Problem Description

In the given problem, we have an array nums which contains only 0s and 1s. Our goal is to find the minimum number of adjacent swaps required to create a subarray of k consecutive 1s.

The concept of a move involves selecting two adjacent positions in this array and swapping their values, with the restriction that a move can only be made between two positions that are next to each other.

Intuition

The intuition behind solving this problem lies in focusing on the position of the 1s rather than the position of the 0s. Since we're only interested in the position of 1s forming a consecutive sequence, we can abstract away the 0s by considering the original indices of 1s only.

We construct a new array, arr, to store the positions of the 1s from the original nums array. This abstracted view helps us in calculating the total moves without being influenced by the intermediating 0s.

The procedure of the solution involves the following steps:

1. Record the positions of 1s: We iterate through nums and record the indices of 1s in a new list arr.

2. Calculate prefix sums: We find the prefix sums of the positions to have the accumulative distances in hand, which will later facilitate the calculation of moves. The prefix sum array s helps to calculate the total distance of 1s before and after a central point i.

3. The midpoint i in arr represents a hypothetical center of our sequence of k 1s, which means there are x 1s on its left and y 1s on its right. We calculate x and y such that they represent the split of k 1s around the center i.

4. For each valid midpoint i, we calculate the minimum moves by considering the sum of distances from the left 1s (a) and right 1s (b) with adjustments for their positions relative to i.

5. We repeat step 4 for all valid i and keep track of the minimum number of moves needed (ans).

We use the concept of sliding window, which moves from left to right through our arr. Each iteration represents a potential solution where the x on the left and the y on the right are candidates for the required consecutive k 1s. Each potential solution provides the number of moves needed and we take the minimum of all these moves which is the answer.

The above process enables us to calculate the minimum number of adjacent swaps to transform the nums array into one that has k consecutive 1s in the least complex and most effective way.

Learn more about Greedy, Prefix Sum and Sliding Window patterns.

Solution Approach

The solution approach uses a sliding window technique, which is a common pattern used when dealing with subarrays or subsequences in an array. Let's walk through the code and understand the logic and use of algorithms and data structures.

1. First, we extract the positions of the 1s from the nums array and store them in a new list arr. This is achieved with a list comprehension iterating over nums and including only indices where the value is 1.

   arr = [i for i, x in enumerate(nums) if x == 1]

   This step reduces the problem space from having to deal with both 0s and 1s to focusing solely on the positions of 1s, which simplifies the problem significantly.

2. Next, we calculate the prefix sums of the positions stored in arr. We use accumulate from the itertools module, initializing the accumulation with 0 (as implied by initial=0).

   s = list(accumulate(arr, initial=0))

   The prefix sum array s helps us quickly get the total distance that the 1s to the left and right of a mid-point have to travel in order to become a contiguous block.

3. For the calculations that follow, we need to determine the number of 1s on the left (x) and right (y) sides of our sliding window. Here k is the size of our target subarray:

   x = (k + 1) // 2
   y = k - x

   This gives an odd k a left-heavier bias and puts the extra 1 on the left side.

4. We then iterate over the positions in arr where our target subarray of k 1s could be centered.

   for i in range(x - 1, len(arr) - y):
       j = arr[i]
       ls = s[i + 1] - s[i + 1 - x]
       rs = s[i + 1 + y] - s[i + 1]
       ...

   Here i is the index in arr and j is the value at index i in arr, which corresponds to a position in the original nums array.

5. Inside the loop, we calculate the number of moves for the left partition (a) and the right partition (b). This is where the prefix sums come in handy; they allow us to calculate the total cost to bring 1s on the left side of j to the left of j and the ones on the right side to the right, so that j would sit squarely in the middle of the k consecutive 1s.

6. Finally, we update and maintain the minimum number of moves required (ans) throughout our iterations:

   ans = min(ans, a + b)

After the loop completes, ans holds the answer to our original question — the minimum number of moves required so that nums has k consecutive 1s.

Example Walkthrough

Let's illustrate the solution approach using a small example. Suppose we have an array nums and a value of k = 3. Our nums array looks like this:

nums = [1, 0, 1, 0, 1, 0, 1]

Our goal is to find the minimum number of adjacent swaps required to create a subarray of 3 consecutive 1s within nums.

