Problem Description

The problem presents a binary array nums, comprised of 0s and 1s. The task is to find the size of the largest contiguous subarray of 1s that could be obtained by removing exactly one element from the array. You are also informed that if there is no such subarray made up entirely of 1s after removing an element, the function should return 0.

Intuition

The intuition for solving this problem involves looking at each segment of 1s as a potential candidate and checking the length we can obtain after removing an element. We want to maximize this length by strategically picking an element for removal. The key insight here is that if there are consecutive 1s, removing one 1 does not impact the length of the subarray of 1s on either side of it.

We can approach the solution by precomputing and storing the lengths of continuous 1s at any index from the left and right directions separately. This is done in two linear scans using the left and right auxiliary arrays:

1. From Left to Right: Starting at the index 1, if the previous element is 1, increment the value in the left array at the current index by 1 plus the value at the previous index. This will give us the number of continuous 1s to the left of each index in the array nums.

2. From Right to Left: Starting from the second to last index and going backwards, if the next element is 1, increment the value in the right array at the current index by 1 plus the value at the next index. This gives the number of continuous 1s to the right of each index.

The final result, after having populated both left and right arrays, is the maximum sum of corresponding elements from left and right arrays across all indices. This effectively simulates the removal of one element (either a 1 or 0) and joining the contiguous 1s from both left and right sides. However, the result should be the length after the removal of one element, so another -1 is implicitly considered in the max operation [since we don't actually remove the element but calculate what the length would be as if we did].

By applying this method, we can find the longest subarray containing only 1s after deleting exactly one element.

Learn more about Dynamic Programming and Sliding Window patterns.

Solution Approach

The solution to this problem relies primarily on dynamic programming to keep track of the length of continuous 1s on both sides of each index in the array nums.

Here is a breakdown of the implementation details:

1. Initialization: Two lists, left and right, each of the same length as nums, are created and initialized with zeroes. These lists will store the length of continuous 1s to the left and right of each index, respectively.

2. Populating the left list: We start iterating over the array nums from the second element onwards (index 1 because we consider 0 as the base case which is already initialized to zero). For each index i, if the element at the previous index (i - 1) is 1, we set left[i] to left[i - 1] + 1. Effectively, this records the length of a stretch of 1s ending at the current index.

3. Populating the right list: We do a similar iteration, but this time from right to left, starting from the second-to-last element (index n - 2 where n is the length of nums). For each index i, if the element at the next index (i + 1) is 1, we set right[i] to right[i + 1] + 1. This captures the length of a stretch of 1s starting right after the current index.

4. Finding the longest subarray: Once we have both left and right lists populated, we iterate over all possible remove positions (these positions do not have to be 1; they can be 0 as well, as we will merge stretches of 1s around them). For each index, we calculate the sum of left[i] + right[i], which approximates the length of the subarray of 1s if we removed the element at the current index. However, since we are supposed to actually remove an element to create the longest subsequence, the sum implicitly considers this by not adding an additional 1 even though we are summing the lengths from both sides.

5. Returning the result: The maximum value from these sums corresponds to the size of the longest subarray containing only 1s after one deletion. This is found using the max() function applied to the sums of left[i] + right[i] for each index i.

The algorithm completes this in linear time with two passes through nums, and the space complexity is governed by the two additional arrays left and right, which are linear in the size of the input array nums.

Example Walkthrough

Let's use an example to illustrate the solution approach.

Suppose the input binary array nums is [1, 1, 0, 1, 1, 1, 0, 1, 1].

Step by Step Solution

1. Initialization:
   We initialize two lists, left and right, with the same length as nums.
   left: [0, 0, 0, 0, 0, 0, 0, 0, 0]
right: [0, 0, 0, 0, 0, 0, 0, 0, 0]

2. Populating the left list:
   We iterate from left to right over the array nums, starting from index 1.

   • nums[1] is 1, and nums[0] is 1, so left[1] = left[0] + 1 -> 0 + 1 = 1.
   • As we continue, left becomes [0, 1, 0, 1, 2, 3, 0, 1, 2].

3. Populating the right list:
   Iterating from right to left, starting from index 7.

   • nums[7] is 1, and nums[8] is 1, so right[7] = right[8] + 1 -> 0 + 1 = 1.
   • Continuing this process, the right list fills up as [2, 1, 0, 4, 3, 2, 0, 1, 0].

4. Finding the longest subarray:
   We calculate the potential longest subarrays after a deletion for each index, by summing the left and right values at each index.

   • At index 2 (the first 0), the sum is left[2] + right[2] = 0 + 0 = 0.
   • At index 6 (the second 0), the sum is left[6] + right[6] = 3 + 0 = 3.
   • The longest subarray occurs at either index 2 or 6, where we would be removing the 0 to join the 1s.
   • Considering every index, we would have sums as [2, 1, 4, 5, 3, 2, 4, 1, 2].

5. Returning the result:
   The maximum value from the sum of left and right for all indexes gives us the result.
   In this case, the maximum is 5, corresponding to index 3. Thus, the size of the longest subarray containing only 1s after one deletion is 5.

This approach allows us to solve the problem with a single pass for left and another for right, totaling linear time complexity, O(n), with n being the number of elements in nums. The use of the two lists left and right gives us a space complexity of O(n) as well.

Solution Implementation

Time and Space Complexity

The given code aims to find the length of the longest subarray of 1s after deleting one element from the array. Here's the analysis of its computational complexity:

Time Complexity:

• The time complexity of this algorithm is O(n), where n is the length of the input array nums. The reasoning behind this is that there are two separate for loops that each iterate over the array exactly once. The first loop starts from index 1 and goes up to n-1, incrementing by 1 on each iteration. The second loop starts from n-2 and goes down to 0, decrementing by 1 on each iteration. In both of these loops, only a constant amount of work is done (simple arithmetic and array accesses), so the time complexity for each loop is linear with respect to the size of the array.

• Furthermore, there's a final step that combines the results of the left and right arrays using max and zip, which is also a linear operation since it involves a single pass over the combined arrays, making it O(n).

Adding up these linear time operations (3 * O(n)) still results in an overall time complexity of O(n).

Space Complexity:

• The space complexity of this algorithm is O(n) as well. This is because two additional arrays, left and right, are created to store the lengths of continuous ones to the left and right of each index, respectively. Each of these arrays is the same length as the input array nums.

• Besides the two arrays left and right, there are only a constant number of variables used (i.e., n, i, a, b), so they do not add more than O(1) to the space complexity.

Thus, the total space complexity is determined by the size of the two additional arrays, which gives us O(n).

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