Problem Description

The problem presents an integer array nums and an integer threshold. The task is to determine the length of the longest subarray starting at index l and ending at index r, where 0 <= l <= r < nums.length. The subarray should satisfy these conditions:

1. The first element of the subarray, nums[l], must be even (nums[l] % 2 == 0).
2. Adjacent elements in the subarray must have different parity – that is, one is even and the other is odd. For any two consecutive elements nums[i] and nums[i + 1] within the subarray, their mod 2 results must be different (nums[i] % 2 != nums[i + 1] % 2).
3. Every element within the subarray must be less than or equal to the threshold (nums[i] <= threshold).

The goal is to return the length of such the longest subarray meeting these criteria.

Intuition

To find the longest subarray that satisfies the problem conditions, we can iterate through the array. For each potential starting index l of a subarray, we check whether the starting element is even and less than or equal to the threshold. If it is, we try to extend the subarray by moving the right index r as far as possible while maintaining the alternating even-odd sequence and ensuring all elements are within the threshold.

The process is as follows:

1. We iterate over each index l in the array nums as a potential start of the subarray.
2. If nums[l] is even and less than or equal to the threshold, we have a potential subarray starting at l. We initialize r as l + 1 because a subarray must be non-empty.
3. We now extend the subarray by incrementing r as long as nums[r] alternates in parity with nums[r-1] and nums[r] is less than or equal to the threshold.
4. Once we can no longer extend r because the next element violates one of our conditions, we calculate the current subarray length as r - l. We track the maximum length found throughout this process with the variable ans.
5. Continue the same process for each index l in the array and return ans, which will hold the maximum length of the longest subarray that meets all the criteria.

By following this approach, we can ensure that we check all possible subarrays that start with an even number and have alternating parities until we either reach the end of the array or encounter elements that do not fulfill the conditions.

Learn more about Sliding Window patterns.

Solution Approach

The implementation uses a straightforward approach, which essentially follows the sliding window pattern. Sliding window is a common pattern used in array/string problems where a subarray or substring is processed, and then the window either expands or contracts to satisfy certain conditions.

Here's a step-by-step breakdown of the algorithm based on the given solution approach:

1. Initialize a variable ans to store the maximum length found for any subarray that satisfies the problem conditions. Also, determine the length n of the input array nums.

2. Start the first for-loop to iterate over the array using the index l, which represents the potential start of the subarray.

3. Inside this loop, first check whether the element at index l is both even (nums[l] % 2 == 0) and less than or equal to the given threshold. Only if these conditions are met, the element at l can be the starting element of a subarray.

4. If the starting element is suitable, initialize the end pointer r for the subarray to l + 1. This is where the window starts to slide.

5. Use a while loop to try expanding this window. The loop will continue as long as r is less than n, the array length, ensuring array bounds are not exceeded.

6. In each iteration of the while loop, check two conditions: whether the parity at nums[r] is different from nums[r - 1] (nums[r] % 2 != nums[r - 1] % 2) and if nums[r] is less than or equal to threshold. This ensures the subarray keeps alternating between even and odd numbers with values within the threshold.

7. As long as these conditions are satisfied, increment r to include the next element in the subarray. If either condition fails, the while loop breaks, and the current window cannot be expanded further.

8. Once the while loop ends, calculate the length of the current subarray by r - l and update ans with the maximum of its current value and this newly found length.

9. After processing the potential starting index l, the for loop moves to the next possible start index and repeats the steps until the whole array is scanned.

10. At the end of the for loop, ans contains the length of the longest subarray satisfying the given conditions, and it's returned as the final answer.

The solution does not use any additional data structures, and its space complexity is O(1), which represents constant space aside from the input array. The time complexity is O(n^2) in the worst case, where n is the number of elements in nums, because for each element in nums as a starting point, we might need to check up to n elements to find the end of the subarray.

Example Walkthrough

Let's walk through an example to illustrate the solution approach.

Consider an array nums = [2, 3, 4, 6, 7] and a threshold = 5.

Following the steps outlined in the solution approach:

1. Initialize ans to 0. The length of the array n is 5.
2. Start iterating over the array with index l ranging from 0 to n-1.
3. At l = 0, we find nums[0] = 2, which is even and less than the threshold, so this can be the start of a subarray.
4. Initiate r to l+1, which is 1.
5. We enter the while loop to expand the window starting at l = 0:
   • r = 1: nums[1] = 3 which is different in parity from nums[0] and is also below the threshold. Increment r to 2.
   • r = 2: nums[2] = 4, which fails the alternating parity condition. Break the while loop.
6. Calculate the current subarray length: r - l = 2 - 0 = 2. Update ans to max(ans, 2) = 2.
7. Increment l to 1 and repeat the steps:
   • nums[1] = 3 is odd, so we skip this as it can't be the start of a subarray.
8. Increment l to 2. No suitable subarray starting from here since nums[2] = 4 does not meet the parity alternation condition with any element afterward, and it's above the threshold.
9. Move on to l = 3. Again, nums[3] = 6 is above the threshold, so we skip this index.
10. Lastly, l = 4. Since nums[4] = 7 is not even, we skip this index.

After iterating through all elements, the maximum subarray length that satisfies the conditions is stored in ans, which is 2 in this example. So the function returns 2.

This illustrates the entire process of checking each potential starting index and trying to expand the window to find the longest subarray that satisfies the given criteria.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code can be analyzed by considering that there are two nested loops: an outer loop that runs from l from 0 to n - 1, and an inner while loop that potentially runs from l + 1 to n in the worst case when all the elements conform to the specified condition (alternating even and odd values under the threshold).

In the worst-case scenario, the inner loop can iterate n times for the first run, then n-1 times, n-2 times, and so forth until 1 time. This forms an arithmetic progression that sums up to (n * (n + 1)) / 2 iterations in total.

Hence, the worst-case time complexity of the function is O(n^2).

Space Complexity

The space complexity of the given code is constant, O(1), as it only uses a fixed number of variables (ans, n, l, r) and does not allocate any additional space that grows with the input size.

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