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algorithms/001438-longest-continuous-subarray-with-absolute-diff-less-than-or-equal-to-limit/README.md

@@ -1,6 +1,8 @@
 1438\. Longest Continuous Subarray With Absolute Diff Less Than or Equal to Limit
 
-Medium
+**Difficulty:** Medium
+
+**Topics:** `Array`, `Queue`, `Sliding Window`, `Heap (Priority Queue)`, `Ordered Set`, `Monotonic Queue`
 
 Given an array of integers `nums` and an integer `limit`, return the size of the longest **non-empty** subarray such that the absolute difference between any two elements of this subarray is less than or equal to `limit`.
 
@@ -43,3 +45,118 @@ Given an array of integers `nums` and an integer `limit`, return the size of the
 - <code>1 <= nums[i] <= 10<sup>9</sup></code>
 - <code>0 <= limit <= 10<sup>9</sup></code>
 
+
+**Hint:**
+1. Use a sliding window approach keeping the maximum and minimum value using a data structure like a multiset from STL in C++.
+2. More specifically, use the two pointer technique, moving the right pointer as far as possible to the right until the subarray is not valid (maxValue - minValue > limit), then moving the left pointer until the subarray is valid again (maxValue - minValue <= limit). Keep repeating this process.
+
+
+
+**Solution:**
+
+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](https://github.com/mah-shamim/leet-code-in-php/tree/main/algorithms/001438-longest-continuous-subarray-with-absolute-diff-less-than-or-equal-to-limit/solution.php)**
+
+```php
+<?php
+/**
+ * @param Integer[] $nums
+ * @param Integer $limit
+ * @return Integer
+ */
+function longestSubarray($nums, $limit) {
+    ...
+    ...
+    ...
+    /**
+     * go to ./solution.php
+     */
+}
+
+// Example usage:
+$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**:
+```plaintext
+Input: nums = [8, 2, 4, 7], limit = 4
+Output: 2
+```
+
+For **Example 2**:
+```plaintext
+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.
+
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+
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