3013. 将数组分成最小总代价的子数组 II

题目描述

给你一个下标从 0 开始长度为 n 的整数数组 nums 和两个 正 整数 k 和 dist 。

一个数组的 代价 是数组中的 第一个 元素。比方说，[1,2,3] 的代价为 1 ，[3,4,1] 的代价为 3 。

你需要将 nums 分割成 k 个 连续且互不相交 的子数组，满足 第二 个子数组与第 k 个子数组中第一个元素的下标距离 不超过 dist 。换句话说，如果你将 nums 分割成子数组 nums[0..(i1 - 1)], nums[i1..(i2 - 1)], ..., nums[ik-1..(n - 1)] ，那么它需要满足 ik-1 - i1 <= dist 。

请你返回这些子数组的 最小 总代价。

 

示例 1：

输入：nums = [1,3,2,6,4,2], k = 3, dist = 3
输出：5
解释：将数组分割成 3 个子数组的最优方案是：[1,3] ，[2,6,4] 和 [2] 。这是一个合法分割，因为 ik-1 - i1 等于 5 - 2 = 3 ，等于 dist 。总代价为 nums[0] + nums[2] + nums[5] ，也就是 1 + 2 + 2 = 5 。
5 是分割成 3 个子数组的最小总代价。

示例 2：

输入：nums = [10,1,2,2,2,1], k = 4, dist = 3
输出：15
解释：将数组分割成 4 个子数组的最优方案是：[10] ，[1] ，[2] 和 [2,2,1] 。这是一个合法分割，因为 ik-1 - i1 等于 3 - 1 = 2 ，小于 dist 。总代价为 nums[0] + nums[1] + nums[2] + nums[3] ，也就是 10 + 1 + 2 + 2 = 15 。
分割 [10] ，[1] ，[2,2,2] 和 [1] 不是一个合法分割，因为 ik-1 和 i1 的差为 5 - 1 = 4 ，大于 dist 。
15 是分割成 4 个子数组的最小总代价。

示例 3：

输入：nums = [10,8,18,9], k = 3, dist = 1
输出：36
解释：将数组分割成 4 个子数组的最优方案是：[10] ，[8] 和 [18,9] 。这是一个合法分割，因为 ik-1 - i1 等于 2 - 1 = 1 ，等于 dist 。总代价为 nums[0] + nums[1] + nums[2] ，也就是 10 + 8 + 18 = 36 。
分割 [10] ，[8,18] 和 [9] 不是一个合法分割，因为 ik-1 和 i1 的差为 3 - 1 = 2 ，大于 dist 。
36 是分割成 3 个子数组的最小总代价。

 

提示：

• 3 <= n <= 105
• 1 <= nums[i] <= 109
• 3 <= k <= n
• k - 2 <= dist <= n - 2

解法

方法一：有序集合

题目需要我们将数组 $\textit{nums}$ 分割成 $k$ 个连续且不相交的子数组，并且第二个子数组的第一个元素与第 $k$ 个子数组的第一个元素的下标距离不超过 $\textit{dist}$，这等价于让我们从 $\textit{nums}$ 下标为 $1$ 的元素开始，找到一个窗口大小为 $\textit{dist}+1$ 的子数组，求其中前 $k-1$ 的最小元素之和。我们将 $k$ 减 $1$，这样我们只需要求出 $k$ 个最小元素之和，再加上 $\textit{nums}[0]$ 即可。

我们可以使用两个有序集合 $\textit{l}$ 和 $\textit{r}$ 分别维护大小为 $\textit{dist} + 1$ 的窗口元素，其中 $\textit{l}$ 维护最小的 $k$ 个元素，而 $\textit{r}$ 维护窗口的剩余元素。我们维护一个变量 $\textit{s}$ 表示 $\textit{nums}[0]$ 与 $l$ 中元素之和。初始时，我们将前 $\textit{dist}+2$ 个元素之和累加到 $\textit{s}$ 中，并将下标为 $[1, \textit{dist} + 1]$ 的所有元素加入到 $\textit{l}$ 中。如果 $\textit{l}$ 的大小大于 $k$，我们循环将 $\textit{l}$ 中的最大元素移动到 $\textit{r}$ 中，直到 $\textit{l}$ 的大小等于 $k$，过程中更新 $\textit{s}$ 的值。

