Count Partitions With Max-Min Difference at Most K
<p>You are given an integer array <code>nums</code> and an integer <code>k</code>. Your task is to partition <code>nums</code> into one or more <strong>non-empty</strong> contiguous segments such that in each segment, the difference between its <strong>maximum</strong> and <strong>minimum</strong> elements is <strong>at most</strong> <code>k</code>.</p>

<p>Return the total number of ways to partition <code>nums</code> under this condition.</p>

<p>Since the answer may be too large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [9,4,1,3,7], k = 4</span></p>

<p><strong>Output:</strong> <span class="example-io">6</span></p>

<p><strong>Explanation:</strong></p>

<p>There are 6 valid partitions where the difference between the maximum and minimum elements in each segment is at most <code>k = 4</code>:</p>

<ul>
	<li><code>[[9], [4], [1], [3], [7]]</code></li>
	<li><code>[[9], [4], [1], [3, 7]]</code></li>
	<li><code>[[9], [4], [1, 3], [7]]</code></li>
	<li><code>[[9], [4, 1], [3], [7]]</code></li>
	<li><code>[[9], [4, 1], [3, 7]]</code></li>
	<li><code>[[9], [4, 1, 3], [7]]</code></li>
</ul>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [3,3,4], k = 0</span></p>

<p><strong>Output:</strong> <span class="example-io">2</span></p>

<p><strong>Explanation:</strong></p>

<p>There are 2 valid partitions that satisfy the given conditions:</p>

<ul>
	<li><code>[[3], [3], [4]]</code></li>
	<li><code>[[3, 3], [4]]</code></li>
</ul>
</div>

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

<ul>
	<li><code>2 &lt;= nums.length &lt;= 5 * 10<sup>4</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>
	<li><code>0 &lt;= k &lt;= 10<sup>9</sup></code></li>
</ul>
Queue; Array; Dynamic Programming; Prefix Sum; Sliding Window; Monotonic Queue
func countPartitions(nums []int, k int) int {
	const mod int = 1e9 + 7
	sl := redblacktree.New[int, int]()
	merge := func(st *redblacktree.Tree[int, int], x, v int) {
		c, _ := st.Get(x)
		if c+v == 0 {
			st.Remove(x)
		} else {
			st.Put(x, c+v)
		}
	}
	n := len(nums)
	f := make([]int, n+1)
	g := make([]int, n+1)
	f[0], g[0] = 1, 1
	for l, r := 1, 1; r <= n; r++ {
		merge(sl, nums[r-1], 1)
		for sl.Right().Key-sl.Left().Key > k {
			merge(sl, nums[l-1], -1)
			l++
		}
		f[r] = g[r-1]
		if l >= 2 {
			f[r] = (f[r] - g[l-2] + mod) % mod
		}
		g[r] = (g[r-1] + f[r]) % mod
	}
	return f[n]
}