#926

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Flip String to Monotone Increasing

Minimize the number of flips needed to make a binary string monotone increasing using dynamic programming.

Problem Statement

You are given a binary string s. A binary string is monotone increasing if it consists of some number of 0's (possibly none), followed by some number of 1's (also possibly none). You are allowed to flip any character s[i] from '0' to '1' or from '1' to '0'. Your task is to determine the minimum number of flips required to make the string monotone increasing.

For example, given s = '00110', the string can be flipped at the last digit to obtain '00111'. The minimum number of flips is 1. Your goal is to find the most efficient way to perform the flips using dynamic programming.

Examples

Example 1

Input: s = "00110"

Output: 1

We flip the last digit to get 00111.

Example 2

Input: s = "010110"

Output: 2

We flip to get 011111, or alternatively 000111.

Example 3

Input: s = "00011000"

Output: 2

We flip to get 00000000.

Constraints

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

Solution Approach

State Transition Dynamic Programming

The problem can be solved using a dynamic programming approach where we track the minimum number of flips needed to achieve a monotone increasing string. We maintain two variables: one for the minimum flips if the string ends with '0' and another for flips if it ends with '1'. The transitions involve deciding whether flipping a current character or leaving it as is minimizes the flip count.

Iterative DP Update

We iterate over the string once, updating the flip count for each character based on the current state and previously computed values. For each index i, we compute the minimum number of flips needed to ensure the string remains monotone increasing by considering the state transitions for both '0' and '1'. This allows for a time complexity of O(n), where n is the length of the string.

Space Optimization

To optimize space, we can reduce the space complexity from O(n) to O(1) by only keeping track of the last two values (for '0' and '1' end states) instead of maintaining the entire DP table. This greatly reduces memory usage without affecting the time complexity.

Complexity Analysis

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

The time complexity is O(n) because we only iterate through the string once, where n is the length of the string. The space complexity is O(1) if we optimize by storing only the necessary previous state values for the dynamic programming transitions, instead of using an entire DP table.

What Interviewers Usually Probe

Candidate should demonstrate a solid understanding of dynamic programming, especially state transitions.
Look for understanding of space optimization techniques in dynamic programming problems.
Candidate should discuss the importance of minimizing flip operations and trade-offs involved in the solution approach.

Common Pitfalls or Variants

Common pitfalls

Failing to optimize space complexity by keeping unnecessary state values.
Confusing the problem with one where no flips are allowed, missing the need for careful state transitions.
Not correctly accounting for the two possible states ('0' end or '1' end) during the iteration, leading to incorrect results.

Follow-up variants

Consider variations with larger binary strings where optimized space handling is critical.
Extend the problem by introducing additional operations, such as flipping multiple characters at once.
Add constraints where the number of flips allowed is limited, adding complexity to the dynamic programming approach.

FAQ

What is the minimum number of flips needed for the string '00110'?

The minimum number of flips is 1. Flipping the last '0' to '1' gives '00111'.

How can dynamic programming help in solving this problem?

Dynamic programming helps by tracking the minimum flips required at each position in the string, allowing for optimal decisions about whether to flip or leave characters as is.

What is the time complexity of solving the 'Flip String to Monotone Increasing' problem?

The time complexity is O(n), where n is the length of the string, as we only need to iterate through the string once.

What is the space complexity of the solution?

The space complexity can be optimized to O(1) by keeping track of only the necessary state values during iteration.

How does GhostInterview help with this type of problem?

GhostInterview provides step-by-step explanations, optimized solutions, and real-time practice for dynamic programming problems like 'Flip String to Monotone Increasing'.

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Solution

Solution 1: Prefix Sum + Enumeration

First, we count the number of '0's in string $s$, denoted as $tot$. We define a variable $ans$ for the answer, initially set $ans = tot$, which represents the number of flips to change all '0's to '1's.

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class Solution:
    def minFlipsMonoIncr(self, s: str) -> int:
        tot = s.count("0")
        ans, cur = tot, 0
        for i, c in enumerate(s, 1):
            cur += int(c == "0")
            ans = min(ans, i - cur + tot - cur)
        return ans

Continue Practicing

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