那么此时初始答案 $\textit{ans} = \textit{s}$。

接下来我们从 $\textit{dist}+2$ 开始遍历 $\textit{nums}$，对于每一个元素 $\textit{nums}[i]$，我们需要将 $\textit{nums}[i-\textit{dist}-1]$ 从 $\textit{l}$ 或 $\textit{r}$ 中移除，然后将 $\textit{nums}[i]$ 加入到 $\textit{l}$ 或 $\textit{r}$ 中。如果 $\textit{nums}[i]$ 小于 $\textit{l}$ 中的最大元素，我们将 $\textit{nums}[i]$ 加入到 $\textit{l}$ 中，否则加入到 $\textit{r}$ 中。如果此时 $\textit{l}$ 的大小小于 $k$，我们将 $\textit{r}$ 中的最小元素移动到 $\textit{l}$ 中，直到 $\textit{l}$ 的大小等于 $k$。如果此时 $\textit{l}$ 的大小大于 $k$，我们将 $\textit{l}$ 中的最大元素移动到 $\textit{r}$ 中，直到 $\textit{l}$ 的大小等于 $k$。过程中更新 $\textit{s}$ 的值，并更新 $\textit{ans} = \min(\textit{ans}, \textit{s})$。

最终答案即为 $\textit{ans}$。

时间复杂度 $O(n \times \log \textit{dist})$，空间复杂度 $O(\textit{dist})$。其中 $n$ 为数组 $\textit{nums}$ 的长度。

Python3JavaC++GoTypeScript

from sortedcontainers import SortedList


class Solution:
    def minimumCost(self, nums: List[int], k: int, dist: int) -> int:
        def l2r():
            nonlocal s
            x = l.pop()
            s -= x
            r.add(x)

        def r2l():
            nonlocal s
            x = r.pop(0)
            l.add(x)
            s += x

        k -= 1
        s = sum(nums[: dist + 2])
        l = SortedList(nums[1 : dist + 2])
        r = SortedList()
        while len(l) > k:
            l2r()
        ans = s
        for i in range(dist + 2, len(nums)):
            x = nums[i - dist - 1]
            if x in l:
                l.remove(x)
                s -= x
            else:
                r.remove(x)
            y = nums[i]
            if y < l[-1]:
                l.add(y)
                s += y
            else:
                r.add(y)
            while len(l) < k:
                r2l()
            while len(l) > k:
                l2r()
            ans = min(ans, s)
        return ans

class Solution {
    private final TreeMap<Integer, Integer> l = new TreeMap<>();
    private final TreeMap<Integer, Integer> r = new TreeMap<>();
    private long s;
    private int size;

    public long minimumCost(int[] nums, int k, int dist) {
        --k;
        s = nums[0];
        for (int i = 1; i < dist + 2; ++i) {
            s += nums[i];
            l.merge(nums[i], 1, Integer::sum);
        }
        size = dist + 1;
        while (size > k) {
            l2r();
        }
        long ans = s;
        for (int i = dist + 2; i < nums.length; ++i) {
            int x = nums[i - dist - 1];
            if (l.containsKey(x)) {
                if (l.merge(x, -1, Integer::sum) == 0) {
                    l.remove(x);
                }
                s -= x;
                --size;
            } else if (r.merge(x, -1, Integer::sum) == 0) {
                r.remove(x);
            }
            int y = nums[i];
            if (y < l.lastKey()) {
                l.merge(y, 1, Integer::sum);
                ++size;
                s += y;
            } else {
                r.merge(y, 1, Integer::sum);
            }
            while (size < k) {
                r2l();
            }
            while (size > k) {
                l2r();
            }
            ans = Math.min(ans, s);
        }
        return ans;
    }

    private void l2r() {
        int x = l.lastKey();
        s -= x;
        if (l.merge(x, -1, Integer::sum) == 0) {
            l.remove(x);
        }
        --size;
        r.merge(x, 1, Integer::sum);
    }

