2336. 无限集中的最小数字

题目描述

现有一个包含所有正整数的集合 [1, 2, 3, 4, 5, ...] 。

实现 SmallestInfiniteSet 类：

• SmallestInfiniteSet() 初始化 SmallestInfiniteSet 对象以包含 所有 正整数。
• int popSmallest() 移除 并返回该无限集中的最小整数。
• void addBack(int num) 如果正整数 num 不 存在于无限集中，则将一个 num 添加 到该无限集中。

 

示例：

输入
["SmallestInfiniteSet", "addBack", "popSmallest", "popSmallest", "popSmallest", "addBack", "popSmallest", "popSmallest", "popSmallest"]
[[], [2], [], [], [], [1], [], [], []]
输出
[null, null, 1, 2, 3, null, 1, 4, 5]

解释
SmallestInfiniteSet smallestInfiniteSet = new SmallestInfiniteSet();
smallestInfiniteSet.addBack(2);    // 2 已经在集合中，所以不做任何变更。
smallestInfiniteSet.popSmallest(); // 返回 1 ，因为 1 是最小的整数，并将其从集合中移除。
smallestInfiniteSet.popSmallest(); // 返回 2 ，并将其从集合中移除。
smallestInfiniteSet.popSmallest(); // 返回 3 ，并将其从集合中移除。
smallestInfiniteSet.addBack(1);    // 将 1 添加到该集合中。
smallestInfiniteSet.popSmallest(); // 返回 1 ，因为 1 在上一步中被添加到集合中，
                                   // 且 1 是最小的整数，并将其从集合中移除。
smallestInfiniteSet.popSmallest(); // 返回 4 ，并将其从集合中移除。
smallestInfiniteSet.popSmallest(); // 返回 5 ，并将其从集合中移除。

 

提示：

• 1 <= num <= 1000
• 最多调用 popSmallest 和 addBack 方法 共计 1000 次

解法

方法一：有序集合 + 模拟

我们注意到，题目中集合的元素范围是 $[1, 1000]$，并且我们需要支持的操作有：

• popSmallest：弹出集合中的最小元素
• addBack：向集合中添加元素

因此，我们可以使用有序集合来模拟，不妨记有序集合为 $s$，集合中的元素为 $s_1, s_2, \cdots, s_n$，其中 $n$ 为有序集合中的元素个数。本题中 $n \le 1000$。

我们在初始化时，将 $[1, 1000]$ 中的所有元素加入有序集合中。时间复杂度 $O(n \times \log n)$。

在 popSmallest 操作中，我们只需要弹出有序集合中的第一个元素即可。单次操作时间复杂度 $O(\log n)$。

在 addBack 操作中，我们只需要将元素加入有序集合中即可。单次操作时间复杂度 $O(\log n)$。

空间复杂度 $O(n)$。

Python3JavaC++GoTypeScriptRust

from sortedcontainers import SortedSet


class SmallestInfiniteSet:
    def __init__(self):
        self.s = SortedSet(range(1, 1001))

    def popSmallest(self) -> int:
        x = self.s[0]
        self.s.remove(x)
        return x

    def addBack(self, num: int) -> None:
        self.s.add(num)


# Your SmallestInfiniteSet object will be instantiated and called as such:
# obj = SmallestInfiniteSet()
# param_1 = obj.popSmallest()
# obj.addBack(num)

class SmallestInfiniteSet {
    private TreeSet<Integer> s = new TreeSet<>();

    public SmallestInfiniteSet() {
        for (int i = 1; i <= 1000; ++i) {
            s.add(i);
        }
    }

    public int popSmallest() {
        return s.pollFirst();
    }

    public void addBack(int num) {
        s.add(num);
    }
}

/**
 * Your SmallestInfiniteSet object will be instantiated and called as such:
 * SmallestInfiniteSet obj = new SmallestInfiniteSet();
 * int param_1 = obj.popSmallest();
 * obj.addBack(num);
 */

class SmallestInfiniteSet {
public:
    SmallestInfiniteSet() {
        for (int i = 1; i <= 1000; ++i) {
            s.insert(i);
        }
    }

    int popSmallest() {
        int x = *s.begin();
        s.erase(s.begin());
        return x;
    }

    void addBack(int num) {
        s.insert(num);
    }

private:
    set<int> s;
};

