<p>You are given an integer array <code>nums</code> and an integer <code>k</code>.</p>

<p>Find the longest subsequence of <code>nums</code> that meets the following requirements:</p>

<ul>
	<li>The subsequence is <strong>strictly increasing</strong> and</li>
	<li>The difference between adjacent elements in the subsequence is <strong>at most</strong> <code>k</code>.</li>
</ul>

<p>Return<em> the length of the <strong>longest</strong> <strong>subsequence</strong> that meets the requirements.</em></p>

<p>A <strong>subsequence</strong> is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [4,2,1,4,3,4,5,8,15], k = 3
<strong>Output:</strong> 5
<strong>Explanation:</strong>
The longest subsequence that meets the requirements is [1,3,4,5,8].
The subsequence has a length of 5, so we return 5.
Note that the subsequence [1,3,4,5,8,15] does not meet the requirements because 15 - 8 = 7 is larger than 3.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [7,4,5,1,8,12,4,7], k = 5
<strong>Output:</strong> 4
<strong>Explanation:</strong>
The longest subsequence that meets the requirements is [4,5,8,12].
The subsequence has a length of 4, so we return 4.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,5], k = 1
<strong>Output:</strong> 1
<strong>Explanation:</strong>
The longest subsequence that meets the requirements is [1].
The subsequence has a length of 1, so we return 1.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= nums[i], k &lt;= 10<sup>5</sup></code></li>
</ul>

class BIT:
    def __init__(self, n):
        self.layers = []
        while n > 0:
            self.layers.append([0]*n)
            n = n//2
        
    def update(self, idx, value):
        for layer in self.layers:
            if idx < len(layer) and layer[idx] < value:
                layer[idx] = value
            idx >>= 1
        
    def max_value(self, a, b):
        value = 0
        b -= 1
        for layer in self.layers:
            if a > b:
                break

            if a&1:
                if layer[a] > value:
                    value = layer[a]
                a = (a >> 1) + 1
            else:
                a >>= 1

            if b&1 == 0: 
                if layer[b] > value:
                    value = layer[b]
                b = (b >> 1) -1
            else:
                b >>= 1
        
        return value
            

class Solution:
    def lengthOfLIS(self, nums: List[int], k: int) -> int:
        bit = BIT(max(nums)+1)
        lis = 0
        
        for num in nums:
            curr_lis = bit.max_value(max(0, num-k), num) + 1
            if curr_lis > lis:
                lis = curr_lis
            bit.update(num, curr_lis)
            # print(f"{num=}, {curr_lis=}")
        
        return lis
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