2817. Minimum Absolute Difference Between Elements With Constraint

Description

You are given a 0-indexed integer array nums and an integer x.

Find the minimum absolute difference between two elements in the array that are at least x indices apart.

In other words, find two indices i and j such that abs(i - j) >= x and abs(nums[i] - nums[j]) is minimized.

Return an integer denoting the minimum absolute difference between two elements that are at least x indices apart.

Example 1:

Input: nums = [4,3,2,4], x = 2
Output: 0
Explanation: We can select nums[0] = 4 and nums[3] = 4.
They are at least 2 indices apart, and their absolute difference is the minimum, 0.
It can be shown that 0 is the optimal answer.

Example 2:

Input: nums = [5,3,2,10,15], x = 1
Output: 1
Explanation: We can select nums[1] = 3 and nums[2] = 2.
They are at least 1 index apart, and their absolute difference is the minimum, 1.
It can be shown that 1 is the optimal answer.

Example 3:

Input: nums = [1,2,3,4], x = 3
Output: 3
Explanation: We can select nums[0] = 1 and nums[3] = 4.
They are at least 3 indices apart, and their absolute difference is the minimum, 3.
It can be shown that 3 is the optimal answer.

Constraints:

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

Solutions

Solution 1: Ordered Set

We create an ordered set to store the elements whose distance to the current index is at least $x$.

Next, we enumerate from index $i = x$, each time we add $nums[i - x]$ into the ordered set. Then we find the two elements in the ordered set which are closest to $nums[i]$, and the minimum absolute difference between them is the answer.

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

• class Solution {
public int minAbsoluteDifference(List<Integer> nums, int x) {
TreeMap<Integer, Integer> tm = new TreeMap<>();
int ans = 1 << 30;
for (int i = x; i < nums.size(); ++i) {
tm.merge(nums.get(i - x), 1, Integer::sum);
Integer key = tm.ceilingKey(nums.get(i));
if (key != null) {
ans = Math.min(ans, key - nums.get(i));
}
key = tm.floorKey(nums.get(i));
if (key != null) {
ans = Math.min(ans, nums.get(i) - key);
}
}
return ans;
}
}

• class Solution {
public:
int minAbsoluteDifference(vector<int>& nums, int x) {
int ans = 1 << 30;
multiset<int> s;
for (int i = x; i < nums.size(); ++i) {
s.insert(nums[i - x]);
auto it = s.lower_bound(nums[i]);
if (it != s.end()) {
ans = min(ans, *it - nums[i]);
}
if (it != s.begin()) {
--it;
ans = min(ans, nums[i] - *it);
}
}
return ans;
}
};

• from sortedcontainers import SortedList

class Solution:
def minAbsoluteDifference(self, nums: List[int], x: int) -> int:
sl = SortedList()
ans = inf
for i in range(x, len(nums)):
j = bisect_left(sl, nums[i])
if j < len(sl):
ans = min(ans, sl[j] - nums[i])
if j:
ans = min(ans, nums[i] - sl[j - 1])
return ans

• func minAbsoluteDifference(nums []int, x int) int {
rbt := redblacktree.NewWithIntComparator()
ans := 1 << 30
for i := x; i < len(nums); i++ {
rbt.Put(nums[i-x], nil)
c, _ := rbt.Ceiling(nums[i])
f, _ := rbt.Floor(nums[i])
if c != nil {
ans = min(ans, c.Key.(int)-nums[i])
}
if f != nil {
ans = min(ans, nums[i]-f.Key.(int))
}
}
return ans
}

• function minAbsoluteDifference(nums: number[], x: number): number {
const s = new TreeMultiSet<number>();
const inf = 1 << 30;
let ans = inf;
for (let i = x; i < nums.length; ++i) {
const c = s.ceil(nums[i]);
const f = s.floor(nums[i]);
if (c) {
ans = Math.min(ans, c - nums[i]);
}
if (f) {
ans = Math.min(ans, nums[i] - f);
}
}
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 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;
}
}