splay_tree.dart

// Copyright (c) 2012, the Dart project authors.  Please see the AUTHORS file
// for details. All rights reserved. Use of this source code is governed by a
// BSD-style license that can be found in the LICENSE file.

part of dart.collection;

typedef _Predicate<T> = bool Function(T value);

/// A node in a splay tree. It holds the sorting key and the left
/// and right children in the tree.
class _SplayTreeNode<K, Node extends _SplayTreeNode<K, Node>> {
  final K key;

  Node? left;
  Node? right;

  _SplayTreeNode(this.key);
}

/// A node in a splay tree based set.
class _SplayTreeSetNode<K> extends _SplayTreeNode<K, _SplayTreeSetNode<K>> {
  _SplayTreeSetNode(K key) : super(key);
}

/// A node in a splay tree based map.
///
/// A [_SplayTreeNode] that also contains a value
class _SplayTreeMapNode<K, V>
    extends _SplayTreeNode<K, _SplayTreeMapNode<K, V>> {
  V value;
  _SplayTreeMapNode(K key, this.value) : super(key);
}

/// A splay tree is a self-balancing binary search tree.
///
/// It has the additional property that recently accessed elements
/// are quick to access again.
/// It performs basic operations such as insertion, look-up and
/// removal, in O(log(n)) amortized time.
abstract class _SplayTree<K, Node extends _SplayTreeNode<K, Node>> {
  // The root node of the splay tree. It will contain either the last
  // element inserted or the last element looked up.
  Node? get _root;
  set _root(Node? newValue);

  // Number of elements in the splay tree.
  int _count = 0;

  /// Counter incremented whenever the keys in the map changes.
  ///
  /// Used to detect concurrent modifications.
  int _modificationCount = 0;

  /// Counter incremented whenever the tree structure changes.
  ///
  /// Used to detect that an in-place traversal cannot use
  /// cached information that relies on the tree structure.
  int _splayCount = 0;

  /// The comparator that is used for this splay tree.
  Comparator<K> get _compare;

  /// The predicate to determine that a given object is a valid key.
  _Predicate get _validKey;

  /// Perform the splay operation for the given key. Moves the node with
  /// the given key to the top of the tree.  If no node has the given
  /// key, the last node on the search path is moved to the top of the
  /// tree. This is the simplified top-down splaying algorithm from:
  /// "Self-adjusting Binary Search Trees" by Sleator and Tarjan.
  ///
  /// Returns the result of comparing the new root of the tree to [key].
  /// Returns -1 if the table is empty.
  int _splay(K key) {
    if (_root == null) return -1;

    // The right and newTreeRight variables start out null, and are set
    // after the first move left.  The right node is the destination
    // for subsequent left rebalances, and newTreeRight holds the left
    // child of the final tree.  The newTreeRight variable is set at most
    // once, after the first move left, and is null iff right is null.
    // The left and newTreeLeft variables play the corresponding role for
    // right rebalances.
    Node? right;
    Node? newTreeRight;
    Node? left;
    Node? newTreeLeft;
    var current = _root!;
    // Hoist the field read out of the loop.
    var compare = _compare;
    int comp;
    while (true) {
      comp = compare(current.key, key);
      if (comp > 0) {
        var currentLeft = current.left;
        if (currentLeft == null) break;
        comp = compare(currentLeft.key, key);
        if (comp > 0) {
          // Rotate right.
          current.left = currentLeft.right;
          currentLeft.right = current;
          current = currentLeft;
          currentLeft = current.left;
          if (currentLeft == null) break;
        }
        // Link right.
        if (right == null) {
          // First left rebalance, store the eventual right child
          newTreeRight = current;
        } else {
          right.left = current;
        }
        right = current;
        current = currentLeft;
      } else if (comp < 0) {
        var currentRight = current.right;
        if (currentRight == null) break;
        comp = compare(currentRight.key, key);
        if (comp < 0) {
          // Rotate left.
          current.right = currentRight.left;
          currentRight.left = current;
          current = currentRight;
          currentRight = current.right;
          if (currentRight == null) break;
        }
        // Link left.
        if (left == null) {
          // First right rebalance, store the eventual left child
          newTreeLeft = current;
        } else {
          left.right = current;
        }
        left = current;
        current = currentRight;
      } else {
        break;
      }
    }
    // Assemble.
    if (left != null) {
      left.right = current.left;
      current.left = newTreeLeft;
    }
    if (right != null) {
      right.left = current.right;
      current.right = newTreeRight;
    }
    _root = current;

