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This approach utilizes prefix sums to calculate the minimum number of flips required to make a binary string monotone increasing. The idea is to traverse the string while maintaining a prefix count of '1's and suffix count of '0's. For each position i, calculate how many flips would be necessary to make all characters before i '0' + make all characters from i '1'.
Time Complexity: O(n) where n is the length of the string.
Space Complexity: O(n) due to the prefix ones array.
1def minFlipsMonoIncr(s: str) -> int:
2 n = len(s)
3 prefix_ones = [0] * (n + 1)
4 for i in range(n):
5 prefix_ones[i + 1] = prefix_ones[i] + (s[i] == '1')
6 min_flips = float('inf')
7 for i in range(n + 1):
8 flips = prefix_ones[i] + (n - i - (prefix_ones[n] - prefix_ones[i]))
9 min_flips = min(min_flips, flips)
10 return min_flipsThis solution iterates through the string and calculates prefix sums for '1's. Then it computes the potential flips required at each breakpoint to make the string monotone increasing and returns the minimum.
This approach leverages dynamic programming to find the minimum flips. Two states are maintained for any position: one representing the minimal flips to have a monotone ending with '0' and the other with '1'. Update these states as we traverse the string.
Time Complexity: O(n) as we traverse across the string once.
Space Complexity: O(1) since only fixed states are maintained.
1public class Solution {
2 public int MinFlipsMonoIncr(string s) {
int flip0 = 0, flip1 = 0;
foreach (char c in s) {
int newFlip1 = Math.Min(flip0, flip1) + (c == '0' ? 1 : 0);
flip0 = flip0 + (c == '1' ? 1 : 0);
flip1 = newFlip1;
}
return Math.Min(flip0, flip1);
}
}A C# method dynamically updating flip-hold states to minimize monotone sequence transformation, designed for straightforward computation and update through enumeration.