    private void r2l() {
        int x = r.firstKey();
        if (r.merge(x, -1, Integer::sum) == 0) {
            r.remove(x);
        }
        l.merge(x, 1, Integer::sum);
        s += x;
        ++size;
    }
}

class Solution {
public:
    long long minimumCost(vector<int>& nums, int k, int dist) {
        --k;
        multiset<int> l(nums.begin() + 1, nums.begin() + dist + 2), r;
        long long s = accumulate(nums.begin(), nums.begin() + dist + 2, 0LL);
        while (l.size() > k) {
            int x = *l.rbegin();
            l.erase(l.find(x));
            s -= x;
            r.insert(x);
        }
        long long ans = s;
        for (int i = dist + 2; i < nums.size(); ++i) {
            int x = nums[i - dist - 1];
            auto it = l.find(x);
            if (it != l.end()) {
                l.erase(it);
                s -= x;
            } else {
                r.erase(r.find(x));
            }
            int y = nums[i];
            if (y < *l.rbegin()) {
                l.insert(y);
                s += y;
            } else {
                r.insert(y);
            }
            while (l.size() == k - 1) {
                int x = *r.begin();
                r.erase(r.find(x));
                l.insert(x);
                s += x;
            }
            while (l.size() == k + 1) {
                int x = *l.rbegin();
                l.erase(l.find(x));
                s -= x;
                r.insert(x);
            }
            ans = min(ans, s);
        }
        return ans;
    }
};

func minimumCost(nums []int, k int, dist int) int64 {
    k--
    l := redblacktree.New[int, int]()
    r := 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)
        }
    }

    s := nums[0]
    for _, x := range nums[1 : dist+2] {
        s += x
        merge(l, x, 1)
    }
    size := dist + 1

    l2r := func() {
        x := l.Right().Key
        merge(l, x, -1)
        s -= x
        size--
        merge(r, x, 1)
    }
    r2l := func() {
        x := r.Left().Key
        merge(r, x, -1)
        merge(l, x, 1)
        s += x
        size++
    }

    for size > k {
        l2r()
    }

    ans := s
    for i := dist + 2; i < len(nums); i++ {
        x := nums[i-dist-1]
        if _, ok := l.Get(x); ok {
            merge(l, x, -1)
            s -= x
            size--
        } else {
            merge(r, x, -1)
        }
        y := nums[i]
        if y < l.Right().Key {
            merge(l, y, 1)
            s += y
            size++
        } else {
            merge(r, y, 1)
        }
        for size < k {
            r2l()
        }
        for size > k {
            l2r()
        }
        ans = min(ans, s)

    }
    return int64(ans)
}

function minimumCost(nums: number[], k: number, dist: number): number {
    --k;
    const l = new TreapMultiSet<number>((a, b) => a - b);
    const r = new TreapMultiSet<number>((a, b) => a - b);
    let s = nums[0];
    for (let i = 1; i < dist + 2; ++i) {
        s += nums[i];
        l.add(nums[i]);
    }
    const l2r = () => {
        const x = l.pop()!;
        s -= x;
        r.add(x);
    };
    const r2l = () => {
        const x = r.shift()!;
        l.add(x);
        s += x;
    };
    while (l.size > k) {
        l2r();
    }
    let ans = s;
    for (let i = dist + 2; i < nums.length; ++i) {
        const x = nums[i - dist - 1];
        if (l.has(x)) {
            l.delete(x);
            s -= x;
        } else {
            r.delete(x);
        }
        const y = nums[i];
        if (y < l.last()!) {
            l.add(y);
            s += y;
        } else {
            r.add(y);
        }
        while (l.size < k) {
            r2l();
        }
        while (l.size > k) {
            l2r();
        }
        ans = Math.min(ans, s);
    }
    return ans;
}

type CompareFunction<T, R extends 'number' | 'boolean'> = (
    a: T,
    b: T,
) => R extends 'number' ? number : boolean;