    _splayCount++;
    return comp;
  }

  // Emulates splaying with a key that is smaller than any in the subtree
  // anchored at [node].
  // and that node is returned. It should replace the reference to [node]
  // in any parent tree or root pointer.
  Node _splayMin(Node node) {
    var current = node;
    var nextLeft = current.left;
    while (nextLeft != null) {
      var left = nextLeft;
      current.left = left.right;
      left.right = current;
      current = left;
      nextLeft = current.left;
    }
    return current;
  }

  // Emulates splaying with a key that is greater than any in the subtree
  // anchored at [node].
  // After this, the largest element in the tree is the root of the subtree,
  // and that node is returned. It should replace the reference to [node]
  // in any parent tree or root pointer.
  Node _splayMax(Node node) {
    var current = node;
    var nextRight = current.right;
    while (nextRight != null) {
      var right = nextRight;
      current.right = right.left;
      right.left = current;
      current = right;
      nextRight = current.right;
    }
    return current;
  }

  Node? _remove(K key) {
    if (_root == null) return null;
    int comp = _splay(key);
    if (comp != 0) return null;
    var root = _root!;
    var result = root;
    var left = root.left;
    _count--;
    // assert(_count >= 0);
    if (left == null) {
      _root = root.right;
    } else {
      var right = root.right;
      // Splay to make sure that the new root has an empty right child.
      root = _splayMax(left);

      // Insert the original right child as the right child of the new
      // root.
      root.right = right;
      _root = root;
    }
    _modificationCount++;
    return result;
  }

  /// Adds a new root node with the given [key] or [value].
  ///
  /// The [comp] value is the result of comparing the existing root's key
  /// with key.
  void _addNewRoot(Node node, int comp) {
    _count++;
    _modificationCount++;
    var root = _root;
    if (root == null) {
      _root = node;
      return;
    }
    // assert(_count >= 0);
    if (comp < 0) {
      node.left = root;
      node.right = root.right;
      root.right = null;
    } else {
      node.right = root;
      node.left = root.left;
      root.left = null;
    }
    _root = node;
  }

  Node? get _first {
    var root = _root;
    if (root == null) return null;
    _root = _splayMin(root);
    return _root;
  }

  Node? get _last {
    var root = _root;
    if (root == null) return null;
    _root = _splayMax(root);
    return _root;
  }

  void _clear() {
    _root = null;
    _count = 0;
    _modificationCount++;
  }
}

int _dynamicCompare(dynamic a, dynamic b) => Comparable.compare(a, b);

Comparator<K> _defaultCompare<K>() {
  // If K <: Comparable, then we can just use Comparable.compare
  // with no casts.
  Object compare = Comparable.compare;
  if (compare is Comparator<K>) {
    return compare;
  }
  // Otherwise wrap and cast the arguments on each call.
  return _dynamicCompare;
}

/// A [Map] of objects that can be ordered relative to each other.
///
/// The map is based on a self-balancing binary tree. It allows most operations
/// in amortized logarithmic time.
///
/// Keys of the map are compared using the `compare` function passed in
/// the constructor, both for ordering and for equality.
/// If the map contains only the key `a`, then `map.containsKey(b)`
/// will return `true` if and only if `compare(a, b) == 0`,
/// and the value of `a == b` is not even checked.
/// If the compare function is omitted, the objects are assumed to be
/// [Comparable], and are compared using their [Comparable.compareTo] method.
/// Non-comparable objects (including `null`) will not work as keys
/// in that case.
///
/// To allow calling [operator []], [remove] or [containsKey] with objects
/// that are not supported by the `compare` function, an extra `isValidKey`
/// predicate function can be supplied. This function is tested before
/// using the `compare` function on an argument value that may not be a [K]
/// value. If omitted, the `isValidKey` function defaults to testing if the
/// value is a [K].
class SplayTreeMap<K, V> extends _SplayTree<K, _SplayTreeMapNode<K, V>>
    with MapMixin<K, V> {
  _SplayTreeMapNode<K, V>? _root;

