2.3.5 Size Balanced Tree

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/*

Maintain an ordered map, that is, an ordered collection of key-value pairs such that each possible
key appears at most once in the collection. In addition, support queries for keys given their ranks
as well as queries for the ranks of given keys. A size balanced tree augments each node with the
size of its subtree, using it to maintain balance and compute order statistics. After each update,
rotations restore the invariant that every subtree is at least as large as each of its sibling's
child subtrees, keeping the height logarithmic.

The comparator `comp` defines the key ordering: `comp(a, b)` is true when `a` precedes `b`. It
defaults to `std::less<K>`; to customize the ordering, instantiate `SBTree<K, V, Compare>` and pass
the comparator to the constructor.

- `SBTree<K, V>()` constructs an empty map.
- `size()` returns the size of the map.
- `empty()` returns whether the map is empty.
- `insert(k, v)` adds an entry with key `k` and value `v` to the map, returning `true` if a new
  entry was added or `false` if the key already exists (in which case the map is unchanged and the
  old value associated with the key is preserved).
- `erase(k)` removes the entry with key `k` from the map, returning `true` if the removal was
  successful or `false` if the key to be removed was not found.
- `find(k)` returns a pointer to a const value associated with key `k`, or `nullptr` if the key was
  not found.
- `find_by_order(k)` returns a key-value pair of the node with a key of 0-based rank `k`, which must
  lie in the range $[0, `size()`)$.
- `order_of_key(x)` returns the number of keys that precede `x` in comparator order. The key does
  not need to be present in the map.
- `entries()` returns all key-value entries in comparator order.

The comparator-aware navigation routines `min()`, `max()`, `lower_bound(k)`, `upper_bound(k)`,
`prev(k)`, and `next(k)` from the treap in 2.3.1 depend only on the BST property and may be adapted
here as needed. For contest use on GNU C++ judges, PBDS ordered trees provide the same
order-statistic operations with much less code; see 8.6. This implementation is useful when PBDS is
unavailable or when customizing tree internals.

The order-statistic API matches GNU PBDS naming and 0-based rank conventions.

Time Complexity:
- O(1) per call to the constructor, `size()`, and `empty()`.
- O(log n) per call to `insert()`, `erase()`, `find()`, `find_by_order()`, and `order_of_key()`,
  where $n$ is the number of entries currently in the map.
- O(n) per call to `entries()`.

Space Complexity:
- O(n) for storage of the map elements.
- O(log n) auxiliary stack space for `insert()`, `erase()`, `find_by_order()`, `order_of_key()`,
  `entries()`, and destruction.
- O(n) for the vector returned by `entries()`.
- O(1) auxiliary for all other operations.

*/

#include <cassert>
#include <functional>
#include <utility>
#include <vector>

template<typename K, typename V, typename Compare = std::less<K>>
class SBTree {
  struct Node {
    K key;
    V value;
    int size;
    Node *left, *right;

    Node(const K &k, const V &v) : key(k), value(v), size(1), left(nullptr), right(nullptr) {}

    inline Node *&child(int c) { return (c == 0) ? left : right; }

    void update() {
      size = 1;
      if (left != nullptr) {
        size += left->size;
      }
      if (right != nullptr) {
        size += right->size;
      }
    }
  } *root;

  Compare comp;

  static inline int size(Node *n) { return (n == nullptr) ? 0 : n->size; }

  static void rotate(Node *&n, int c) {
    Node *tmp = n->child(c);
    n->child(c) = tmp->child(!c);
    tmp->child(!c) = n;
    n->update();
    tmp->update();
    n = tmp;
  }

  static void maintain(Node *&n, int c) {
    if (n == nullptr || n->child(c) == nullptr) {
      return;
    }
    Node *&tmp = n->child(c);
    if (size(tmp->child(c)) > size(n->child(!c))) {
      rotate(n, c);
    } else if (size(tmp->child(!c)) > size(n->child(!c))) {
      rotate(tmp, !c);
      rotate(n, c);
    } else {
      return;
    }
    maintain(n->left, 0);
    maintain(n->right, 1);
    maintain(n, 0);
    maintain(n, 1);
  }

  bool insert(Node *&n, const K &k, const V &v) {
    if (n == nullptr) {
      n = new Node(k, v);
      return true;
    }
    bool found;
    if (comp(k, n->key)) {
      found = insert(n->left, k, v);
      maintain(n, 0);
    } else if (comp(n->key, k)) {
      found = insert(n->right, k, v);
      maintain(n, 1);
    } else {
      return false;
    }
    n->update();
    return found;
  }

