Minimum Number of K Consecutive Bit Flips

You are given a binary array nums and an integer k.

A k-bit flip is choosing a subarray of length k from nums and simultaneously changing every 0 in the subarray to 1, and every 1 in the subarray to 0.

Return the minimum number of k-bit flips required so that there is no 0 in the array. If it is not possible, return -1.

A subarray is a contiguous part of an array.

For example:

Input: nums = [0,1,0], k = 1 Output: 2 Explanation: Flip nums[0], then flip nums[2].

Input: nums = [1,1,0], k = 2 Output: -1 Explanation: No matter how we flip subarrays of size 2, we cannot make the array become [1,1,1].

Input: nums = [0,0,0,1,0,1,1,0], k = 3 Output: 3 Explanation: Flip nums[0],nums[1],nums[2]: nums becomes [1,1,1,1,0,1,1,0] Flip nums[4],nums[5],nums[6]: nums becomes [1,1,1,1,1,0,0,0] Flip nums[5],nums[6],nums[7]: nums becomes [1,1,1,1,1,1,1,1]

def min_k_bit_flips(nums, k):
    n = len(nums)
    flips = 0
    flipped = [0] * n  # flipped[i] indicates if the subarray ending at i has been flipped
    curr_flips = 0

    for i in range(n):
        if i >= k:
            curr_flips -= flipped[i - k]

        if (nums[i] + curr_flips) % 2 == 0:
            if i + k > n:
                return -1
            flips += 1
            curr_flips += 1
            flipped[i] = 1

    return flips

# Example 1
nums1 = [0, 1, 0]
k1 = 1
print(f"Example 1: nums = {nums1}, k = {k1}, Output = {min_k_bit_flips(nums1, k1)}")

# Example 2
nums2 = [1, 1, 0]
k2 = 2
print(f"Example 2: nums = {nums2}, k = {k2}, Output = {min_k_bit_flips(nums2, k2)}")

# Example 3
nums3 = [0, 0, 0, 1, 0, 1, 1, 0]
k3 = 3
print(f"Example 3: nums = {nums3}, k = {k3}, Output = {min_k_bit_flips(nums3, k3)}")

# Edge Case 1: k is larger than nums length
nums4 = [0,1]
k4 = 3
print(f"Edge Case 1: nums = {nums4}, k = {k4}, Output = {min_k_bit_flips(nums4, k4)}")

# Edge Case 2: nums is already all 1s
nums5 = [1,1,1]
k5 = 2
print(f"Edge Case 2: nums = {nums5}, k = {k5}, Output = {min_k_bit_flips(nums5, k5)}")

Functionality

The function min_k_bit_flips(nums, k) calculates the minimum number of k-bit flips required to make all elements in the nums array equal to 1. It returns -1 if it's impossible.

Explanation:

1. Initialization:

   • flips: Counts the number of k-bit flips.
   • flipped: A list to track if a flip has been performed ending at index i. This helps to keep track of the current number of active flips.
   • curr_flips: Keeps track of the number of active flips affecting the current element.

2. Iteration:

   • Iterate through the nums array.
   • If the current index i is greater than or equal to k, it means a previous flip's effect might no longer be active. Remove flipped[i - k] from curr_flips.
   • Check if the current element nums[i] plus the current number of active flips curr_flips results in an even number (which means 0). If so, a flip is needed.
   • If a flip is needed, check if there are enough elements to perform a k-bit flip. If not, return -1.
   • Increment flips and curr_flips, and mark flipped[i] as 1 to indicate that a flip has been performed ending at the current index.

3. Return:

   • Return the total number of flips.

Big O Runtime Analysis

The time complexity of the solution is O(n), where n is the length of the nums array. This is because the code iterates through the array once.

Big O Space Analysis

The space complexity of the solution is O(n), where n is the length of the nums array. This is due to the flipped array that keeps track of the flips performed. Note that curr_flips is only an integer so does not impact the overall space complexity.

Edge Cases

1. k is larger than nums length:

   • If k is larger than the length of nums, it's impossible to perform a k-bit flip on the entire array. It should return -1 if a zero remains in the array after one attempted flip.

2. nums is already all 1s:

   • If nums is already all 1s, no flips are needed, and the function should return 0.

3. Array cannot be converted to all 1s:

   • If there is no way to convert the array to all 1s with k-bit flips, the function returns -1.