  Comparator<K> _compare;
  _Predicate _validKey;

  SplayTreeMap(
      [int Function(K key1, K key2)? compare,
      bool Function(dynamic potentialKey)? isValidKey])
      : _compare = compare ?? _defaultCompare<K>(),
        _validKey = isValidKey ?? ((dynamic v) => v is K);

  /// Creates a [SplayTreeMap] that contains all key/value pairs of [other].
  ///
  /// The keys must all be instances of [K] and the values of [V].
  /// The [other] map itself can have any type.
  factory SplayTreeMap.from(Map<dynamic, dynamic> other,
      [int Function(K key1, K key2)? compare,
      bool Function(dynamic potentialKey)? isValidKey]) {
    if (other is Map<K, V>) {
      return SplayTreeMap<K, V>.of(other, compare, isValidKey);
    }
    SplayTreeMap<K, V> result = SplayTreeMap<K, V>(compare, isValidKey);
    other.forEach((dynamic k, dynamic v) {
      result[k] = v;
    });
    return result;
  }

  /// Creates a [SplayTreeMap] that contains all key/value pairs of [other].
  factory SplayTreeMap.of(Map<K, V> other,
          [int Function(K key1, K key2)? compare,
          bool Function(dynamic potentialKey)? isValidKey]) =>
      SplayTreeMap<K, V>(compare, isValidKey)..addAll(other);

  /// Creates a [SplayTreeMap] where the keys and values are computed from the
  /// [iterable].
  ///
  /// For each element of the [iterable] this constructor computes a key/value
  /// pair, by applying [key] and [value] respectively.
  ///
  /// The keys of the key/value pairs do not need to be unique. The last
  /// occurrence of a key will simply overwrite any previous value.
  ///
  /// If no functions are specified for [key] and [value] the default is to
  /// use the iterable value itself.
  factory SplayTreeMap.fromIterable(Iterable iterable,
      {K Function(dynamic element)? key,
      V Function(dynamic element)? value,
      int Function(K key1, K key2)? compare,
      bool Function(dynamic potentialKey)? isValidKey}) {
    SplayTreeMap<K, V> map = SplayTreeMap<K, V>(compare, isValidKey);
    MapBase._fillMapWithMappedIterable(map, iterable, key, value);
    return map;
  }

  /// Creates a [SplayTreeMap] associating the given [keys] to [values].
  ///
  /// This constructor iterates over [keys] and [values] and maps each element
  /// of [keys] to the corresponding element of [values].
  ///
  /// If [keys] contains the same object multiple times, the last occurrence
  /// overwrites the previous value.
  ///
  /// It is an error if the two [Iterable]s don't have the same length.
  factory SplayTreeMap.fromIterables(Iterable<K> keys, Iterable<V> values,
      [int Function(K key1, K key2)? compare,
      bool Function(dynamic potentialKey)? isValidKey]) {
    SplayTreeMap<K, V> map = SplayTreeMap<K, V>(compare, isValidKey);
    MapBase._fillMapWithIterables(map, keys, values);
    return map;
  }

  V? operator [](Object? key) {
    if (!_validKey(key)) return null;
    if (_root != null) {
      int comp = _splay(key as dynamic);
      if (comp == 0) {
        return _root!.value;
      }
    }
    return null;
  }

  V? remove(Object? key) {
    if (!_validKey(key)) return null;
    _SplayTreeMapNode<K, V>? mapRoot = _remove(key as dynamic);
    if (mapRoot != null) return mapRoot.value;
    return null;
  }

  void operator []=(K key, V value) {
    if (key == null) throw ArgumentError(key);
    // Splay on the key to move the last node on the search path for
    // the key to the root of the tree.
    int comp = _splay(key);
    if (comp == 0) {
      _root!.value = value;
      return;
    }
    _addNewRoot(_SplayTreeMapNode(key, value), comp);
  }