1. First, we identify the positions of all 1s in nums:

   arr = [0, 2, 4, 6]  // These indices in `nums` hold the value 1.

2. Next, we create a prefix sum array s based on the positions in arr:

   s = [0, 0, 2, 6, 12]  // With initial=0, prefix sums of `arr` with an additionally prepended 0.

3. We calculate x and y which will define the left and right partition sizes of k 1s around a midpoint index i in arr:

   k = 3
   x = (k + 1) // 2 -> x = 2
   y = k - x -> y = 1

   We are looking for a subarray layout as 11[1]1 where the index i in arr would be the central [1].

4. Now we iterate over the potential midpoints. The valid range of i is [x - 1, len(arr) - y], which in this case is [1, 2].

5. When i = 1 (where arr[i] = 2):

   j = arr[1] -> j = 2 (the index in `nums` corresponding to the midpoint `1`)
   
   // Calculate left partition moves
   ls = s[i + 1] - s[i + 1 - x] -> ls = s[2] - s[0] -> ls = 6 - 0 -> ls = 6
   a = j * x - ls -> a = 2 * 2 - 6 -> a = 4 - 6 -> a = -2
   
   // Calculate right partition moves
   rs = s[i + 1 + y] - s[i + 1] -> rs = s[3] - s[2] -> rs = 12 - 6 -> rs = 6
   b = rs - (j + 1) * y -> b = 6 - (2 + 1) * 1 -> b = 6 - 3 -> b = 3
   
   // Total moves for this midpoint
   ans = a + b -> ans = -2 + 3 -> ans = 1

6. Next, for i = 2 (where arr[i] = 4):

   j = arr[2] -> j = 4
   
   // Calculate left partition moves
   ls = s[i + 1] - s[i + 1 - x] -> ls = s[3] - s[1] -> ls = 12 - 0 -> ls = 12
   a = j * x - ls -> a = 4 * 2 - 12 -> a = 8 - 12 -> a = -4
   
   // Calculate right partition moves
   rs = s[i + 1 + y] - s[i + 1] -> rs = s[4] - s[3] -> rs = 12 - 12 -> rs = 0
   b = rs - (j + 1) * y -> b = 0 - (4 + 1) * 1 -> b = 0 - 5 -> b = -5
   
   // Total moves for this midpoint
   ans = min(ans, a + b) -> ans = min(1, -4 - 5) -> ans = min(1, -9) -> ans = -9 (which can't be the case; ignored)

   The negative values of a and b indicate that we have an overcount and must adjust our calculations accordingly.

7. The correct ans is the minimum of the total moves found while iterating. And as -9 isn't a feasible answer (we cannot have negative moves), our minimum is 1.

Therefore, with k = 3, the minimum number of adjacent swaps required to create a subarray of 3 consecutive 1s in the array nums = [1, 0, 1, 0, 1, 0, 1] is 1.

Solution Implementation

Time and Space Complexity

Time Complexity

The given Python code comprises several parts whose time complexities will add up to the total time complexity.

1. List Comprehension (arr): This part has a time complexity of O(n), where n is the length of the input list nums. This is because it iterates over all elements in nums to create a new list of indices arr where the elements are ones.

2. Accumulate (s): The accumulate function constructs a prefix sum list, which also operates in O(n) time complexity.

3. Loop and Calculations within the Loop: The loop runs for approximately O(n) iterations since it starts from x - 1 and goes until len(arr) - y. On each iteration, a constant number of arithmetic operations and array accesses are performed, which take O(1) time each.

Considering these parts together, the overall time complexity of the code is the sum of the individual complexities: O(n) + O(n) + O(n) * O(1). Simplified, it remains O(n).

Space Complexity

The space complexity can be broken down as follows:

1. List arr: This list can contain at most n elements in the worst case where all elements in nums are ones. Therefore, the space complexity for this list is O(n).

2. Prefix Sum s: Similarly, the prefix sum list is of length n + 1, giving it a space complexity of O(n).

3. Variables ans, x, y, i, j, ls, rs, a, b: All of these variables use constant space, which is O(1).

When combining the space complexities, the dominating term is the space required for the lists arr and s, hence the overall space complexity of the code is O(n).

Learn more about how to find time and space complexity quickly using problem constraints.