Question

2454. 下一个更大元素 IV

English Version

题目描述

给你一个下标从 0 开始的非负整数数组 nums 。对于 nums 中每一个整数，你必须找到对应元素的 第二大 整数。

如果 nums[j] 满足以下条件，那么我们称它为 nums[i] 的 第二大 整数：

• j > i
• nums[j] > nums[i]
• 恰好存在 一个 k 满足 i < k < j 且 nums[k] > nums[i] 。

如果不存在 nums[j] ，那么第二大整数为 -1 。

• 比方说，数组 [1, 2, 4, 3] 中，1 的第二大整数是 4 ，2 的第二大整数是 3 ，3 和 4 的第二大整数是 -1 。

请你返回一个整数数组_ _answer ，其中_ _answer[i]是_ _nums[i] 的第二大整数。

 

示例 1：

输入：nums = [2,4,0,9,6]
输出：[9,6,6,-1,-1]
解释：
下标为 0 处：2 的右边，4 是大于 2 的第一个整数，9 是第二个大于 2 的整数。
下标为 1 处：4 的右边，9 是大于 4 的第一个整数，6 是第二个大于 4 的整数。
下标为 2 处：0 的右边，9 是大于 0 的第一个整数，6 是第二个大于 0 的整数。
下标为 3 处：右边不存在大于 9 的整数，所以第二大整数为 -1 。
下标为 4 处：右边不存在大于 6 的整数，所以第二大整数为 -1 。
所以我们返回 [9,6,6,-1,-1] 。

示例 2：

输入：nums = [3,3]
输出：[-1,-1]
解释：
由于每个数右边都没有更大的数，所以我们返回 [-1,-1] 。

 

提示：

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

解法

方法一：排序 + 有序集合

我们可以将数组中的元素转成二元组 $(x, i)$，其中 $x$ 为元素的值，$i$ 为元素的下标。然后按照元素的值从大到小排序。

接下来，我们遍历排序后的数组，维护一个有序集合，其中存储的是元素的下标。当我们遍历到元素 $(x, i)$ 时，所有大于 $x$ 的元素的下标都已经在有序集合中了。我们只需要在有序集合中，找到 $i$ 的下下一个元素的下标 $j$，那么 $j$ 对应的元素就是 $x$ 的第二大元素。然后，我们将 $i$ 加入到有序集合中。继续遍历下一个元素。

时间复杂度 $O(n \times \log n)$，空间复杂度 $O(n)$。其中 $n$ 为数组的长度。

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]]
      sl.add(i)
    return ans

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)];
      }
      ts.add(i);
    }
    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;
  }
};

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];
      }
    }
    ts.add(i);
  }
  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);
  }

  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;
  }
}