  V putIfAbsent(K key, V ifAbsent()) {
    if (key == null) throw ArgumentError(key);
    int comp = _splay(key);
    if (comp == 0) {
      return _root!.value;
    }
    int modificationCount = _modificationCount;
    int splayCount = _splayCount;
    V value = ifAbsent();
    if (modificationCount != _modificationCount) {
      throw ConcurrentModificationError(this);
    }
    if (splayCount != _splayCount) {
      comp = _splay(key);
      // Key is still not there, otherwise _modificationCount would be changed.
      assert(comp != 0);
    }
    _addNewRoot(_SplayTreeMapNode(key, value), comp);
    return value;
  }

  void addAll(Map<K, V> other) {
    other.forEach((K key, V value) {
      this[key] = value;
    });
  }

  bool get isEmpty {
    return (_root == null);
  }

  bool get isNotEmpty => !isEmpty;

  void forEach(void f(K key, V value)) {
    Iterator<_SplayTreeMapNode<K, V>> nodes =
        _SplayTreeNodeIterator<K, _SplayTreeMapNode<K, V>>(this);
    while (nodes.moveNext()) {
      _SplayTreeMapNode<K, V> node = nodes.current;
      f(node.key, node.value);
    }
  }

  int get length {
    return _count;
  }

  void clear() {
    _clear();
  }

  bool containsKey(Object? key) {
    return _validKey(key) && _splay(key as dynamic) == 0;
  }

  bool containsValue(Object? value) {
    int initialSplayCount = _splayCount;
    bool visit(_SplayTreeMapNode<K, V>? node) {
      while (node != null) {
        if (node.value == value) return true;
        if (initialSplayCount != _splayCount) {
          throw ConcurrentModificationError(this);
        }
        if (node.right != null && visit(node.right)) {
          return true;
        }
        node = node.left;
      }
      return false;
    }

    return visit(_root);
  }

  Iterable<K> get keys =>
      _SplayTreeKeyIterable<K, _SplayTreeMapNode<K, V>>(this);

  Iterable<V> get values => _SplayTreeValueIterable<K, V>(this);

  /// Get the first key in the map. Returns `null` if the map is empty.
  K? firstKey() {
    if (_root == null) return null;
    return _first!.key;
  }

  /// Get the last key in the map. Returns `null` if the map is empty.
  K? lastKey() {
    if (_root == null) return null;
    return _last!.key;
  }

  /// Get the last key in the map that is strictly smaller than [key]. Returns
  /// `null` if no key was not found.
  K? lastKeyBefore(K key) {
    if (key == null) throw ArgumentError(key);
    if (_root == null) return null;
    int comp = _splay(key);
    if (comp < 0) return _root!.key;
    _SplayTreeMapNode<K, V>? node = _root!.left;
    if (node == null) return null;
    var nodeRight = node.right;
    while (nodeRight != null) {
      node = nodeRight;
      nodeRight = node.right;
    }
    return node!.key;
  }

  /// Get the first key in the map that is strictly larger than [key]. Returns
  /// `null` if no key was not found.
  K? firstKeyAfter(K key) {
    if (key == null) throw ArgumentError(key);
    if (_root == null) return null;
    int comp = _splay(key);
    if (comp > 0) return _root!.key;
    _SplayTreeMapNode<K, V>? node = _root!.right;
    if (node == null) return null;
    var nodeLeft = node.left;
    while (nodeLeft != null) {
      node = nodeLeft;
      nodeLeft = node.left;
    }
    return node!.key;
  }
}

abstract class _SplayTreeIterator<K, Node extends _SplayTreeNode<K, Node>, T>
    implements Iterator<T> {
  final _SplayTree<K, Node> _tree;

  /// Worklist of nodes to visit.
  ///
  /// These nodes have been passed over on the way down in a
  /// depth-first left-to-right traversal. Visiting each node,
  /// and their right subtrees will visit the remainder of
  /// the nodes of a full traversal.
  ///
  /// Only valid as long as the original tree isn't reordered.
  final List<Node> _workList = [];

  /// Original modification counter of [_tree].
  ///
  /// Incremented on [_tree] when a key is added or removed.
  /// If it changes, iteration is aborted.
  ///
  /// Not final because some iterators may modify the tree knowingly,
  /// and they update the modification count in that case.
  int _modificationCount;

  /// Count of splay operations on [_tree] when [_workList] was built.
  ///
  /// If the splay count on [_tree] increases, [_workList] becomes invalid.
  int _splayCount;

