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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.
1public classThe Java-based solution adopts a dynamic programming method to selectively compute necessary bit flips ensuring efficiency by utilizing two rolling state variables.