#1888

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Number of Flips to Make the Binary String Alternating

Find the minimum number of flips to make a binary string alternate, using state transition dynamic programming.

string dynamic programming sliding window

Problem Statement

You are given a binary string, and you are allowed to perform two operations: flipping a bit to '0' or flipping a bit to '1'. Your task is to return the minimum number of operations required to convert the string into an alternating sequence, where no two adjacent characters are the same.

A string is considered alternating if each adjacent pair of characters differs. For example, '101010' is alternating, while '111000' is not. The string's length can go up to 100,000, so a time-efficient approach is needed to find the solution.

Examples

Example 1

Input: s = "111000"

Output: 2

Use the first operation two times to make s = "100011". Then, use the second operation on the third and sixth elements to make s = "101010".

Example 2

Input: s = "010"

Output: 0

The string is already alternating.

Example 3

Input: s = "1110"

Output: 1

Use the second operation on the second element to make s = "1010".

Constraints

1 <= s.length <= 105
s[i] is either '0' or '1'.

Solution Approach

State Transition Dynamic Programming

The key approach is to use dynamic programming to track the minimum number of flips required for each possible alternating sequence. By focusing on transitions between '0' and '1' at even and odd indices, the algorithm efficiently calculates the minimum flips needed.

Sliding Window Approach

A sliding window can be used to evaluate the string in chunks, ensuring that operations are minimized for each segment. By considering each position and the required flip to alternate it, you can reduce unnecessary operations while maintaining an optimal solution.

Track Odd and Even Indexed Flips

Track the number of '0's and '1's in odd and even positions separately. This helps to efficiently decide which bit to flip based on its position and the desired alternating pattern, minimizing the number of operations.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the final approach, but typically it will be O(n), where n is the length of the string. The space complexity also depends on the approach, but an optimal solution can achieve O(1) space using only a few variables for tracking positions and flips.

What Interviewers Usually Probe

Look for understanding of state transitions and their optimization in a string manipulation context.
Check if the candidate can connect dynamic programming principles with practical application, especially in minimizing operations.
Assess whether the candidate focuses on the efficiency of the algorithm, especially for large input sizes.

Common Pitfalls or Variants

Common pitfalls

Failing to recognize the importance of tracking odd and even positions separately can lead to inefficient solutions.
Not optimizing for large strings may result in time complexity that doesn't meet the problem's constraints.
Overcomplicating the solution with unnecessary operations can lead to excessive space or time complexity.

Follow-up variants

Consider variants where the string contains characters other than '0' and '1', such as binary sequences with additional symbols.
Adapt the problem to consider non-binary strings where adjacent characters must still differ, like alternating colors in a grid.
Change the problem to allow more complex operations, such as flipping multiple bits at once or limiting the number of allowed flips.

FAQ

How do I approach the Minimum Number of Flips to Make the Binary String Alternating problem?

Start by understanding the alternating string pattern and use dynamic programming to track flips needed at odd and even positions.

What dynamic programming technique is used in this problem?

State transition dynamic programming is used to track the minimum flips for each possible alternating pattern based on the string's current state.

What are common mistakes when solving the Minimum Number of Flips to Make the Binary String Alternating problem?

Failing to optimize for large input sizes and not efficiently tracking the number of flips for odd and even indexed positions can lead to inefficient solutions.

How can I optimize my solution for large strings in this problem?

Focus on tracking only essential values like the number of flips needed at odd and even indices, and avoid unnecessary operations or additional data structures.

Can the problem be adapted to other types of strings?

Yes, you can adapt the problem to non-binary strings or different operations, but the underlying pattern of alternating characters remains the same.

terminal

Solution

Solution 1: Sliding Window

We notice that operation $1$ effectively turns the string into a cycle, and operation $2$ makes a substring of length $n$ within the cycle into an alternating binary string.

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class Solution:
    def minFlips(self, s: str) -> int:
        n = len(s)
        target = "01"
        cnt = sum(c != target[i & 1] for i, c in enumerate(s))
        ans = min(cnt, n - cnt)
        for i in range(n):
            cnt -= s[i] != target[i & 1]
            cnt += s[i] != target[(i + n) & 1]
            ans = min(ans, cnt, n - cnt)
        return ans

Continue Practicing

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