/**
 * Your SmallestInfiniteSet object will be instantiated and called as such:
 * SmallestInfiniteSet* obj = new SmallestInfiniteSet();
 * int param_1 = obj->popSmallest();
 * obj->addBack(num);
 */

type SmallestInfiniteSet struct {
    s *treemap.Map
}

func Constructor() SmallestInfiniteSet {
    s := treemap.NewWithIntComparator()
    for i := 1; i <= 1000; i++ {
        s.Put(i, nil)
    }
    return SmallestInfiniteSet{s}
}

func (this *SmallestInfiniteSet) PopSmallest() int {
    x, _ := this.s.Min()
    this.s.Remove(x.(int))
    return x.(int)
}

func (this *SmallestInfiniteSet) AddBack(num int) {
    this.s.Put(num, nil)
}

/**
 * Your SmallestInfiniteSet object will be instantiated and called as such:
 * obj := Constructor();
 * param_1 := obj.PopSmallest();
 * obj.AddBack(num);
 */

class SmallestInfiniteSet {
    private s: TreeSet<number>;

    constructor() {
        this.s = new TreeSet();
        for (let i = 1; i <= 1000; ++i) {
            this.s.add(i);
        }
    }

    popSmallest(): number {
        return this.s.shift()!;
    }

    addBack(num: number): void {
        this.s.add(num);
    }
}

type Compare<T> = (lhs: T, rhs: T) => number;

class RBTreeNode<T = number> {
    data: T;
    count: number;
    left: RBTreeNode<T> | null;
    right: RBTreeNode<T> | null;
    parent: RBTreeNode<T> | null;
    color: number;
    constructor(data: T) {
        this.data = data;
        this.left = this.right = this.parent = null;
        this.color = 0;
        this.count = 1;
    }

    sibling(): RBTreeNode<T> | null {
        if (!this.parent) return null; // sibling null if no parent
        return this.isOnLeft() ? this.parent.right : this.parent.left;
    }

    isOnLeft(): boolean {
        return this === this.parent!.left;
    }

    hasRedChild(): boolean {
        return (
            Boolean(this.left && this.left.color === 0) ||
            Boolean(this.right && this.right.color === 0)
        );
    }
}

class RBTree<T> {
    root: RBTreeNode<T> | null;
    lt: (l: T, r: T) => boolean;
    constructor(compare: Compare<T> = (l: T, r: T) => (l < r ? -1 : l > r ? 1 : 0)) {
        this.root = null;
        this.lt = (l: T, r: T) => compare(l, r) < 0;
    }

    rotateLeft(pt: RBTreeNode<T>): void {
        const right = pt.right!;
        pt.right = right.left;

        if (pt.right) pt.right.parent = pt;
        right.parent = pt.parent;

        if (!pt.parent) this.root = right;
        else if (pt === pt.parent.left) pt.parent.left = right;
        else pt.parent.right = right;

        right.left = pt;
        pt.parent = right;
    }

    rotateRight(pt: RBTreeNode<T>): void {
        const left = pt.left!;
        pt.left = left.right;

        if (pt.left) pt.left.parent = pt;
        left.parent = pt.parent;

        if (!pt.parent) this.root = left;
        else if (pt === pt.parent.left) pt.parent.left = left;
        else pt.parent.right = left;

        left.right = pt;
        pt.parent = left;
    }

    swapColor(p1: RBTreeNode<T>, p2: RBTreeNode<T>): void {
        const tmp = p1.color;
        p1.color = p2.color;
        p2.color = tmp;
    }

    swapData(p1: RBTreeNode<T>, p2: RBTreeNode<T>): void {
        const tmp = p1.data;
        p1.data = p2.data;
        p2.data = tmp;
    }

    fixAfterInsert(pt: RBTreeNode<T>): void {
        let parent = null;
        let grandParent = null;

        while (pt !== this.root && pt.color !== 1 && pt.parent?.color === 0) {
            parent = pt.parent;
            grandParent = pt.parent.parent;

            /*  Case : A
                Parent of pt is left child of Grand-parent of pt */
            if (parent === grandParent?.left) {
                const uncle = grandParent.right;

                /* Case : 1
                   The uncle of pt is also red
                   Only Recoloring required */
                if (uncle && uncle.color === 0) {
                    grandParent.color = 0;
                    parent.color = 1;
                    uncle.color = 1;
                    pt = grandParent;
                } else {
                    /* Case : 2
                       pt is right child of its parent
                       Left-rotation required */
                    if (pt === parent.right) {
                        this.rotateLeft(parent);
                        pt = parent;
                        parent = pt.parent;
                    }

                    /* Case : 3
                       pt is left child of its parent
                       Right-rotation required */
                    this.rotateRight(grandParent);
                    this.swapColor(parent!, grandParent);
                    pt = parent!;
                }
            } else {
                /* Case : B
               Parent of pt is right child of Grand-parent of pt */
                const uncle = grandParent!.left;