  bool erase(Node *&n, const K &k) {
    if (n == nullptr) {
      return false;
    }
    bool found;
    int c = comp(k, n->key);
    if (comp(k, n->key)) {
      found = erase(n->left, k);
    } else if (comp(n->key, k)) {
      found = erase(n->right, k);
    } else {
      if (n->right == nullptr || n->left == nullptr) {
        Node *tmp = n;
        n = (n->right == nullptr) ? n->left : n->right;
        delete tmp;
        return true;
      }
      Node *p = n->right;
      while (p->left != nullptr) {
        p = p->left;
      }
      K successor_key = p->key;
      n->key = successor_key;
      n->value = p->value;
      found = erase(n->right, successor_key);
    }
    maintain(n, c);
    n->update();
    return found;
  }

  static std::pair<K, V> find_by_order(Node *n, int k) {
    int left_size = size(n->left);
    if (k < left_size) {
      return find_by_order(n->left, k);
    } else if (k > left_size) {
      return find_by_order(n->right, k - left_size - 1);
    }
    return {n->key, n->value};
  }

  int order_of_key(Node *n, const K &x) const {
    if (n == nullptr) {
      return 0;
    }
    if (comp(x, n->key)) {
      return order_of_key(n->left, x);
    } else if (comp(n->key, x)) {
      return order_of_key(n->right, x) + size(n->left) + 1;
    }
    return size(n->left);
  }

  static void collect_entries(Node *n, std::vector<std::pair<K, V>> &res) {
    if (n != nullptr) {
      collect_entries(n->left, res);
      res.emplace_back(n->key, n->value);
      collect_entries(n->right, res);
    }
  }

  static void clean_up(Node *n) {
    if (n != nullptr) {
      clean_up(n->left);
      clean_up(n->right);
      delete n;
    }
  }

 public:
  explicit SBTree(Compare comp = Compare{}) : root(nullptr), comp(std::move(comp)) {}

  ~SBTree() { clean_up(root); }
  SBTree(const SBTree &) = delete;
  SBTree &operator=(const SBTree &) = delete;
  int size() const { return size(root); }
  bool empty() const { return root == nullptr; }
  bool insert(const K &k, const V &v) { return insert(root, k, v); }
  bool erase(const K &k) { return erase(root, k); }
  int order_of_key(const K &x) const { return order_of_key(root, x); }

  const V *find(const K &k) const {
    Node *n = root;
    while (n != nullptr) {
      if (comp(k, n->key)) {
        n = n->left;
      } else if (comp(n->key, k)) {
        n = n->right;
      } else {
        return &(n->value);
      }
    }
    return nullptr;
  }

  std::pair<K, V> find_by_order(int k) const {
    assert(0 <= k && k < size(root));
    return find_by_order(root, k);
  }

  std::vector<std::pair<K, V>> entries() const {
    std::vector<std::pair<K, V>> res;
    res.reserve(size(root));
    collect_entries(root, res);
    return res;
  }
};

/*** Example Usage ***/

using namespace std;

int main() {
  SBTree<int, char> t;
  t.insert(2, 'b');
  t.insert(1, 'a');
  t.insert(3, 'c');
  t.insert(5, 'e');
  assert(t.insert(4, 'd'));
  assert(*t.find(4) == 'd');
  assert(!t.insert(4, 'd'));
  assert(
      (t.entries() == vector<pair<int, char>>{{1, 'a'}, {2, 'b'}, {3, 'c'}, {4, 'd'}, {5, 'e'}})
  );
  assert(t.erase(1));
  assert(!t.erase(1));
  assert(t.find(1) == nullptr);
  assert((t.entries() == vector<pair<int, char>>{{2, 'b'}, {3, 'c'}, {4, 'd'}, {5, 'e'}}));
  assert(t.order_of_key(2) == 0);
  assert(t.order_of_key(3) == 1);
  assert(t.order_of_key(5) == 3);
  assert(t.order_of_key(6) == 4);
  assert(t.find_by_order(0).first == 2);
  assert(t.find_by_order(1).first == 3);
  assert(t.find_by_order(2).first == 4);

  SBTree<int, char> replacement;
  replacement.insert(2, 'b');
  replacement.insert(1, 'a');
  replacement.insert(3, 'c');
  assert(replacement.erase(2));
  assert(*replacement.find(3) == 'c');

  SBTree<int, char, greater<int>> descending;
  for (int key : {2, 1, 3}) {
    descending.insert(key, '0' + key);
  }
  assert((descending.entries() == vector<pair<int, char>>{{3, '3'}, {2, '2'}, {1, '1'}}));
  assert(descending.order_of_key(2) == 1);
  assert(descending.find_by_order(0).first == 3);
  assert(descending.erase(2) && descending.find(2) == nullptr);
  return 0;
}