interface ITreapMultiSet<T> extends Iterable<T> {
    add: (...value: T[]) => this;
    has: (value: T) => boolean;
    delete: (value: T) => void;

    bisectLeft: (value: T) => number;
    bisectRight: (value: T) => number;

    indexOf: (value: T) => number;
    lastIndexOf: (value: T) => number;

    at: (index: number) => T | undefined;
    first: () => T | undefined;
    last: () => T | undefined;

    lower: (value: T) => T | undefined;
    higher: (value: T) => T | undefined;
    floor: (value: T) => T | undefined;
    ceil: (value: T) => T | undefined;

    shift: () => T | undefined;
    pop: (index?: number) => T | undefined;

    count: (value: T) => number;

    keys: () => IterableIterator<T>;
    values: () => IterableIterator<T>;
    rvalues: () => IterableIterator<T>;
    entries: () => IterableIterator<[number, T]>;

    readonly size: number;
}

class TreapNode<T = number> {
    value: T;
    count: number;
    size: number;
    priority: number;
    left: TreapNode<T> | null;
    right: TreapNode<T> | null;

    constructor(value: T) {
        this.value = value;
        this.count = 1;
        this.size = 1;
        this.priority = Math.random();
        this.left = null;
        this.right = null;
    }

    static getSize(node: TreapNode<any> | null): number {
        return node?.size ?? 0;
    }

    static getFac(node: TreapNode<any> | null): number {
        return node?.priority ?? 0;
    }

    pushUp(): void {
        let tmp = this.count;
        tmp += TreapNode.getSize(this.left);
        tmp += TreapNode.getSize(this.right);
        this.size = tmp;
    }

    rotateRight(): TreapNode<T> {
        // eslint-disable-next-line @typescript-eslint/no-this-alias
        let node: TreapNode<T> = this;
        const left = node.left;
        node.left = left?.right ?? null;
        left && (left.right = node);
        left && (node = left);
        node.right?.pushUp();
        node.pushUp();
        return node;
    }

    rotateLeft(): TreapNode<T> {
        // eslint-disable-next-line @typescript-eslint/no-this-alias
        let node: TreapNode<T> = this;
        const right = node.right;
        node.right = right?.left ?? null;
        right && (right.left = node);
        right && (node = right);
        node.left?.pushUp();
        node.pushUp();
        return node;
    }
}

class TreapMultiSet<T = number> implements ITreapMultiSet<T> {
    private readonly root: TreapNode<T>;
    private readonly compareFn: CompareFunction<T, 'number'>;
    private readonly leftBound: T;
    private readonly rightBound: T;

    constructor(compareFn?: CompareFunction<T, 'number'>);
    constructor(compareFn: CompareFunction<T, 'number'>, leftBound: T, rightBound: T);
    constructor(
        compareFn: CompareFunction<T, any> = (a: any, b: any) => a - b,
        leftBound: any = -Infinity,
        rightBound: any = Infinity,
    ) {
        this.root = new TreapNode<T>(rightBound);
        this.root.priority = Infinity;
        this.root.left = new TreapNode<T>(leftBound);
        this.root.left.priority = -Infinity;
        this.root.pushUp();

        this.leftBound = leftBound;
        this.rightBound = rightBound;
        this.compareFn = compareFn;
    }

    get size(): number {
        return this.root.size - 2;
    }

    get height(): number {
        const getHeight = (node: TreapNode<T> | null): number => {
            if (node == null) return 0;
            return 1 + Math.max(getHeight(node.left), getHeight(node.right));
        };

        return getHeight(this.root);
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Returns true if value is a member.
     */
    has(value: T): boolean {
        const compare = this.compareFn;
        const dfs = (node: TreapNode<T> | null, value: T): boolean => {
            if (node == null) return false;
            if (compare(node.value, value) === 0) return true;
            if (compare(node.value, value) < 0) return dfs(node.right, value);
            return dfs(node.left, value);
        };

