Question

3073. 最大递增三元组

English Version

题目描述

给定一个数组 nums，返回满足 i < j < k 且 nums[i] < nums[j] < nums[k] 的三元组 (i, j, k) _ _的 最大值。

三元组 (i, j, k)  的 值 为 nums[i] - nums[j] + nums[k]。

 

 

示例 1:

输入：nums = [5,6,9]
输出：8
解释：对于一个递增的三元组，我们只有一个选择，那就是选择所有三个元素。三元组的值为 5 - 6 + 9 = 8。

示例 2:

输入：nums = [1,5,3,6]
输出：4
解释：只有两个递增三元组：
(0, 1, 3)：这个三元组的值为 nums[0] - nums[1] + nums[3] = 1 - 5 + 6 = 2。
(0, 2, 3)：这个三元组的值为 nums[0] - nums[2] + nums[3] = 1 - 3 + 6 = 4。
因此答案是 4。

 

提示:

• 3 <= nums.length <= 105
• 1 <= nums[i] <= 109
• 输入数据保证至少一个三元组满足给定条件。

解法

方法一：后缀最大值 + 有序集合

我们不妨考虑枚举 $nums[j]$，那么我们需要在 $j$ 的左侧找到一个最大的 $nums[i]$，使得 $nums[i] < nums[j]$，并且在 $j$ 的右侧找到一个最大的 $nums[k]$，使得 $nums[k] > nums[j]$。

因此，我们可以预处理出数组 $right$，其中 $right[i]$ 表示 $nums[i]$ 右侧的最大值。然后我们可以使用有序集合来维护 $nums[j]$ 左侧的值，这样我们就可以在 $O(\log n)$ 的时间内找到最大的小于 $nums[j]$ 的 $nums[i]$。

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

from sortedcontainers import SortedList


class Solution:
  def maximumTripletValue(self, nums: List[int]) -> int:
    n = len(nums)
    right = [nums[-1]] * n
    for i in range(n - 2, -1, -1):
      right[i] = max(nums[i], right[i + 1])
    sl = SortedList([nums[0]])
    ans = 0
    for j in range(1, n - 1):
      if right[j + 1] > nums[j]:
        i = sl.bisect_left(nums[j]) - 1
        if i >= 0:
          ans = max(ans, sl[i] - nums[j] + right[j + 1])
      sl.add(nums[j])
    return ans

class Solution {
  public int maximumTripletValue(int[] nums) {
    int n = nums.length;
    int[] right = new int[n];
    right[n - 1] = nums[n - 1];
    for (int i = n - 2; i >= 0; --i) {
      right[i] = Math.max(nums[i], right[i + 1]);
    }
    TreeSet<Integer> ts = new TreeSet<>();
    ts.add(nums[0]);
    int ans = 0;
    for (int j = 1; j < n - 1; ++j) {
      if (right[j + 1] > nums[j]) {
        Integer it = ts.lower(nums[j]);
        if (it != null) {
          ans = Math.max(ans, it - nums[j] + right[j + 1]);
        }
      }
      ts.add(nums[j]);
    }
    return ans;
  }
}

class Solution {
public:
  int maximumTripletValue(vector<int>& nums) {
    int n = nums.size();
    vector<int> right(n, nums.back());
    for (int i = n - 2; ~i; --i) {
      right[i] = max(nums[i], right[i + 1]);
    }
    set<int> ts;
    ts.insert(nums[0]);
    int ans = 0;
    for (int j = 1; j < n - 1; ++j) {
      if (right[j + 1] > nums[j]) {
        auto it = ts.lower_bound(nums[j]);
        if (it != ts.begin()) {
          --it;
          ans = max(ans, *it - nums[j] + right[j + 1]);
        }
      }
      ts.insert(nums[j]);
    }
    return ans;
  }
};

func maximumTripletValue(nums []int) (ans int) {
  n := len(nums)
  right := make([]int, n)
  right[n-1] = nums[n-1]
  for i := n - 2; i >= 0; i-- {
    right[i] = max(nums[i], right[i+1])
  }
  ts := treemap.NewWithIntComparator()
  ts.Put(nums[0], nil)
  for j := 1; j < n-1; j++ {
    if right[j+1] > nums[j] {
      val, _ := ts.Floor(nums[j] - 1)
      if val != nil {
        ans = max(ans, val.(int)-nums[j]+right[j+1])
      }
    }
    ts.Put(nums[j], nil)
  }
  return
}

function maximumTripletValue(nums: number[]): number {
  const n = nums.length;
  const right: number[] = Array(n).fill(nums[n - 1]);
  for (let i = n - 2; ~i; --i) {
    right[i] = Math.max(nums[i], right[i + 1]);
  }
  const ts = new TreeSet<number>();
  ts.add(nums[0]);
  let ans = 0;
  for (let j = 1; j < n - 1; ++j) {
    if (right[j + 1] > nums[j]) {
      const val = ts.lower(nums[j]);
      if (val !== undefined) {
        ans = Math.max(ans, val - nums[j] + right[j + 1]);
      }
    }
    ts.add(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);
  }

  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;
  }
}