Minimum Number of K Consecutive Bit Flips

Minimum Number of K Consecutive Bit Flips

1. What is this problem about?

The Minimum Number of K Consecutive Bit Flips coding problem is a binary array challenge. You are given an array of 0s and 1s and an integer $K$. You can perform a "K-bit flip" by choosing any contiguous subarray of length $K$ and flipping all its bits (0 becomes 1, and 1 becomes 0). Your objective is to find the minimum number of flips needed to make every element in the array equal to 1. If it is impossible, you should return -1.

2. Why is this asked in interviews?

This question is a favorite at companies like Bloomberg and Google because it tests a candidate's ability to optimize a greedy approach. A naive simulation would take $O(N \cdot K)$ time, which is too slow. The challenge is to use a Sliding Window or a Difference Array technique to track the number of flips affecting each bit in $O(1)$ time per element, demonstrating high-level algorithmic thinking.

3. Algorithmic pattern used

This problem follows the Greedy interview pattern with Lazy Propagation. The greedy choice is simple: if you are at index i and the bit is currently 0, you must flip the subarray starting at i and ending at i+K-1. There is no other way to fix that 0 without affecting bits to its left. To do this efficiently:

1. Keep track of how many flips are currently "active" (affecting the current index).
2. If the number of active flips makes the current bit 0, start a new flip (if possible).
3. Use a queue or a marker in the array to know when a flip's $K$-length range has expired.

4. Example explanation

Input: [0, 1, 0], $K = 2$.

1. Index 0: The bit is 0. We must flip the range $[0, 1]$. Total flips = 1. Active flips = 1.
2. Index 1: The bit was 1, but we have 1 active flip, so it's now 0. We must flip the range $[1, 2]$. Total flips = 2. Active flips = 2.
3. Index 2: The bit was 0. The first flip (at index 0) has expired. We have 1 active flip (from index 1). The bit becomes 1. All bits are now 1. Result: 2 flips.

5. Common mistakes candidates make

• Inefficient Flipping: Actually iterating through $K$ elements to flip them every time a 0 is found. This leads to $O(N \cdot K)$ time complexity.
• Boundary Errors: Failing to check if there are at least $K$ elements remaining before starting a flip. If you need to flip but only have $K-1$ elements left, it's impossible.
• State Tracking: Miscalculating the "parity" of flips. Since flipping twice brings you back to the original state, only the even/odd count of active flips matters.

6. Interview preparation tip

Practice "in-place" state tracking. For problems where an action affects a range, see if you can use the array itself (by adding a large number or flipping a sign) to mark the end of that action's influence. This is a common trick in Sliding Window interview patterns to achieve $O(1)$ extra space.