  /// Current node.
  Node? _currentNode;

  _SplayTreeIterator(_SplayTree<K, Node> tree)
      : _tree = tree,
        _modificationCount = tree._modificationCount,
        _splayCount = tree._splayCount {
    _findLeftMostDescendent(tree._root);
  }

  _SplayTreeIterator.startAt(_SplayTree<K, Node> tree, K startKey)
      : _tree = tree,
        _modificationCount = tree._modificationCount,
        _splayCount = -1 {
    if (tree._root == null) return;
    int compare = tree._splay(startKey);
    _splayCount = tree._splayCount;
    if (compare < 0) {
      // Don't include the root, start at the next element after the root.
      _findLeftMostDescendent(tree._root!.right);
    } else {
      _workList.add(tree._root!);
    }
  }

  T get current {
    var node = _currentNode;
    if (node == null) return null as T;
    return _getValue(node);
  }

  void _findLeftMostDescendent(Node? node) {
    while (node != null) {
      _workList.add(node);
      node = node.left;
    }
  }

  /// Called when the tree structure of the tree has changed.
  ///
  /// This can be caused by a splay operation.
  /// If the key-set changes, iteration is aborted before getting
  /// here, so we know that the keys are the same as before, it's
  /// only the tree that has been reordered.
  void _rebuildWorkList(Node currentNode) {
    assert(_workList.isNotEmpty);
    _workList.clear();
    _tree._splay(currentNode.key);
    _findLeftMostDescendent(_tree._root!.right);
    assert(_workList.isNotEmpty);
  }

  bool moveNext() {
    if (_modificationCount != _tree._modificationCount) {
      throw ConcurrentModificationError(_tree);
    }
    // Picks the next element in the worklist as current.
    // Updates the worklist with the left-most path of the current node's
    // right-hand child.
    // If the worklist is no longer valid (after a splay), it is rebuild
    // from scratch.
    if (_workList.isEmpty) {
      _currentNode = null;
      return false;
    }
    if (_tree._splayCount != _splayCount && _currentNode != null) {
      _rebuildWorkList(_currentNode!);
    }
    _currentNode = _workList.removeLast();
    _findLeftMostDescendent(_currentNode!.right);
    return true;
  }

  T _getValue(Node node);
}

class _SplayTreeKeyIterable<K, Node extends _SplayTreeNode<K, Node>>
    extends EfficientLengthIterable<K> {
  _SplayTree<K, Node> _tree;
  _SplayTreeKeyIterable(this._tree);
  int get length => _tree._count;
  bool get isEmpty => _tree._count == 0;
  Iterator<K> get iterator => _SplayTreeKeyIterator<K, Node>(_tree);

  Set<K> toSet() {
    SplayTreeSet<K> set = SplayTreeSet<K>(_tree._compare, _tree._validKey);
    set._count = _tree._count;
    set._root = set._copyNode<Node>(_tree._root);
    return set;
  }
}

class _SplayTreeValueIterable<K, V> extends EfficientLengthIterable<V> {
  SplayTreeMap<K, V> _map;
  _SplayTreeValueIterable(this._map);
  int get length => _map._count;
  bool get isEmpty => _map._count == 0;
  Iterator<V> get iterator => _SplayTreeValueIterator<K, V>(_map);
}

class _SplayTreeKeyIterator<K, Node extends _SplayTreeNode<K, Node>>
    extends _SplayTreeIterator<K, Node, K> {
  _SplayTreeKeyIterator(_SplayTree<K, Node> map) : super(map);
  K _getValue(Node node) => node.key;
}

class _SplayTreeValueIterator<K, V>
    extends _SplayTreeIterator<K, _SplayTreeMapNode<K, V>, V> {
  _SplayTreeValueIterator(SplayTreeMap<K, V> map) : super(map);
  V _getValue(_SplayTreeMapNode<K, V> node) => node.value;
}

class _SplayTreeNodeIterator<K, Node extends _SplayTreeNode<K, Node>>
    extends _SplayTreeIterator<K, Node, Node> {
  _SplayTreeNodeIterator(_SplayTree<K, Node> tree) : super(tree);
  _SplayTreeNodeIterator.startAt(_SplayTree<K, Node> tree, K startKey)
      : super.startAt(tree, startKey);
  Node _getValue(Node node) => node;
}

