# 2454. Next Greater Element IV

## Description

You are given a 0-indexed array of non-negative integers nums. For each integer in nums, you must find its respective second greater integer.

The second greater integer of nums[i] is nums[j] such that:

• j > i
• nums[j] > nums[i]
• There exists exactly one index k such that nums[k] > nums[i] and i < k < j.

If there is no such nums[j], the second greater integer is considered to be -1.

• For example, in the array [1, 2, 4, 3], the second greater integer of 1 is 4, 2 is 3, and that of 3 and 4 is -1.

Return an integer array answer, where answer[i] is the second greater integer of nums[i].

Example 1:

Input: nums = [2,4,0,9,6]
Output: [9,6,6,-1,-1]
Explanation:
0th index: 4 is the first integer greater than 2, and 9 is the second integer greater than 2, to the right of 2.
1st index: 9 is the first, and 6 is the second integer greater than 4, to the right of 4.
2nd index: 9 is the first, and 6 is the second integer greater than 0, to the right of 0.
3rd index: There is no integer greater than 9 to its right, so the second greater integer is considered to be -1.
4th index: There is no integer greater than 6 to its right, so the second greater integer is considered to be -1.
Thus, we return [9,6,6,-1,-1].


Example 2:

Input: nums = [3,3]
Output: [-1,-1]
Explanation:
We return [-1,-1] since neither integer has any integer greater than it.


Constraints:

• 1 <= nums.length <= 105
• 0 <= nums[i] <= 109

## Solutions

Solution 1: Sorting + Ordered Set

We can convert the elements in the array into pairs $(x, i)$, where $x$ is the value of the element and $i$ is the index of the element. Then sort by the value of the elements in descending order.

Next, we traverse the sorted array, maintaining an ordered set that stores the indices of the elements. When we traverse to the element $(x, i)$, the indices of all elements greater than $x$ are already in the ordered set. We only need to find the index $j$ of the next element after $i$ in the ordered set, then the element corresponding to $j$ is the second largest element of $x$. Then, we add $i$ to the ordered set. Continue to traverse the next element.

The time complexity is $O(n \times \log n)$, and the space complexity is $O(n)$. Here, $n$ is the length of the array.

• class Solution {
public int[] secondGreaterElement(int[] nums) {
int n = nums.length;
int[] ans = new int[n];
Arrays.fill(ans, -1);
int[][] arr = new int[n][0];
for (int i = 0; i < n; ++i) {
arr[i] = new int[] {nums[i], i};
}
Arrays.sort(arr, (a, b) -> a[0] == b[0] ? a[1] - b[1] : b[0] - a[0]);
TreeSet<Integer> ts = new TreeSet<>();
for (int[] pair : arr) {
int i = pair[1];
Integer j = ts.higher(i);
if (j != null && ts.higher(j) != null) {
ans[i] = nums[ts.higher(j)];
}
}
return ans;
}
}

• class Solution {
public:
vector<int> secondGreaterElement(vector<int>& nums) {
int n = nums.size();
vector<int> ans(n, -1);
vector<pair<int, int>> arr(n);
for (int i = 0; i < n; ++i) {
arr[i] = {-nums[i], i};
}
sort(arr.begin(), arr.end());
set<int> ts;
for (auto& [_, i] : arr) {
auto it = ts.upper_bound(i);
if (it != ts.end() && ts.upper_bound(*it) != ts.end()) {
ans[i] = nums[*ts.upper_bound(*it)];
}
ts.insert(i);
}
return ans;
}
};

• from sortedcontainers import SortedList

class Solution:
def secondGreaterElement(self, nums: List[int]) -> List[int]:
arr = [(x, i) for i, x in enumerate(nums)]
arr.sort(key=lambda x: -x[0])
sl = SortedList()
n = len(nums)
ans = [-1] * n
for _, i in arr:
j = sl.bisect_right(i)
if j + 1 < len(sl):
ans[i] = nums[sl[j + 1]]
return ans


• function secondGreaterElement(nums: number[]): number[] {
const n = nums.length;
const arr: number[][] = [];
for (let i = 0; i < n; ++i) {
arr.push([nums[i], i]);
}
arr.sort((a, b) => (a[0] == b[0] ? a[1] - b[1] : b[0] - a[0]));
const ans = Array(n).fill(-1);
const ts = new TreeSet<number>();
for (const [_, i] of arr) {
let j = ts.higher(i);
if (j !== undefined) {
j = ts.higher(j);
if (j !== undefined) {
ans[i] = nums[j];
}
}
}
return ans;
}

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;
}
}