926 Flip String to Monotone Increasing.py

#!/usr/bin/python3
"""
A string of '0's and '1's is monotone increasing if it consists of some number
of '0's (possibly 0), followed by some number of '1's (also possibly 0.)

We are given a string S of '0's and '1's, and we may flip any '0' to a '1' or a
'1' to a '0'.

Return the minimum number of flips to make S monotone increasing.



Example 1:

Input: "00110"
Output: 1
Explanation: We flip the last digit to get 00111.
Example 2:

Input: "010110"
Output: 2
Explanation: We flip to get 011111, or alternatively 000111.
Example 3:

Input: "00011000"
Output: 2
Explanation: We flip to get 00000000.


Note:

1 <= S.length <= 20000
S only consists of '0' and '1' characters.
"""


class Solution:
    def minFlipsMonoIncr(self, S: str) -> int:
        """
        let S[i] be the flipping point, leftside 0, rightside 1
        count number of 1 from the left,
        count number of 0 from the right
        O(N)
        """
        n = len(S)
        Z = [0 for _ in range(n+1)]  # let Z[i] be #zero in A[i:]
        O = [0 for _ in range(n+1)]  # let O[i] be #one in A[:i]
        for i in range(1, n+1):
            O[i] = O[i-1]
            if S[i-1] == "1":
                O[i] += 1

        for i in range(n-1, -1, -1):
            Z[i] = Z[i+1]
            if S[i] == "0":
                Z[i] += 1

        ret = float('inf')
        for i in range(n):
            ret = min(ret, O[i] + Z[i+1])

        return ret