/// A [Set] of objects that can be ordered relative to each other.
///
/// The set is based on a self-balancing binary tree. It allows most operations
/// in amortized logarithmic time.
///
/// Elements of the set are compared using the `compare` function passed in
/// the constructor, both for ordering and for equality.
/// If the set contains only an object `a`, then `set.contains(b)`
/// will return `true` if and only if `compare(a, b) == 0`,
/// and the value of `a == b` is not even checked.
/// If the compare function is omitted, the objects are assumed to be
/// [Comparable], and are compared using their [Comparable.compareTo] method.
/// Non-comparable objects (including `null`) will not work as an element
/// in that case.
class SplayTreeSet<E> extends _SplayTree<E, _SplayTreeSetNode<E>>
    with IterableMixin<E>, SetMixin<E> {
  _SplayTreeSetNode<E>? _root;

  Comparator<E> _compare;
  _Predicate _validKey;

  /// Create a new [SplayTreeSet] with the given compare function.
  ///
  /// If the [compare] function is omitted, it defaults to [Comparable.compare],
  /// and the elements must be comparable.
  ///
  /// A provided `compare` function may not work on all objects. It may not even
  /// work on all `E` instances.
  ///
  /// For operations that add elements to the set, the user is supposed to not
  /// pass in objects that doesn't work with the compare function.
  ///
  /// The methods [contains], [remove], [lookup], [removeAll] or [retainAll]
  /// are typed to accept any object(s), and the [isValidKey] test can used to
  /// filter those objects before handing them to the `compare` function.
  ///
  /// If [isValidKey] is provided, only values satisfying `isValidKey(other)`
  /// are compared using the `compare` method in the methods mentioned above.
  /// If the `isValidKey` function returns false for an object, it is assumed to
  /// not be in the set.
  ///
  /// If omitted, the `isValidKey` function defaults to checking against the
  /// type parameter: `other is E`.
  SplayTreeSet(
      [int Function(E key1, E key2)? compare,
      bool Function(dynamic potentialKey)? isValidKey])
      : _compare = compare ?? _defaultCompare<E>(),
        _validKey = isValidKey ?? ((dynamic v) => v is E);

  /// Creates a [SplayTreeSet] that contains all [elements].
  ///
  /// The set works as if created by `new SplayTreeSet<E>(compare, isValidKey)`.
  ///
  /// All the [elements] should be instances of [E] and valid arguments to
  /// [compare].
  /// The `elements` iterable itself may have any element type, so this
  /// constructor can be used to down-cast a `Set`, for example as:
  /// ```dart
  /// Set<SuperType> superSet = ...;
  /// Set<SubType> subSet =
  ///     new SplayTreeSet<SubType>.from(superSet.whereType<SubType>());
  /// ```
  factory SplayTreeSet.from(Iterable elements,
      [int Function(E key1, E key2)? compare,
      bool Function(dynamic potentialKey)? isValidKey]) {
    if (elements is Iterable<E>) {
      return SplayTreeSet<E>.of(elements, compare, isValidKey);
    }
    SplayTreeSet<E> result = SplayTreeSet<E>(compare, isValidKey);
    for (var element in elements) {
      result.add(element as dynamic);
    }
    return result;
  }

  /// Creates a [SplayTreeSet] from [elements].
  ///
  /// The set works as if created by `new SplayTreeSet<E>(compare, isValidKey)`.
  ///
  /// All the [elements] should be valid as arguments to the [compare] function.
  factory SplayTreeSet.of(Iterable<E> elements,
          [int Function(E key1, E key2)? compare,
          bool Function(dynamic potentialKey)? isValidKey]) =>
      SplayTreeSet(compare, isValidKey)..addAll(elements);

  Set<T> _newSet<T>() =>
      SplayTreeSet<T>((T a, T b) => _compare(a as E, b as E), _validKey);

  Set<R> cast<R>() => Set.castFrom<E, R>(this, newSet: _newSet);

  // From Iterable.