                /*  Case : 1
                    The uncle of pt is also red
                    Only Recoloring required */
                if (uncle != null && uncle.color === 0) {
                    grandParent!.color = 0;
                    parent.color = 1;
                    uncle.color = 1;
                    pt = grandParent!;
                } else {
                    /* Case : 2
                       pt is left child of its parent
                       Right-rotation required */
                    if (pt === parent.left) {
                        this.rotateRight(parent);
                        pt = parent;
                        parent = pt.parent;
                    }

                    /* Case : 3
                       pt is right child of its parent
                       Left-rotation required */
                    this.rotateLeft(grandParent!);
                    this.swapColor(parent!, grandParent!);
                    pt = parent!;
                }
            }
        }
        this.root!.color = 1;
    }

    delete(val: T): boolean {
        const node = this.find(val);
        if (!node) return false;
        node.count--;
        if (!node.count) this.deleteNode(node);
        return true;
    }

    deleteAll(val: T): boolean {
        const node = this.find(val);
        if (!node) return false;
        this.deleteNode(node);
        return true;
    }

    deleteNode(v: RBTreeNode<T>): void {
        const u = BSTreplace(v);

        // True when u and v are both black
        const uvBlack = (u === null || u.color === 1) && v.color === 1;
        const parent = v.parent!;

        if (!u) {
            // u is null therefore v is leaf
            if (v === this.root) this.root = null;
            // v is root, making root null
            else {
                if (uvBlack) {
                    // u and v both black
                    // v is leaf, fix double black at v
                    this.fixDoubleBlack(v);
                } else {
                    // u or v is red
                    if (v.sibling()) {
                        // sibling is not null, make it red"
                        v.sibling()!.color = 0;
                    }
                }
                // delete v from the tree
                if (v.isOnLeft()) parent.left = null;
                else parent.right = null;
            }
            return;
        }

        if (!v.left || !v.right) {
            // v has 1 child
            if (v === this.root) {
                // v is root, assign the value of u to v, and delete u
                v.data = u.data;
                v.left = v.right = null;
            } else {
                // Detach v from tree and move u up
                if (v.isOnLeft()) parent.left = u;
                else parent.right = u;
                u.parent = parent;
                if (uvBlack) this.fixDoubleBlack(u);
                // u and v both black, fix double black at u
                else u.color = 1; // u or v red, color u black
            }
            return;
        }

        // v has 2 children, swap data with successor and recurse
        this.swapData(u, v);
        this.deleteNode(u);

        // find node that replaces a deleted node in BST
        function BSTreplace(x: RBTreeNode<T>): RBTreeNode<T> | null {
            // when node have 2 children
            if (x.left && x.right) return successor(x.right);
            // when leaf
            if (!x.left && !x.right) return null;
            // when single child
            return x.left ?? x.right;
        }
        // find node that do not have a left child
        // in the subtree of the given node
        function successor(x: RBTreeNode<T>): RBTreeNode<T> {
            let temp = x;
            while (temp.left) temp = temp.left;
            return temp;
        }
    }

    fixDoubleBlack(x: RBTreeNode<T>): void {
        if (x === this.root) return; // Reached root

        const sibling = x.sibling();
        const parent = x.parent!;
        if (!sibling) {
            // No sibiling, double black pushed up
            this.fixDoubleBlack(parent);
        } else {
            if (sibling.color === 0) {
                // Sibling red
                parent.color = 0;
                sibling.color = 1;
                if (sibling.isOnLeft()) this.rotateRight(parent);
                // left case
                else this.rotateLeft(parent); // right case
                this.fixDoubleBlack(x);
            } else {
                // Sibling black
                if (sibling.hasRedChild()) {
                    // at least 1 red children
                    if (sibling.left && sibling.left.color === 0) {
                        if (sibling.isOnLeft()) {
                            // left left
                            sibling.left.color = sibling.color;
                            sibling.color = parent.color;
                            this.rotateRight(parent);
                        } else {
                            // right left
                            sibling.left.color = parent.color;
                            this.rotateRight(sibling);
                            this.rotateLeft(parent);
                        }
                    } else {
                        if (sibling.isOnLeft()) {
                            // left right
                            sibling.right!.color = parent.color;
                            this.rotateLeft(sibling);
                            this.rotateRight(parent);
                        } else {
                            // right right
                            sibling.right!.color = sibling.color;
                            sibling.color = parent.color;
                            this.rotateLeft(parent);
                        }
                    }
                    parent.color = 1;
                } else {
                    // 2 black children
                    sibling.color = 0;
                    if (parent.color === 1) this.fixDoubleBlack(parent);
                    else parent.color = 1;
                }
            }
        }
    }

    insert(data: T): boolean {
        // search for a position to insert
        let parent = this.root;
        while (parent) {
            if (this.lt(data, parent.data)) {
                if (!parent.left) break;
                else parent = parent.left;
            } else if (this.lt(parent.data, data)) {
                if (!parent.right) break;
                else parent = parent.right;
            } else break;
        }

