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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.
1public class Solution {
2 public int MinFlipsMonoIncr(string s) {
3 int n = s.Length;
4 int[] prefixOnes = new int[n + 1];
5 for (int i = 0; i < n; ++i) {
6 prefixOnes[i + 1] = prefixOnes[i] + (s[i] == '1' ? 1 : 0);
7 }
8 int minFlips = int.MaxValue;
9 for (int i = 0; i <= n; ++i) {
10 int flips = prefixOnes[i] + (n - i - (prefixOnes[n] - prefixOnes[i]));
11 minFlips = Math.Min(minFlips, flips);
12 }
13 return minFlips;
14 }
15}The C# version uses an integer array for prefix sums, allowing it to compute and minimize the number of necessary bit-flips through a single loop pass.
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.
1def minFlipsMonoIncrThe dynamic programming approach maintains two counters that track the minimal flips for maintaining monotonic sequences. As the string is parsed, appropriate states are updated.