Maximum Binary String After Change

Maximum Binary String After Change

What is this problem about?

The Maximum Binary String After Change coding problem presents you with a binary string consisting of '0's and '1's. You are allowed to perform two types of operations any number of times:

1. Change "00" to "10".
2. Change "10" to "01".

The objective is to transform the string into the lexicographically largest possible binary string. In binary terms, a larger string is one that has '1's as far to the left as possible.

Why is this asked in interviews?

This problem is frequently used in interviews (notably at Huawei) to test a candidate's grasp of Greedy Algorithms. It requires observing the underlying rules of the transformation:

1. The first operation ("00" -> "10") allows us to "consume" a '0' to turn a leading '0' into a '1'.
2. The second operation ("10" -> "01") allows us to "shift" a '0' to the right through '1's.

By combining these, we can effectively move all '0's together and then reduce them to a single '0' placed as far to the right as possible, maximizing the leading '1's.

Algorithmic pattern used

The core pattern is Greedy logic. Instead of simulating the operations (which could be very slow), we analyze the final state of the string.

• Leading '1's in the original string stay at the front.
• All '0's after the first group of '1's can be moved together.
• Once all '0's are adjacent (e.g., "000"), we can repeatedly apply "00" -> "10" until only one '0' remains (e.g., "110").
• The only way to not have a '0' in the final string is if the original string had fewer than two '0's.

Example explanation

Consider the string "000110".

1. Count the total number of zeros. Here, we have four '0's.
2. Find the index of the first '0'. It's at index 0.
3. We can move all '0's after the first one to be adjacent to it using the "10" -> "01" rule. This conceptually gives us something like "000011".
4. Apply "00" -> "10" repeatedly.
   • "0000" becomes "1000"
   • "1000" becomes "1100"
   • "1100" becomes "1110"
5. The final result is "111011". Note that the single '0' is placed at index_of_first_zero + (total_zeros - 1).

Common mistakes candidates make

• Direct Simulation: Attempting to literally search for "00" and "10" and replace them can lead to infinite loops or suboptimal time complexity.
• Miscounting Zeros: Forgetting that leading '1's before the first '0' cannot be moved or changed.
• Overcomplicating the logic: Missing the greedy observation that any string with at least one '0' will end up with exactly one '0' (unless it was all '1's initially).

Interview preparation tip

For string manipulation problems where you want to maximize the lexicographical value, always look for ways to "push" the smaller characters (like '0') as far to the right as possible. Ask yourself: "What is the best possible version of this string I can imagine?" and then see if the operations allow you to reach it.