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/**
* AVL tree, a Binary Search Tree that satisfies the Height-Balance
* Property.
*
* @example
* var avlTree = require('path-to-algorithms/src/data-structures'+
* '/avl-tree');
* var avl = new avlTree.AVLTree();
*
* avl.insert(2000);
* avl.insert(1989);
* avl.insert(1991);
* avl.insert(2001);
* avl.insert(1966);
*
* @module data-structures/avl-tree
*/
(function (exports) {
'use strict';
/**
* Node of the tree.
*
* @public
* @constructor
* @param {Number|String} value Value of the node.
* @param {Node} left Left sibling.
* @param {Node} right Right sibling.
* @param {Node} parent Parent of the node.
* @param {Number} height Height of the node.
*/
exports.Node = function (value, left, right, parent, height) {
/**
* @member {Number|String}
*/
this.value = value;
this._left = left;
this._right = right;
this._parent = parent;
this._height = height;
};
/**
* AVL Tree.
*
* @public
* @constructor
*/
exports.AVLTree = function () {
this._root = null;
};
/**
* Calculates the height of a node based on height
* property of children.
*
* @public
* @method
* @param {Node} node Given node's height is returned.
*/
exports.AVLTree.prototype._getHeightAtNode = function (node) {
if (node._left !== null && node._right !== null){
var height = Math.max(node._left._height, node._right._height);
height += 1;
return height;
} else if (node._left !== null){
return node._left._height + 1;
} else if (node._right !== null){
return node._right._height + 1;
} else {
return 1;
}
};
/**
* Checks if a given node has an imbalance.
*
* @public
* @method
* @param {Node} node Given node's children are checked for
* imbalance.
*/
exports.AVLTree.prototype._isBalancedAtNode = function (node) {
if (node._left !== null && node._right !== null){
return (Math.abs(node._left._height - node._right._height) <= 1);
}
if (node._right !== null && node._left === null){
return node._right._height < 2;
}
if (node._left !== null && node._right === null){
return node._left._height < 2;
}
return true;
};
/**
* Gets the nodes to be restructured during an AVL restructure
* after a remove/delete takes place.
*
* @public
* @method
* @param {Array} traveledNodes Array of previously traveled nodes
* that are used to help determine the nodes to be restructured.
*/
exports.AVLTree.prototype._getNodesToRestructureRemove = function (traveledNodes) {
// z is last traveled node - imbalance found at z
var zIndex = traveledNodes.length;
zIndex -= 1;
var z = traveledNodes[zIndex];
// y should be child of z with larger height
// (cannot be ancestor of removed node)
var y;
if ((z._left !== null && z._right !== null) || (z._left !== null && z._right === null)){
y = z._left;
} else if (z._right !== null && z._left === null){
y = z._right;
}
// x should be tallest child of y.
// If children same height, x should be child of y
// that has same orientation as z to y.
var x;
if (y._left !== null && y._right !== null){
if (y._left._height > y._right._height){
x = y._left;
} else if (y._left._height < y._right._height){
x = y._right;
} else if (y._left._height === y._right._height){
x = (z._left === y) ? y._left : y._right;
}
} else if (y._left !== null && y._right === null){
x = y._left;
} else if (y._right !== null && y._left === null){
x = y._right;
}
return [x, y, z];
};
/**
* Gets the nodes to be restructured during an AVL restructure
* after an insert takes place.
*
* @public
* @method
* @param {Array} traveledNodes Array of previously traveled nodes
* that are used to help determine the nodes to be restructured.
*/
exports.AVLTree.prototype._getNodesToRestructureInsert = function (traveledNodes) {
// z is last traveled node - imbalance found at z
var zIndex = traveledNodes.length;
zIndex -= 1;
var z = traveledNodes[zIndex];
// y should be child of z with larger height
// (must be ancestor of inserted node)
// therefore, last traveled node is correct.
var yIndex = traveledNodes.length;
yIndex -= 2;
var y = traveledNodes[yIndex];
// x should be tallest child of y.
