Flip String to Monotone Increasing

class Solution {
public:
    int minimumFlipsMonotonic(string input) {
        int result = 0, countOne = 0;
        for (char ch : input) {
            if (ch == '0') {
                result = min(countOne, result + 1);
            } else {
                ++countOne;
            }
        }
        return result;
    }
};

In the provided C++ code, a solution class has a method named minimumFlipsMonotonic, which takes a string, input, as parameter and returns an integer. This function aims to determine the minimum number of flips required to make the input string monotonic increasing. Monotonic increasing here implies that all the zeroes in the string come before all the ones.

An explanation of how the code accomplishes this:

• Two integer variables, result and countOne, are initialized to zero. The result will store the minimum number of flips needed, while countOne will keep track of the number of '1's encountered as the string is processed.

• The function then loops through each character in the input string. For each character:

  • If it encounters a '0', it updates result by comparing the current value of countOne (representing flips to change all previous '1' to '0') and result + 1 (representing a flip of the current '0' to '1'), taking the minimum of these two values.

  • If it encounters a '1', the countOne is incremented, updating the potential number of flipped ones.

• Finally, the value of result is returned. This value represents the minimal flip operations required to make the string monotonic increasing.

class Solution {
    public int minimumFlipsToMonotonic(String str) {
        int flips = 0, countOnes = 0;
        for (int index = 0; index < str.length(); ++index) {
            if (str.charAt(index) == '0') {
                flips = Math.min(countOnes, flips + 1);
            } else {
                ++countOnes;
            }
        }
        return flips;
    }
}

The provided Java solution addresses the problem of converting a binary string into a monotone increasing string using the minimum number of flips. A monotone increasing string is one where every character is either the same as or greater than the one before it, which, in the case of a binary string, means the string could transition from any number of '0's to any number of '1's but not vice versa.

• The solution iterates through the input string to count flips needed to turn the string into a monotone increasing string.
• Maintains a flips counter to track the minimal flips required and a countOnes counter to count the number of '1's encountered.
• If a '0' is found in the string, the algorithm computes whether it is better to flip this '0' to '1' or to flip all previously encountered '1's to '0's.
• This decision is based on comparing the current number of flips required if flipping the '0' (flips + 1) versus the total '1's seen so far (countOnes).
• At the end, the function returns the computed minimum number of flips (flips) that would turn the input string into a monotone increasing string.

Ensure to provide a valid binary string input to avoid unexpected behaviors since the function assumes a valid input consisting only of '0's and '1's.

class Solution:
    def minimumFlipsToMonotonic(self, binary_str: str) -> int:
        flip_count = 0
        one_count = 0
        for char in binary_str:
            if char == '0':
                flip_count = min(one_count, flip_count + 1)
            else:
                one_count += 1
        return flip_count

The given Python solution addresses the problem of flipping a binary string to make it monotone increasing. The approach utilizes dynamic programming to efficiently determine the minimal number of flips required. Below is an outline of the solution steps:

• Initialize two counters: flip_count to track the number of flips, and one_count to track the number of '1's encountered.
• Traverse the binary string character by character using a loop.
  • If the current character is '0', update flip_count as the minimal value between one_count (flipping all previous '1's to '0's) and flip_count + 1 (flipping the current '0' to '1').
  • If the current character is '1', increment one_count.
• Return the flip_count finally determined, which represents the minimum number of flips required to make the string monotone increasing.

This strategy ensures an optimal solution by considering flipping sequences dynamically, counting flips and ones separately to make decisions at each step. The time complexity of this solution is O(n), where n is the length of the string, due to the single traversal of the characters.