        return dfs(this.root, value);
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Add value to sorted set.
     */
    add(...values: T[]): this {
        const compare = this.compareFn;
        const dfs = (
            node: TreapNode<T> | null,
            value: T,
            parent: TreapNode<T>,
            direction: 'left' | 'right',
        ): void => {
            if (node == null) return;
            if (compare(node.value, value) === 0) {
                node.count++;
                node.pushUp();
            } else if (compare(node.value, value) > 0) {
                if (node.left) {
                    dfs(node.left, value, node, 'left');
                } else {
                    node.left = new TreapNode(value);
                    node.pushUp();
                }

                if (TreapNode.getFac(node.left) > node.priority) {
                    parent[direction] = node.rotateRight();
                }
            } else if (compare(node.value, value) < 0) {
                if (node.right) {
                    dfs(node.right, value, node, 'right');
                } else {
                    node.right = new TreapNode(value);
                    node.pushUp();
                }

                if (TreapNode.getFac(node.right) > node.priority) {
                    parent[direction] = node.rotateLeft();
                }
            }
            parent.pushUp();
        };

        values.forEach(value => dfs(this.root.left, value, this.root, 'left'));
        return this;
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Remove value from sorted set if it is a member.
     * If value is not a member, do nothing.
     */
    delete(value: T): void {
        const compare = this.compareFn;
        const dfs = (
            node: TreapNode<T> | null,
            value: T,
            parent: TreapNode<T>,
            direction: 'left' | 'right',
        ): void => {
            if (node == null) return;

            if (compare(node.value, value) === 0) {
                if (node.count > 1) {
                    node.count--;
                    node?.pushUp();
                } else if (node.left == null && node.right == null) {
                    parent[direction] = null;
                } else {
                    // 旋到根节点
                    if (
                        node.right == null ||
                        TreapNode.getFac(node.left) > TreapNode.getFac(node.right)
                    ) {
                        parent[direction] = node.rotateRight();
                        dfs(parent[direction]?.right ?? null, value, parent[direction]!, 'right');
                    } else {
                        parent[direction] = node.rotateLeft();
                        dfs(parent[direction]?.left ?? null, value, parent[direction]!, 'left');
                    }
                }
            } else if (compare(node.value, value) > 0) {
                dfs(node.left, value, node, 'left');
            } else if (compare(node.value, value) < 0) {
                dfs(node.right, value, node, 'right');
            }

            parent?.pushUp();
        };

        dfs(this.root.left, value, this.root, 'left');
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Returns an index to insert value in the sorted set.
     * If the value is already present, the insertion point will be before (to the left of) any existing values.
     */
    bisectLeft(value: T): number {
        const compare = this.compareFn;
        const dfs = (node: TreapNode<T> | null, value: T): number => {
            if (node == null) return 0;

            if (compare(node.value, value) === 0) {
                return TreapNode.getSize(node.left);
            } else if (compare(node.value, value) > 0) {
                return dfs(node.left, value);
            } else if (compare(node.value, value) < 0) {
                return dfs(node.right, value) + TreapNode.getSize(node.left) + node.count;
            }

            return 0;
        };

        return dfs(this.root, value) - 1;
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Returns an index to insert value in the sorted set.
     * If the value is already present, the insertion point will be before (to the right of) any existing values.
     */
    bisectRight(value: T): number {
        const compare = this.compareFn;
        const dfs = (node: TreapNode<T> | null, value: T): number => {
            if (node == null) return 0;

            if (compare(node.value, value) === 0) {
                return TreapNode.getSize(node.left) + node.count;
            } else if (compare(node.value, value) > 0) {
                return dfs(node.left, value);
            } else if (compare(node.value, value) < 0) {
                return dfs(node.right, value) + TreapNode.getSize(node.left) + node.count;
            }

            return 0;
        };
        return dfs(this.root, value) - 1;
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Returns the index of the first occurrence of a value in the set, or -1 if it is not present.
     */
    indexOf(value: T): number {
        const compare = this.compareFn;
        let isExist = false;

        const dfs = (node: TreapNode<T> | null, value: T): number => {
            if (node == null) return 0;

