Flip String to Monotone Increasing

Flip String to Monotone Increasing

1. What is this problem about?

The Flip String to Monotone Increasing interview question asks for the minimum number of flips (0 to 1 or 1 to 0) to make a binary string "monotone increasing." A string is monotone increasing if it consists of some number of 0s followed by some number of 1s (e.g., "00011").

2. Why is this asked in interviews?

Companies like Meta and Amazon ask the Flip String coding problem to evaluate a candidate's ability to use Dynamic Programming or single-pass optimizations. It tests if you can identify that at any point in the string, you only need to know how many 1s came before and the minimum flips needed so far. It evaluation your String interview pattern skills.

3. Algorithmic pattern used

This problem can be solved with a One-Pass Greedy/DP approach.

1. Counters: Maintain onesCount (number of 1s seen so far) and flipCount (min flips to make the prefix monotone).
2. Logic: Iterate through the string:
   • If you see a '1', just increment onesCount. No flips needed to keep it monotone (it stays at the end).
   • If you see a '0', you have two choices to maintain monotonicity:
     • Flip this '0' to '1' (cost = flipCount + 1).
     • Keep this '0' and flip all previous '1's to '0' (cost = onesCount).
   • Update flipCount = min(flipCount + 1, onesCount).

4. Example explanation

String: "00110"

1. '0': ones=0, flip=0.
2. '0': ones=0, flip=0.
3. '1': ones=1, flip=0.
4. '1': ones=2, flip=0.
5. '0': flip = min(0+1, 2) = 1. Result: 1 flip. (Flip the last 0 to 1 -> "00111").

5. Common mistakes candidates make

• $O(N^2)$ prefix sums: Calculating total flips for every possible split point $(0...i)$ and $(i...n)$. While $O(N)$ with prefix sums, the one-pass approach is cleaner and uses less space.
• Greedy Error: Thinking you should always flip the smaller count. You must track the global minimum cumulative cost.
• Wrong target: Trying to make the string all 0s or all 1s instead of specifically monotone increasing.

6. Interview preparation tip

Whenever you see "Minimum flips/changes to reach a target state," think of Prefix DP. If the state is simple (like monotone increasing), you can often reduce the DP table to just one or two variables.