        // insert node into parent
        const node = new RBTreeNode(data);
        if (!parent) this.root = node;
        else if (this.lt(node.data, parent.data)) parent.left = node;
        else if (this.lt(parent.data, node.data)) parent.right = node;
        else {
            parent.count++;
            return false;
        }
        node.parent = parent;
        this.fixAfterInsert(node);
        return true;
    }

    find(data: T): RBTreeNode<T> | null {
        let p = this.root;
        while (p) {
            if (this.lt(data, p.data)) {
                p = p.left;
            } else if (this.lt(p.data, data)) {
                p = p.right;
            } else break;
        }
        return p ?? null;
    }

    *inOrder(root: RBTreeNode<T> = this.root!): Generator<T, undefined, void> {
        if (!root) return;
        for (const v of this.inOrder(root.left!)) yield v;
        yield root.data;
        for (const v of this.inOrder(root.right!)) yield v;
    }

    *reverseInOrder(root: RBTreeNode<T> = this.root!): Generator<T, undefined, void> {
        if (!root) return;
        for (const v of this.reverseInOrder(root.right!)) yield v;
        yield root.data;
        for (const v of this.reverseInOrder(root.left!)) yield v;
    }
}

class TreeSet<T = number> {
    _size: number;
    tree: RBTree<T>;
    compare: Compare<T>;
    constructor(
        collection: T[] | Compare<T> = [],
        compare: Compare<T> = (l: T, r: T) => (l < r ? -1 : l > r ? 1 : 0),
    ) {
        if (typeof collection === 'function') {
            compare = collection;
            collection = [];
        }
        this._size = 0;
        this.compare = compare;
        this.tree = new RBTree(compare);
        for (const val of collection) this.add(val);
    }

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

    has(val: T): boolean {
        return !!this.tree.find(val);
    }

    add(val: T): boolean {
        const successful = this.tree.insert(val);
        this._size += successful ? 1 : 0;
        return successful;
    }

    delete(val: T): boolean {
        const deleted = this.tree.deleteAll(val);
        this._size -= deleted ? 1 : 0;
        return deleted;
    }

    ceil(val: T): T | undefined {
        let p = this.tree.root;
        let higher = null;
        while (p) {
            if (this.compare(p.data, val) >= 0) {
                higher = p;
                p = p.left;
            } else {
                p = p.right;
            }
        }
        return higher?.data;
    }

    floor(val: T): T | undefined {
        let p = this.tree.root;
        let lower = null;
        while (p) {
            if (this.compare(val, p.data) >= 0) {
                lower = p;
                p = p.right;
            } else {
                p = p.left;
            }
        }
        return lower?.data;
    }

    higher(val: T): T | undefined {
        let p = this.tree.root;
        let higher = null;
        while (p) {
            if (this.compare(val, p.data) < 0) {
                higher = p;
                p = p.left;
            } else {
                p = p.right;
            }
        }
        return higher?.data;
    }

    lower(val: T): T | undefined {
        let p = this.tree.root;
        let lower = null;
        while (p) {
            if (this.compare(p.data, val) < 0) {
                lower = p;
                p = p.right;
            } else {
                p = p.left;
            }
        }
        return lower?.data;
    }

    first(): T | undefined {
        return this.tree.inOrder().next().value;
    }

    last(): T | undefined {
        return this.tree.reverseInOrder().next().value;
    }

    shift(): T | undefined {
        const first = this.first();
        if (first === undefined) return undefined;
        this.delete(first);
        return first;
    }

    pop(): T | undefined {
        const last = this.last();
        if (last === undefined) return undefined;
        this.delete(last);
        return last;
    }

    *[Symbol.iterator](): Generator<T, void, void> {
        for (const val of this.values()) yield val;
    }

    *keys(): Generator<T, void, void> {
        for (const val of this.values()) yield val;
    }

    *values(): Generator<T, undefined, void> {
        for (const val of this.tree.inOrder()) yield val;
        return undefined;
    }