// If children same height, x should be ancestor
// of inserted node (in traveled path).
var x;
if (y._left !== null && y._right !== null){
if (y._left._height > y._right._height){
x = y._left;
} else if (y._left._height < y._right._height){
x = y._right;
} else if (y._left._height === y._right._height){
var xIndex = traveledNodes.length;
xIndex -= 3;
x = traveledNodes[xIndex];
}
} else if (y._left !== null && y._right === null){
x = y._left;
} else if (y._right !== null && y._left === null){
x = y._right;
}
return [x, y, z];
};
/**
* Maintains the height balance property by
* walking to root and checking for invalid height
* differences between children and restructuring
* appropriately.
*
* @public
* @method
* @param {Node} node Started node.
* @param {Boolean} isRemove Represents if method was called after remove.
*/
exports.AVLTree.prototype._maintainHeightBalanceProperty = function (node, isRemove) {
var current = node;
var traveledNodes = [];
while (current !== null){
traveledNodes.push(current);
current._height = this._getHeightAtNode(current);
if (!this._isBalancedAtNode(current)){
var nodesToBeRestructured = (isRemove)
? this._getNodesToRestructureRemove(traveledNodes)
: this._getNodesToRestructureInsert(traveledNodes);
this._restructure(nodesToBeRestructured);
}
current = current._parent;
}
};
/**
* Identifies the pattern of given nodes, then calls
* the appropriate pattern rotator.
*
* @public
* @method
* @param {Array} nodesToBeRestructured
* array of nodes, in format, [x, y, z], to be restructured
*/
exports.AVLTree.prototype._restructure = function (nodesToBeRestructured) {
var x = nodesToBeRestructured[0];
var y = nodesToBeRestructured[1];
var z = nodesToBeRestructured[2];
//Determine Rotation Pattern
if (z._right === y && y._right === x){
this._rightRight(x, y, z);
} else if (z._left === y && y._left === x){
this._leftLeft(x, y, z);
} else if (z._right === y && y._left === x){
this._rightLeft(x, y, z);
} else if (z._left === y && y._right === x){
this._leftRight(x, y, z);
}
};
/**
* Rotates the given nodes from a right right pattern
* to a parent, with 2 children.
*
* @public
* @method
* @param {Node} x node with lowest height to be restructured.
* @param {Node} y parent of x parameter.
* @param {Node} z grandparent of x, largest height.
*/
exports.AVLTree.prototype._rightRight = function (x, y, z) {
/*
z
y => y
x z x
*/
// pass z parent to y and move y's left to z's right
if (z._parent !== null){
var orientation = (z._parent._left === z) ? '_left' : '_right';
z._parent[orientation] = y;
y._parent = z._parent;
} else {
this._root = y;
y._parent = null;
}
// z adopts y's left.
z._right = y._left;
if (z._right !== null){
z._right._parent = z;
}
// y adopts z
y._left = z;
z._parent = y;
// Correct each nodes height - order matters, children first
x._height = this._getHeightAtNode(x);
z._height = this._getHeightAtNode(z);
y._height = this._getHeightAtNode(y);
};
/**
* Rotates the given nodes from a left left pattern
* to a parent, with 2 children.
*
* @public
* @method
* @param {Node} x node with lowest height to be restructured.
* @param {Node} y parent of x parameter.
* @param {Node} z grandparent of x, largest height.
*/
exports.AVLTree.prototype._leftLeft = function (x, y, z) {
/*
z
y => y
x x z
*/
//pass z parent to y and move y's right to z's left
if (z._parent !== null){
var orientation = (z._parent._left === z) ? '_left' : '_right';
z._parent[orientation] = y;
y._parent = z._parent;
} else {
this._root = y;
y._parent = null;
}
z._left = y._right;
if (z._left !== null) {
z._left._parent = z;
}
//fix y right child
y._right = z;
z._parent = y;
// Correct each nodes height - order matters, children first
x._height = this._getHeightAtNode(x);
z._height = this._getHeightAtNode(z);
y._height = this._getHeightAtNode(y);
};
/**
* Rotates the given nodes from a right left pattern
* to a parent, with 2 children.
*
* @public
* @method
* @param {Node} x node with lowest height to be restructured.
* @param {Node} y parent of x parameter.
* @param {Node} z grandparent of x, largest height.