            if (compare(node.value, value) === 0) {
                isExist = true;
                return TreapNode.getSize(node.left);
            } else if (compare(node.value, value) > 0) {
                return dfs(node.left, value);
            } else if (compare(node.value, value) < 0) {
                return dfs(node.right, value) + TreapNode.getSize(node.left) + node.count;
            }

            return 0;
        };
        const res = dfs(this.root, value) - 1;
        return isExist ? res : -1;
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Returns the index of the last occurrence of a value in the set, or -1 if it is not present.
     */
    lastIndexOf(value: T): number {
        const compare = this.compareFn;
        let isExist = false;

        const dfs = (node: TreapNode<T> | null, value: T): number => {
            if (node == null) return 0;

            if (compare(node.value, value) === 0) {
                isExist = true;
                return TreapNode.getSize(node.left) + node.count - 1;
            } else if (compare(node.value, value) > 0) {
                return dfs(node.left, value);
            } else if (compare(node.value, value) < 0) {
                return dfs(node.right, value) + TreapNode.getSize(node.left) + node.count;
            }

            return 0;
        };

        const res = dfs(this.root, value) - 1;
        return isExist ? res : -1;
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Returns the item located at the specified index.
     * @param index The zero-based index of the desired code unit. A negative index will count back from the last item.
     */
    at(index: number): T | undefined {
        if (index < 0) index += this.size;
        if (index < 0 || index >= this.size) return undefined;

        const dfs = (node: TreapNode<T> | null, rank: number): T | undefined => {
            if (node == null) return undefined;

            if (TreapNode.getSize(node.left) >= rank) {
                return dfs(node.left, rank);
            } else if (TreapNode.getSize(node.left) + node.count >= rank) {
                return node.value;
            } else {
                return dfs(node.right, rank - TreapNode.getSize(node.left) - node.count);
            }
        };

        const res = dfs(this.root, index + 2);
        return ([this.leftBound, this.rightBound] as any[]).includes(res) ? undefined : res;
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Find and return the element less than `val`, return `undefined` if no such element found.
     */
    lower(value: T): T | undefined {
        const compare = this.compareFn;
        const dfs = (node: TreapNode<T> | null, value: T): T | undefined => {
            if (node == null) return undefined;
            if (compare(node.value, value) >= 0) return dfs(node.left, value);

            const tmp = dfs(node.right, value);
            if (tmp == null || compare(node.value, tmp) > 0) {
                return node.value;
            } else {
                return tmp;
            }
        };

        const res = dfs(this.root, value) as any;
        return res === this.leftBound ? undefined : res;
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Find and return the element greater than `val`, return `undefined` if no such element found.
     */
    higher(value: T): T | undefined {
        const compare = this.compareFn;
        const dfs = (node: TreapNode<T> | null, value: T): T | undefined => {
            if (node == null) return undefined;
            if (compare(node.value, value) <= 0) return dfs(node.right, value);

            const tmp = dfs(node.left, value);

            if (tmp == null || compare(node.value, tmp) < 0) {
                return node.value;
            } else {
                return tmp;
            }
        };

        const res = dfs(this.root, value) as any;
        return res === this.rightBound ? undefined : res;
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Find and return the element less than or equal to `val`, return `undefined` if no such element found.
     */
    floor(value: T): T | undefined {
        const compare = this.compareFn;
        const dfs = (node: TreapNode<T> | null, value: T): T | undefined => {
            if (node == null) return undefined;
            if (compare(node.value, value) === 0) return node.value;
            if (compare(node.value, value) >= 0) return dfs(node.left, value);

            const tmp = dfs(node.right, value);
            if (tmp == null || compare(node.value, tmp) > 0) {
                return node.value;
            } else {
                return tmp;
            }
        };

        const res = dfs(this.root, value) as any;
        return res === this.leftBound ? undefined : res;
    }

