Question

1878. Get Biggest Three Rhombus Sums in a Grid

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Description

You are given an m x n integer matrix grid​​​.

A rhombus sum is the sum of the elements that form the border of a regular rhombus shape in grid​​​. The rhombus must have the shape of a square rotated 45 degrees with each of the corners centered in a grid cell. Below is an image of four valid rhombus shapes with the corresponding colored cells that should be included in each rhombus sum:

Note that the rhombus can have an area of 0, which is depicted by the purple rhombus in the bottom right corner.

Return _the biggest three distinct rhombus sums in the _grid_ in descending order__. If there are less than three distinct values, return all of them_.

 

Example 1:

Input: grid = [[3,4,5,1,3],[3,3,4,2,3],[20,30,200,40,10],[1,5,5,4,1],[4,3,2,2,5]]
Output: [228,216,211]
Explanation: The rhombus shapes for the three biggest distinct rhombus sums are depicted above.
- Blue: 20 + 3 + 200 + 5 = 228
- Red: 200 + 2 + 10 + 4 = 216
- Green: 5 + 200 + 4 + 2 = 211

Example 2:

Input: grid = [[1,2,3],[4,5,6],[7,8,9]]
Output: [20,9,8]
Explanation: The rhombus shapes for the three biggest distinct rhombus sums are depicted above.
- Blue: 4 + 2 + 6 + 8 = 20
- Red: 9 (area 0 rhombus in the bottom right corner)
- Green: 8 (area 0 rhombus in the bottom middle)

Example 3:

Input: grid = [[7,7,7]]
Output: [7]
Explanation: All three possible rhombus sums are the same, so return [7].

 

Constraints:

• m == grid.length
• n == grid[i].length
• 1 <= m, n <= 50
• 1 <= grid[i][j] <= 105

Solutions

Solution 1: Enumerate Diamond Center + Prefix Sum + Ordered Set

We can preprocess to get two prefix sum arrays $s_1$ and $s_2$, where $s_1[i][j]$ represents the sum of the elements on the upper left diagonal ending at $(i, j)$, and $s_2[i][j]$ represents the sum of the elements on the upper right diagonal ending at $(i, j)$.

Next, we enumerate each position $(i, j)$, first add $grid[i][j]$ to the ordered set $ss$, and then enumerate the length $k$ of the diamond. The sum of the diamond with $(i, j)$ as the center and a side length of $k$ is:

$$ \begin{aligned} &\quad s_1[i + k][j] - s_1[i][j - k] + s_1[i][j + k] - s_1[i - k][j] \ &+ s_2[i][j - k] - s_2[i - k][j] + s_2[i + k][j] - s_2[i][j + k] \ &- grid[i + k - 1][j - 1] + grid[i - k - 1][j - 1] \end{aligned} $$

We add this value to the ordered set $ss$, while ensuring that the size of the ordered set $ss$ does not exceed $3$. Finally, we output the elements in the ordered set $ss$ in reverse order.

The time complexity is $O(m \times n \times \min(m, n))$, and the space complexity is $O(m \times n)$. Here, $m$ and $n$ are the number of rows and columns of the matrix, respectively.

from sortedcontainers import SortedSet


class Solution:
  def getBiggestThree(self, grid: List[List[int]]) -> List[int]:
    m, n = len(grid), len(grid[0])
    s1 = [[0] * (n + 2) for _ in range(m + 1)]
    s2 = [[0] * (n + 2) for _ in range(m + 1)]
    for i, row in enumerate(grid, 1):
      for j, x in enumerate(row, 1):
        s1[i][j] = s1[i - 1][j - 1] + x
        s2[i][j] = s2[i - 1][j + 1] + x
    ss = SortedSet()
    for i, row in enumerate(grid, 1):
      for j, x in enumerate(row, 1):
        l = min(i - 1, m - i, j - 1, n - j)
        ss.add(x)
        for k in range(1, l + 1):
          a = s1[i + k][j] - s1[i][j - k]
          b = s1[i][j + k] - s1[i - k][j]
          c = s2[i][j - k] - s2[i - k][j]
          d = s2[i + k][j] - s2[i][j + k]
          ss.add(
            a + b + c + d - grid[i + k - 1][j - 1] + grid[i - k - 1][j - 1]
          )
        while len(ss) > 3:
          ss.remove(ss[0])
    return list(ss)[::-1]