    /**
     * Return a generator for reverse order traversing the set
     */
    *rvalues(): Generator<T, undefined, void> {
        for (const val of this.tree.reverseInOrder()) yield val;
        return undefined;
    }
}

class TreeMultiSet<T = number> {
    _size: number;
    tree: RBTree<T>;
    compare: Compare<T>;
    constructor(
        collection: T[] | Compare<T> = [],
        compare: Compare<T> = (l: T, r: T) => (l < r ? -1 : l > r ? 1 : 0),
    ) {
        if (typeof collection === 'function') {
            compare = collection;
            collection = [];
        }
        this._size = 0;
        this.compare = compare;
        this.tree = new RBTree(compare);
        for (const val of collection) this.add(val);
    }

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

    has(val: T): boolean {
        return !!this.tree.find(val);
    }

    add(val: T): boolean {
        const successful = this.tree.insert(val);
        this._size++;
        return successful;
    }

    delete(val: T): boolean {
        const successful = this.tree.delete(val);
        if (!successful) return false;
        this._size--;
        return true;
    }

    count(val: T): number {
        const node = this.tree.find(val);
        return node ? node.count : 0;
    }

    ceil(val: T): T | undefined {
        let p = this.tree.root;
        let higher = null;
        while (p) {
            if (this.compare(p.data, val) >= 0) {
                higher = p;
                p = p.left;
            } else {
                p = p.right;
            }
        }
        return higher?.data;
    }

    floor(val: T): T | undefined {
        let p = this.tree.root;
        let lower = null;
        while (p) {
            if (this.compare(val, p.data) >= 0) {
                lower = p;
                p = p.right;
            } else {
                p = p.left;
            }
        }
        return lower?.data;
    }

    higher(val: T): T | undefined {
        let p = this.tree.root;
        let higher = null;
        while (p) {
            if (this.compare(val, p.data) < 0) {
                higher = p;
                p = p.left;
            } else {
                p = p.right;
            }
        }
        return higher?.data;
    }

    lower(val: T): T | undefined {
        let p = this.tree.root;
        let lower = null;
        while (p) {
            if (this.compare(p.data, val) < 0) {
                lower = p;
                p = p.right;
            } else {
                p = p.left;
            }
        }
        return lower?.data;
    }

    first(): T | undefined {
        return this.tree.inOrder().next().value;
    }

    last(): T | undefined {
        return this.tree.reverseInOrder().next().value;
    }

    shift(): T | undefined {
        const first = this.first();
        if (first === undefined) return undefined;
        this.delete(first);
        return first;
    }

    pop(): T | undefined {
        const last = this.last();
        if (last === undefined) return undefined;
        this.delete(last);
        return last;
    }

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

    *keys(): Generator<T, void, void> {
        for (const val of this.values()) yield val;
    }

    *values(): Generator<T, undefined, void> {
        for (const val of this.tree.inOrder()) {
            let count = this.count(val);
            while (count--) yield val;
        }
        return undefined;
    }

    /**
     * Return a generator for reverse order traversing the multi-set
     */
    *rvalues(): Generator<T, undefined, void> {
        for (const val of this.tree.reverseInOrder()) {
            let count = this.count(val);
            while (count--) yield val;
        }
        return undefined;
    }
}

/**
 * Your SmallestInfiniteSet object will be instantiated and called as such:
 * var obj = new SmallestInfiniteSet()
 * var param_1 = obj.popSmallest()
 * obj.addBack(num)
 */

use std::collections::BTreeSet;

struct SmallestInfiniteSet {
    s: BTreeSet<i32>,
}

impl SmallestInfiniteSet {
    fn new() -> Self {
        let mut set = BTreeSet::new();
        for i in 1..=1000 {
            set.insert(i);
        }
        SmallestInfiniteSet { s: set }
    }

    fn pop_smallest(&mut self) -> i32 {
        let x = *self.s.iter().next().unwrap();
        self.s.remove(&x);
        x
    }

    fn add_back(&mut self, num: i32) {
        self.s.insert(num);
    }
}

方法二

TypeScript

class SmallestInfiniteSet {
    private pq: typeof MinPriorityQueue;
    private s: Set<number>;

    constructor() {
        this.pq = new MinPriorityQueue();
        this.s = new Set();
        for (let i = 1; i <= 1000; i++) {
            this.pq.enqueue(i, i);
            this.s.add(i);
        }
    }

    popSmallest(): number {
        const x = this.pq.dequeue()?.element;
        this.s.delete(x);
        return x;
    }

    addBack(num: number): void {
        if (!this.s.has(num)) {
            this.pq.enqueue(num, num);
            this.s.add(num);
        }
    }
}

/**
 * Your SmallestInfiniteSet object will be instantiated and called as such:
 * var obj = new SmallestInfiniteSet()
 * var param_1 = obj.popSmallest()
 * obj.addBack(num)
 */

评论