Minimum K Consecutive Bit Flips

Summary

Given a binary array, find the minimum number of bit flips (i.e., changing 0 to 1 or vice versa) required to make all subarrays of length K have at least one 1. This problem involves prefix sums and sliding windows, making it a challenging problem that requires careful planning and execution.

Detailed Explanation

To solve this problem, we can use a combination of prefix sums and sliding windows. The idea is to maintain a running count of the number of 0s in the current window of size K, as well as the total number of 1s seen so far. When we encounter a 1, we update our running count and increment our total count.

Here's a step-by-step breakdown of the solution:

1. Initialize two variables: count to keep track of the number of 0s in the current window, and total_ones to keep track of the total number of 1s seen so far.
2. Iterate through the binary array using a sliding window of size K.
3. For each element in the array, update our running count count based on whether it's a 0 or 1. If it's a 0, increment count. If it's a 1, reset count to 0.
4. If we've reached the end of the window, check if the number of 1s in the window is less than K-1. If so, update our total count by adding the length of the current window minus one (since we need at least one 1).
5. Repeat steps 2-4 until the end of the array.
6. The minimum number of bit flips required is equal to the total count.

Time complexity: O(n), where n is the length of the binary array. Space complexity: O(1), since we only use a constant amount of space to store our running counts and totals.

Optimized Solutions

Java