class Solution {
  public int[] getBiggestThree(int[][] grid) {
    int m = grid.length, n = grid[0].length;
    int[][] s1 = new int[m + 1][n + 2];
    int[][] s2 = new int[m + 1][n + 2];
    for (int i = 1; i <= m; ++i) {
      for (int j = 1; j <= n; ++j) {
        s1[i][j] = s1[i - 1][j - 1] + grid[i - 1][j - 1];
        s2[i][j] = s2[i - 1][j + 1] + grid[i - 1][j - 1];
      }
    }
    TreeSet<Integer> ss = new TreeSet<>();
    for (int i = 1; i <= m; ++i) {
      for (int j = 1; j <= n; ++j) {
        int l = Math.min(Math.min(i - 1, m - i), Math.min(j - 1, n - j));
        ss.add(grid[i - 1][j - 1]);
        for (int k = 1; k <= l; ++k) {
          int a = s1[i + k][j] - s1[i][j - k];
          int b = s1[i][j + k] - s1[i - k][j];
          int c = s2[i][j - k] - s2[i - k][j];
          int d = s2[i + k][j] - s2[i][j + k];
          ss.add(a + b + c + d - grid[i + k - 1][j - 1] + grid[i - k - 1][j - 1]);
        }
        while (ss.size() > 3) {
          ss.pollFirst();
        }
      }
    }
    int[] ans = new int[ss.size()];
    for (int i = 0; i < ans.length; ++i) {
      ans[i] = ss.pollLast();
    }
    return ans;
  }
}

class Solution {
public:
  vector<int> getBiggestThree(vector<vector<int>>& grid) {
    int m = grid.size(), n = grid[0].size();
    vector<vector<int>> s1(m + 1, vector<int>(n + 2));
    vector<vector<int>> s2(m + 1, vector<int>(n + 2));
    for (int i = 1; i <= m; ++i) {
      for (int j = 1; j <= n; ++j) {
        s1[i][j] = s1[i - 1][j - 1] + grid[i - 1][j - 1];
        s2[i][j] = s2[i - 1][j + 1] + grid[i - 1][j - 1];
      }
    }
    set<int> ss;
    for (int i = 1; i <= m; ++i) {
      for (int j = 1; j <= n; ++j) {
        int l = min({i - 1, m - i, j - 1, n - j});
        ss.insert(grid[i - 1][j - 1]);
        for (int k = 1; k <= l; ++k) {
          int a = s1[i + k][j] - s1[i][j - k];
          int b = s1[i][j + k] - s1[i - k][j];
          int c = s2[i][j - k] - s2[i - k][j];
          int d = s2[i + k][j] - s2[i][j + k];
          ss.insert(a + b + c + d - grid[i + k - 1][j - 1] + grid[i - k - 1][j - 1]);
        }
        while (ss.size() > 3) {
          ss.erase(ss.begin());
        }
      }
    }
    return vector<int>(ss.rbegin(), ss.rend());
  }
};

func getBiggestThree(grid [][]int) []int {
  m, n := len(grid), len(grid[0])
  s1 := make([][]int, m+1)
  s2 := make([][]int, m+1)
  for i := range s1 {
    s1[i] = make([]int, n+2)
    s2[i] = make([]int, n+2)
  }
  for i := 1; i <= m; i++ {
    for j := 1; j <= n; j++ {
      s1[i][j] = s1[i-1][j-1] + grid[i-1][j-1]
      s2[i][j] = s2[i-1][j+1] + grid[i-1][j-1]
    }
  }

  ss := treemap.NewWithIntComparator()
  for i := 1; i <= m; i++ {
    for j := 1; j <= n; j++ {
      l := min(i-1, m-i, j-1, n-j)
      ss.Put(grid[i-1][j-1], nil)
      for k := 1; k <= l; k++ {
        a := s1[i+k][j] - s1[i][j-k]
        b := s1[i][j+k] - s1[i-k][j]
        c := s2[i][j-k] - s2[i-k][j]
        d := s2[i+k][j] - s2[i][j+k]
        ss.Put(a+b+c+d-grid[i+k-1][j-1]+grid[i-k-1][j-1], nil)
      }
      for ss.Size() > 3 {
        x, _ := ss.Min()
        ss.Remove(x.(int))
      }
    }
  }
  ans := make([]int, ss.Size())
  for i, k := range ss.Keys() {
    ans[len(ans)-i-1] = k.(int)
  }
  return ans
}

function getBiggestThree(grid: number[][]): number[] {
  const m = grid.length;
  const n = grid[0].length;
  const s1: number[][] = Array.from({ length: m + 1 }, () => Array(n + 2).fill(0));
  const s2: number[][] = Array.from({ length: m + 1 }, () => Array(n + 2).fill(0));
  for (let i = 1; i <= m; ++i) {
    for (let j = 1; j <= n; ++j) {
      s1[i][j] = s1[i - 1][j - 1] + grid[i - 1][j - 1];
      s2[i][j] = s2[i - 1][j + 1] + grid[i - 1][j - 1];
    }
  }
  const ss = new TreeSet<number>();
  for (let i = 1; i <= m; ++i) {
    for (let j = 1; j <= n; ++j) {
      const l = Math.min(i - 1, m - i, j - 1, n - j);
      ss.add(grid[i - 1][j - 1]);
      for (let k = 1; k <= l; ++k) {
        const a = s1[i + k][j] - s1[i][j - k];
        const b = s1[i][j + k] - s1[i - k][j];
        const c = s2[i][j - k] - s2[i - k][j];
        const d = s2[i + k][j] - s2[i][j + k];
        ss.add(a + b + c + d - grid[i + k - 1][j - 1] + grid[i - k - 1][j - 1]);
      }
      while (ss.size() > 3) {
        ss.shift();
      }
    }
  }
  return [...ss].reverse();
}

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;
  }
}