  Iterator<E> get iterator =>
      _SplayTreeKeyIterator<E, _SplayTreeSetNode<E>>(this);

  int get length => _count;
  bool get isEmpty => _root == null;
  bool get isNotEmpty => _root != null;

  E get first {
    if (_count == 0) throw IterableElementError.noElement();
    return _first!.key;
  }

  E get last {
    if (_count == 0) throw IterableElementError.noElement();
    return _last!.key;
  }

  E get single {
    if (_count == 0) throw IterableElementError.noElement();
    if (_count > 1) throw IterableElementError.tooMany();
    return _root!.key;
  }

  // From Set.
  bool contains(Object? element) {
    return _validKey(element) && _splay(element as E) == 0;
  }

  bool add(E element) {
    int compare = _splay(element);
    if (compare == 0) return false;
    _addNewRoot(_SplayTreeSetNode(element), compare);
    return true;
  }

  bool remove(Object? object) {
    if (!_validKey(object)) return false;
    return _remove(object as E) != null;
  }

  void addAll(Iterable<E> elements) {
    for (E element in elements) {
      int compare = _splay(element);
      if (compare != 0) {
        _addNewRoot(_SplayTreeSetNode(element), compare);
      }
    }
  }

  void removeAll(Iterable<Object?> elements) {
    for (Object? element in elements) {
      if (_validKey(element)) _remove(element as E);
    }
  }

  void retainAll(Iterable<Object?> elements) {
    // Build a set with the same sense of equality as this set.
    SplayTreeSet<E> retainSet = SplayTreeSet<E>(_compare, _validKey);
    int modificationCount = _modificationCount;
    for (Object? object in elements) {
      if (modificationCount != _modificationCount) {
        // The iterator should not have side effects.
        throw ConcurrentModificationError(this);
      }
      // Equivalent to this.contains(object).
      if (_validKey(object) && _splay(object as E) == 0) {
        retainSet.add(_root!.key);
      }
    }
    // Take over the elements from the retained set, if it differs.
    if (retainSet._count != _count) {
      _root = retainSet._root;
      _count = retainSet._count;
      _modificationCount++;
    }
  }

  E? lookup(Object? object) {
    if (!_validKey(object)) return null;
    int comp = _splay(object as E);
    if (comp != 0) return null;
    return _root!.key;
  }

  Set<E> intersection(Set<Object?> other) {
    Set<E> result = SplayTreeSet<E>(_compare, _validKey);
    for (E element in this) {
      if (other.contains(element)) result.add(element);
    }
    return result;
  }

  Set<E> difference(Set<Object?> other) {
    Set<E> result = SplayTreeSet<E>(_compare, _validKey);
    for (E element in this) {
      if (!other.contains(element)) result.add(element);
    }
    return result;
  }

  Set<E> union(Set<E> other) {
    return _clone()..addAll(other);
  }

  SplayTreeSet<E> _clone() {
    var set = SplayTreeSet<E>(_compare, _validKey);
    set._count = _count;
    set._root = _copyNode<_SplayTreeSetNode<E>>(_root);
    return set;
  }

  // Copies the structure of a SplayTree into a new similar structure.
  // Works on _SplayTreeMapNode as well, but only copies the keys,
  _SplayTreeSetNode<E>? _copyNode<Node extends _SplayTreeNode<E, Node>>(
      Node? node) {
    if (node == null) return null;
    // Given a source node and a destination node, copy the left
    // and right subtrees of the source node into the destination node.
    // The left subtree is copied recursively, but the right spine
    // of every subtree is copied iteratively.
    void copyChildren(Node node, _SplayTreeSetNode<E> dest) {
      Node? left;
      Node? right;
      do {
        left = node.left;
        right = node.right;
        if (left != null) {
          var newLeft = _SplayTreeSetNode<E>(left.key);
          dest.left = newLeft;
          // Recursively copy the left tree.
          copyChildren(left, newLeft);
        }
        if (right != null) {
          var newRight = _SplayTreeSetNode<E>(right.key);
          dest.right = newRight;
          // Set node and dest to copy the right tree iteratively.
          node = right;
          dest = newRight;
        }
      } while (right != null);
    }

    var result = _SplayTreeSetNode<E>(node.key);
    copyChildren(node, result);
    return result;
  }

  void clear() {
    _clear();
  }

  Set<E> toSet() => _clone();

  String toString() => IterableBase.iterableToFullString(this, '{', '}');
}