The problem asks for the longest subarray where the absolute difference between any two elements is less than or equal to a given limit. This requires efficiently finding subarrays that satisfy this condition. The challenge lies in the size of the input (nums.length up to _**10^5)), which necessitates an efficient solution that avoids brute force methods.

Key Points:

• Sliding Window: The core idea is to use the sliding window technique with two pointers (l and r), where l is the left pointer and r is the right pointer.
• Maintaining Min and Max: To check whether the subarray is valid, we need to keep track of the minimum and maximum elements in the current subarray. This is done using queues.
• Efficiency: The approach must efficiently manage the minimum and maximum values as the window expands and contracts.

Approach:

1. Sliding Window with Two Pointers: We maintain a sliding window using two pointers (l and r). The window is valid if the difference between the maximum and minimum elements in the window is less than or equal to limit.
2. Deques to Track Min/Max: A deque (double-ended queue) will be used to track the minimum and maximum values in the current window. The sliding window is expanded by moving the right pointer r and contracted by moving the left pointer l when the condition is violated.
3. Adjust Window: For every valid window, we calculate the size of the window and update the answer. When the window is invalid, we increment the left pointer until the condition is satisfied.

Plan:

1. Initialize two deques: minQ for the minimum values and maxQ for the maximum values.
2. Use a loop to expand the window with the right pointer (r), pushing elements into the deques while maintaining their order.
3. For each new element added to the window, check if the absolute difference between the max and min values exceeds limit. If it does, increment the left pointer (l) and adjust the deques.
4. Track the maximum window size where the difference condition holds.

Let's implement this solution in PHP: 1438. Longest Continuous Subarray With Absolute Diff Less Than or Equal to Limit

<?php
/**
 * @param Integer[] $nums
 * @param Integer $limit
 * @return Integer
 */
function longestSubarray($nums, $limit) {
    $ans = 1;
    $minQ = new SplDoublyLinkedList();
    $maxQ = new SplDoublyLinkedList();

    for ($l = 0, $r = 0; $r < count($nums); ++$r) {
        while (!$minQ->isEmpty() && $minQ->top() > $nums[$r])
            $minQ->pop();
        $minQ->push($nums[$r]);
        while (!$maxQ->isEmpty() && $maxQ->top() < $nums[$r])
            $maxQ->pop();
        $maxQ->push($nums[$r]);
        while ($maxQ->bottom() - $minQ->bottom() > $limit) {
            if ($minQ->bottom() == $nums[$l])
                $minQ->shift();
            if ($maxQ->bottom() == $nums[$l])
                $maxQ->shift();
            ++$l;
        }
        $ans = max($ans, $r - $l + 1);
    }

    return $ans;
}

// Example usage:
$nums = [8,2,4,7];
$limit = 4;
echo longestSubarray($nums, $limit) . "\n"; // Output: 2

$nums = [10,1,2,4,7,2];
$limit = 5;
echo longestSubarray($nums, $limit) . "\n"; // Output: 4

$nums = [4,2,2,2,4,4,2,2];
$limit = 0;
echo longestSubarray($nums, $limit) . "\n"; // Output: 3
?>

Explanation:

1. Deques for Min/Max: The deques maintain the elements in the order required to find the min and max in constant time. For minQ, we pop elements from the back as long as they are greater than the current element (nums[r]) to ensure the smallest elements are at the front. Similarly, for maxQ, we pop elements from the back as long as they are smaller than the current element (nums[r]), ensuring the largest elements are at the front.

2. Validating the Window: As we expand the window by moving r forward, we check the difference between the max and min values. If it's greater than limit, we shrink the window from the left by incrementing l until the condition is satisfied.

3. Updating the Result: Every time a valid window is found, we calculate the size of the current subarray and update the result (ans).

Example Walkthrough:

Example 1:

Input: nums = [8, 2, 4, 7], limit = 4
Output: 2

• Start with l = 0 and r = 0. The initial window contains only one element: [8]. The condition holds since the max-min = 0 <= 4.
• Expand the window by moving r to 1. The window becomes [8, 2]. The max-min = 6 > 4, so move l to 1.
• The next valid window is [2], which has size 1.
• Move r to 2. The window becomes [2, 4]. The max-min = 2 <= 4, so it's valid.
• Move r to 3. The window becomes [2, 4, 7]. The max-min = 5 > 4, so move l to 2.
• The valid subarray is [4, 7], with size 2.
• The result is 2, which is the longest subarray.

Example 2:

Input: nums = [10, 1, 2, 4, 7, 2], limit = 5
Output: 4

• Start with l = 0 and expand the window until the subarray [2, 4, 7, 2] is the longest valid one, giving a result of 4.

Time Complexity:

• Time Complexity: O(n), where n is the length of the array. Each element is added and removed from the deques at most once, leading to linear time complexity.
• Space Complexity: O(n), due to the storage of deques that can hold at most n elements.

Output for Example:

For Example 1:

Input: nums = [8, 2, 4, 7], limit = 4
Output: 2

For Example 2:

Input: nums = [10, 1, 2, 4, 7, 2], limit = 5
Output: 4

This approach efficiently finds the longest subarray that satisfies the condition using a sliding window technique with two deques. The sliding window ensures that we only need to traverse the array once, making the solution scalable for large inputs. The deques allow constant-time access to the maximum and minimum values, making this approach optimal for the problem constraints.