Question

220. 存在重复元素 III

English Version

题目描述

给你一个整数数组 nums 和两个整数 indexDiff 和 valueDiff 。

找出满足下述条件的下标对 (i, j)：

• i != j,
• abs(i - j) <= indexDiff
• abs(nums[i] - nums[j]) <= valueDiff

如果存在，返回 true_ ；_否则，返回_ _false_ _。

 

示例 1：

输入：nums = [1,2,3,1], indexDiff = 3, valueDiff = 0
输出：true
解释：可以找出 (i, j) = (0, 3) 。
满足下述 3 个条件：
i != j --> 0 != 3
abs(i - j) <= indexDiff --> abs(0 - 3) <= 3
abs(nums[i] - nums[j]) <= valueDiff --> abs(1 - 1) <= 0

示例 2：

输入：nums = [1,5,9,1,5,9], indexDiff = 2, valueDiff = 3
输出：false
解释：尝试所有可能的下标对 (i, j) ，均无法满足这 3 个条件，因此返回 false 。

 

提示：

• 2 <= nums.length <= 105
• -109 <= nums[i] <= 109
• 1 <= indexDiff <= nums.length
• 0 <= valueDiff <= 109

解法

方法一：滑动窗口 + 有序集合

我们维护一个大小为 $k$ 的滑动窗口，窗口中的元素保持有序。

遍历数组 nums，对于每个元素 $nums[i]$，我们在有序集合中查找第一个大于等于 $nums[i] - t$ 的元素，如果元素存在，并且该元素小于等于 $nums[i] + t$，说明找到了一对符合条件的元素，返回 true。否则，我们将 $nums[i]$ 插入到有序集合中，并且如果有序集合的大小超过了 $k$，我们需要将最早加入有序集合的元素删除。

时间复杂度 $O(n \times \log k)$，其中 $n$ 是数组 nums 的长度。对于每个元素，我们需要 $O(\log k)$ 的时间来查找有序集合中的元素，一共有 $n$ 个元素，因此总时间复杂度是 $O(n \times \log k)$。

from sortedcontainers import SortedSet


class Solution:
  def containsNearbyAlmostDuplicate(
    self, nums: List[int], indexDiff: int, valueDiff: int
  ) -> bool:
    s = SortedSet()
    for i, v in enumerate(nums):
      j = s.bisect_left(v - valueDiff)
      if j < len(s) and s[j] <= v + valueDiff:
        return True
      s.add(v)
      if i >= indexDiff:
        s.remove(nums[i - indexDiff])
    return False

class Solution {
  public boolean containsNearbyAlmostDuplicate(int[] nums, int indexDiff, int valueDiff) {
    TreeSet<Long> ts = new TreeSet<>();
    for (int i = 0; i < nums.length; ++i) {
      Long x = ts.ceiling((long) nums[i] - (long) valueDiff);
      if (x != null && x <= (long) nums[i] + (long) valueDiff) {
        return true;
      }
      ts.add((long) nums[i]);
      if (i >= indexDiff) {
        ts.remove((long) nums[i - indexDiff]);
      }
    }
    return false;
  }
}

class Solution {
public:
  bool containsNearbyAlmostDuplicate(vector<int>& nums, int indexDiff, int valueDiff) {
    set<long> s;
    for (int i = 0; i < nums.size(); ++i) {
      auto it = s.lower_bound((long) nums[i] - valueDiff);
      if (it != s.end() && *it <= (long) nums[i] + valueDiff) return true;
      s.insert((long) nums[i]);
      if (i >= indexDiff) s.erase((long) nums[i - indexDiff]);
    }
    return false;
  }
};

func containsNearbyAlmostDuplicate(nums []int, k int, t int) bool {
  n := len(nums)
  left, right := 0, 0
  rbt := redblacktree.NewWithIntComparator()
  for right < n {
    cur := nums[right]
    right++
    if p, ok := rbt.Floor(cur); ok && cur-p.Key.(int) <= t {
      return true
    }
    if p, ok := rbt.Ceiling(cur); ok && p.Key.(int)-cur <= t {
      return true
    }
    rbt.Put(cur, struct{}{})
    if right-left > k {
      rbt.Remove(nums[left])
      left++
    }
  }
  return false
}

function containsNearbyAlmostDuplicate(
  nums: number[],
  indexDiff: number,
  valueDiff: number,
): boolean {
  const ts = new TreeSet<number>();
  for (let i = 0; i < nums.length; ++i) {
    const x = ts.ceil(nums[i] - valueDiff);
    if (x != null && x <= nums[i] + valueDiff) {
      return true;
    }
    ts.add(nums[i]);
    if (i >= indexDiff) {
      ts.delete(nums[i - indexDiff]);
    }
  }
  return false;
}

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;
  }
}

public class Solution {
  public bool ContainsNearbyAlmostDuplicate(int[] nums, int k, int t) {
    if (k <= 0 || t < 0) return false;
    var index = new SortedList<int, object>();
    for (int i = 0; i < nums.Length; ++i) {
      if (index.ContainsKey(nums[i])) {
        return true;
      }
      index.Add(nums[i], null);
      var j = index.IndexOfKey(nums[i]);
      if (j > 0 && (long)nums[i] - index.Keys[j - 1] <= t) {
        return true;
      }
      if (j < index.Count - 1 && (long)index.Keys[j + 1] - nums[i] <= t) {
        return true;
      }
      if (index.Count > k) {
        index.Remove(nums[i - k]);
      }
    }
    return false;
  }
}