) to hold candidate special substrings.
  • Recursive function calls to process inner special strings.

This recursive decomposition ensures we explore all possible swaps implicitly, and sorting the segments yields the lexicographically largest result possible.

Code Solutions

# Python solution with detailed comments

def makeLargestSpecial(s: str) -> str:
    # Base case: if s is empty, return empty string
    if not s:
        return s

    segments = []  # List to hold valid special substrings
    count = 0      # Counter to track the balance of '1's and '0's
    start = 0      # Starting index for a potential special substring

    # Iterate over each character in string s
    for i, char in enumerate(s):
        # Increase counter for '1' and decrease for '0'
        if char == '1':
            count += 1
        else:
            count -= 1

        # When the count becomes 0, we found a valid special substring
        if count == 0:
            # Recursively process the inner part (excluding the outer '1' and '0')
            inner = makeLargestSpecial(s[start+1:i])
            # Wrap the processed inner substring with outer characters '1' and '0'
            segments.append("1" + inner + "0")
            start = i + 1  # Update start for the next segment

    # Sort segments in reverse lexicographical order to maximize value
    segments.sort(reverse=True)
    # Join sorted segments and return as final answer
    return "".join(segments)

# Example usage:
print(makeLargestSpecial("11011000"))
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