    /**
     *
     * @complexity `O(logn)`
     * @description Find and return the element greater than or equal to `val`, return `undefined` if no such element found.
     */
    ceil(value: T): T | undefined {
        const compare = this.compareFn;
        const dfs = (node: TreapNode<T> | null, value: T): T | undefined => {
            if (node == null) return undefined;
            if (compare(node.value, value) === 0) return node.value;
            if (compare(node.value, value) <= 0) return dfs(node.right, value);

            const tmp = dfs(node.left, value);

            if (tmp == null || compare(node.value, tmp) < 0) {
                return node.value;
            } else {
                return tmp;
            }
        };

        const res = dfs(this.root, value) as any;
        return res === this.rightBound ? undefined : res;
    }

    /**
     * @complexity `O(logn)`
     * @description
     * Returns the last element from set.
     * If the set is empty, undefined is returned.
     */
    first(): T | undefined {
        const iter = this.inOrder();
        iter.next();
        const res = iter.next().value;
        return res === this.rightBound ? undefined : res;
    }

    /**
     * @complexity `O(logn)`
     * @description
     * Returns the last element from set.
     * If the set is empty, undefined is returned .
     */
    last(): T | undefined {
        const iter = this.reverseInOrder();
        iter.next();
        const res = iter.next().value;
        return res === this.leftBound ? undefined : res;
    }

    /**
     * @complexity `O(logn)`
     * @description
     * Removes the first element from an set and returns it.
     * If the set is empty, undefined is returned and the set is not modified.
     */
    shift(): T | undefined {
        const first = this.first();
        if (first === undefined) return undefined;
        this.delete(first);
        return first;
    }

    /**
     * @complexity `O(logn)`
     * @description
     * Removes the last element from an set and returns it.
     * If the set is empty, undefined is returned and the set is not modified.
     */
    pop(index?: number): T | undefined {
        if (index == null) {
            const last = this.last();
            if (last === undefined) return undefined;
            this.delete(last);
            return last;
        }

        const toDelete = this.at(index);
        if (toDelete == null) return;
        this.delete(toDelete);
        return toDelete;
    }

    /**
     *
     * @complexity `O(logn)`
     * @description
     * Returns number of occurrences of value in the sorted set.
     */
    count(value: T): number {
        const compare = this.compareFn;
        const dfs = (node: TreapNode<T> | null, value: T): number => {
            if (node == null) return 0;
            if (compare(node.value, value) === 0) return node.count;
            if (compare(node.value, value) < 0) return dfs(node.right, value);
            return dfs(node.left, value);
        };

        return dfs(this.root, value);
    }

    *[Symbol.iterator](): Generator<T, any, any> {
        yield* this.values();
    }

    /**
     * @description
     * Returns an iterable of keys in the set.
     */
    *keys(): Generator<T, any, any> {
        yield* this.values();
    }

    /**
     * @description
     * Returns an iterable of values in the set.
     */
    *values(): Generator<T, any, any> {
        const iter = this.inOrder();
        iter.next();
        const steps = this.size;
        for (let _ = 0; _ < steps; _++) {
            yield iter.next().value;
        }
    }

    /**
     * @description
     * Returns a generator for reversed order traversing the set.
     */
    *rvalues(): Generator<T, any, any> {
        const iter = this.reverseInOrder();
        iter.next();
        const steps = this.size;
        for (let _ = 0; _ < steps; _++) {
            yield iter.next().value;
        }
    }

    /**
     * @description
     * Returns an iterable of key, value pairs for every entry in the set.
     */
    *entries(): IterableIterator<[number, T]> {
        const iter = this.inOrder();
        iter.next();
        const steps = this.size;
        for (let i = 0; i < steps; i++) {
            yield [i, iter.next().value];
        }
    }

    private *inOrder(root: TreapNode<T> | null = this.root): Generator<T, any, any> {
        if (root == null) return;
        yield* this.inOrder(root.left);
        const count = root.count;
        for (let _ = 0; _ < count; _++) {
            yield root.value;
        }
        yield* this.inOrder(root.right);
    }

    private *reverseInOrder(root: TreapNode<T> | null = this.root): Generator<T, any, any> {
        if (root == null) return;
        yield* this.reverseInOrder(root.right);
        const count = root.count;
        for (let _ = 0; _ < count; _++) {
            yield root.value;
        }
        yield* this.reverseInOrder(root.left);
    }
}

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