# 2336. Smallest Number in Infinite Set

## Description

You have a set which contains all positive integers [1, 2, 3, 4, 5, ...].

Implement the SmallestInfiniteSet class:

• SmallestInfiniteSet() Initializes the SmallestInfiniteSet object to contain all positive integers.
• int popSmallest() Removes and returns the smallest integer contained in the infinite set.
• void addBack(int num) Adds a positive integer num back into the infinite set, if it is not already in the infinite set.

Example 1:

Input
[[], [2], [], [], [], [1], [], [], []]
Output
[null, null, 1, 2, 3, null, 1, 4, 5]

Explanation
SmallestInfiniteSet smallestInfiniteSet = new SmallestInfiniteSet();
smallestInfiniteSet.popSmallest(); // return 1, since 1 is the smallest number, and remove it from the set.
smallestInfiniteSet.popSmallest(); // return 2, and remove it from the set.
smallestInfiniteSet.popSmallest(); // return 3, and remove it from the set.
smallestInfiniteSet.popSmallest(); // return 1, since 1 was added back to the set and
// is the smallest number, and remove it from the set.
smallestInfiniteSet.popSmallest(); // return 4, and remove it from the set.
smallestInfiniteSet.popSmallest(); // return 5, and remove it from the set.


Constraints:

• 1 <= num <= 1000
• At most 1000 calls will be made in total to popSmallest and addBack.

## Solutions

Solution 1: Ordered Set + Simulation

We note that the range of elements in the set given by the problem is $[1, 1000]$, and the operations we need to support are:

• popSmallest: Pop the smallest element from the set
• addBack: Add an element back to the set

Therefore, we can use an ordered set to simulate this. Let’s denote the ordered set as $s$, and the elements in the set as $s_1, s_2, \cdots, s_n$, where $n$ is the number of elements in the ordered set. In this problem, $n \le 1000$.

During initialization, we add all elements in $[1, 1000]$ to the ordered set. The time complexity is $O(n \times \log n)$.

In the popSmallest operation, we just need to pop the first element from the ordered set. The time complexity for a single operation is $O(\log n)$.

In the addBack operation, we just need to add the element back to the ordered set. The time complexity for a single operation is $O(\log n)$.

The space complexity is $O(n)$.

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

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

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

}
}

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

• 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;
}

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();
*/

• 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:

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


• 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();
*/

• class SmallestInfiniteSet {
private s: TreeSet<number>;

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

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

}
}

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);
}

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);
}

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()
*/


• 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);
}
}/**
* Your SmallestInfiniteSet object will be instantiated and called as such:
* let obj = SmallestInfiniteSet::new();
* let ret_1: i32 = obj.pop_smallest();