Constrained Subsequence Sum
<p>Given an integer array <code>nums</code> and an integer <code>k</code>, return the maximum sum of a <strong>non-empty</strong> subsequence of that array such that for every two <strong>consecutive</strong> integers in the subsequence, <code>nums[i]</code> and <code>nums[j]</code>, where <code>i &lt; j</code>, the condition <code>j - i &lt;= k</code> is satisfied.</p>

<p>A <em>subsequence</em> of an array is obtained by deleting some number of elements (can be zero) from the array, leaving the remaining elements in their original order.</p>

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

<pre>
<strong>Input:</strong> nums = [10,2,-10,5,20], k = 2
<strong>Output:</strong> 37
<b>Explanation:</b> The subsequence is [10, 2, 5, 20].
</pre>

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

<pre>
<strong>Input:</strong> nums = [-1,-2,-3], k = 1
<strong>Output:</strong> -1
<b>Explanation:</b> The subsequence must be non-empty, so we choose the largest number.
</pre>

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

<pre>
<strong>Input:</strong> nums = [10,-2,-10,-5,20], k = 2
<strong>Output:</strong> 23
<b>Explanation:</b> The subsequence is [10, -2, -5, 20].
</pre>

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

<ul>
	<li><code>1 &lt;= k &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>-10<sup>4</sup> &lt;= nums[i] &lt;= 10<sup>4</sup></code></li>
</ul>
Queue; Array; Dynamic Programming; Sliding Window; Monotonic Queue; Heap (Priority Queue)
function constrainedSubsetSum(nums: number[], k: number): number {
    const q = new Deque<number>();
    const n = nums.length;
    q.pushBack(0);
    let ans = nums[0];
    const f: number[] = Array(n).fill(0);
    for (let i = 0; i < n; ++i) {
        while (i - q.frontValue()! > k) {
            q.popFront();
        }
        f[i] = Math.max(0, f[q.frontValue()!]!) + nums[i];
        ans = Math.max(ans, f[i]);
        while (!q.isEmpty() && f[q.backValue()!]! <= f[i]) {
            q.popBack();
        }
        q.pushBack(i);
    }
    return ans;
}

class Node<T> {
    value: T;
    next: Node<T> | null;
    prev: Node<T> | null;

    constructor(value: T) {
        this.value = value;
        this.next = null;
        this.prev = null;
    }
}

class Deque<T> {
    private front: Node<T> | null;
    private back: Node<T> | null;
    private size: number;

    constructor() {
        this.front = null;
        this.back = null;
        this.size = 0;
    }

    pushFront(val: T): void {
        const newNode = new Node(val);
        if (this.isEmpty()) {
            this.front = newNode;
            this.back = newNode;
        } else {
            newNode.next = this.front;
            this.front!.prev = newNode;
            this.front = newNode;
        }
        this.size++;
    }

    pushBack(val: T): void {
        const newNode = new Node(val);
        if (this.isEmpty()) {
            this.front = newNode;
            this.back = newNode;
        } else {
            newNode.prev = this.back;
            this.back!.next = newNode;
            this.back = newNode;
        }
        this.size++;
    }

    popFront(): T | undefined {
        if (this.isEmpty()) {
            return undefined;
        }
        const value = this.front!.value;
        this.front = this.front!.next;
        if (this.front !== null) {
            this.front.prev = null;
        } else {
            this.back = null;
        }
        this.size--;
        return value;
    }

    popBack(): T | undefined {
        if (this.isEmpty()) {
            return undefined;
        }
        const value = this.back!.value;
        this.back = this.back!.prev;
        if (this.back !== null) {
            this.back.next = null;
        } else {
            this.front = null;
        }
        this.size--;
        return value;
    }

    frontValue(): T | undefined {
        return this.front?.value;
    }

    backValue(): T | undefined {
        return this.back?.value;
    }

    getSize(): number {
        return this.size;
    }

    isEmpty(): boolean {
        return this.size === 0;
    }
}