*/
exports.AVLTree.prototype._rightLeft = function (x, y, z) {
/*
z
y => x
x z y
*/
//pass z parent to x
if (z._parent !== null){
var orientation = (z._parent._left === z) ? '_left' : '_right';
z._parent[orientation] = x;
x._parent = z._parent;
} else {
this._root = x;
x._parent = null;
}
// Adoptions
z._right = x._left;
if (z._right !== null){
z._right._parent = z;
}
y._left = x._right;
if (y._left !== null){
y._left._parent = y;
}
// Point to new children (x new parent)
x._left = z;
x._right = y;
x._left._parent = x;
x._right._parent = x;
// Correct each nodes height - order matters, children first
y._height = this._getHeightAtNode(y);
z._height = this._getHeightAtNode(z);
x._height = this._getHeightAtNode(x);
};
/**
* Rotates the given nodes from a left right pattern
* to a parent, with 2 children.
*
* @public
* @method
* @param {Node} x node with lowest height to be restructured.
* @param {Node} y parent of x parameter.
* @param {Node} z grandparent of x, largest height.
*/
exports.AVLTree.prototype._leftRight = function (x, y, z) {
/*
z
y => x
x y z
*/
//pass z parent to x
if (z._parent !== null){
var orientation = (z._parent._left === z) ? '_left' : '_right';
z._parent[orientation] = x;
x._parent = z._parent;
} else {
this._root = x;
x._parent = null;
}
// Adoptions
z._left = x._right;
if (z._left !== null){
z._left._parent = z;
}
y._right = x._left;
if (y._right !== null){
y._right._parent = y;
}
// Point to new children (x new parent)
x._right = z;
x._left = y;
x._left._parent = x;
x._right._parent = x;
// Correct each nodes height - order matters, children first
y._height = this._getHeightAtNode(y);
z._height = this._getHeightAtNode(z);
x._height = this._getHeightAtNode(x);
};
/**
* Inserts a node into the AVL Tree.<br><br>
* Time complexity: O(log N) in the average case
* and O(N) in the worst case.
*
* @public
* @method
* @param {Number|String} value Node value.
* @param {Node} current Current node.
*/
exports.AVLTree.prototype.insert = function (value, current) {
if (this._root === null) {
this._root = new exports.Node(value, null, null, null, 1);
this._maintainHeightBalanceProperty(this._root);
return;
}
var insertKey;
current = current || this._root;
if (current.value > value) {
insertKey = '_left';
} else {
insertKey = '_right';
}
if (!current[insertKey]) {
current[insertKey] = new exports.Node(value, null, null, current);
this._maintainHeightBalanceProperty(current[insertKey], false);
} else {
this.insert(value, current[insertKey]);
}
};
/**
* In-order traversal from the given node.
*
* @private
* @param {Node} current Node from which to start the traversal.
* @param {Function} callback Callback which
* will be called for each traversed node.
*/
exports.AVLTree.prototype._inorder = function (current, callback) {
if (!current) {
return;
}
this._inorder(current._left, callback);
if (typeof callback === 'function') {
callback(current);
}
this._inorder(current._right, callback);
};
/**
* In-order traversal of the whole AVL tree.
*
* @public
* @method
* @param {Function} callback Callback which will be
* called for each traversed node.
*/
exports.AVLTree.prototype.inorder = function (callback) {
return this._inorder(this._root, callback);
};
/**
* Post-order traversal from given node.
*
* @private
* @param {Node} current Node from which to start the traversal.
* @param {Function} callback Callback which will
* be called for each traversed node
*/
exports.AVLTree.prototype._postorder = function (current, callback) {
if (!current) {
return;
}
if (typeof callback === 'function') {
callback(current);
}
this._postorder(current._left, callback);
this._postorder(current._right, callback);
};
/**
* Post-order traversal of the whole tree.
*
* @public
* @param {Function} callback Callback which
* will be called for each traversed node.
*/
exports.AVLTree.prototype.postorder = function (callback) {
return this._postorder(this._root, callback);
};
/**
* Pre-order traversal of the tree from given node.
*
* @private
* @param {Node} current Node from which to start the traversal.
* @param {Function} callback Callback which
* will be called for each traversed node.
*/
exports.AVLTree.prototype._preorder = function (current, callback) {
if (!current) {
return;
}
if (typeof callback === 'function') {
callback(current);
}
this._preorder(current._left, callback);
this._preorder(current._right, callback);
};
/**
* Pre-order preorder traversal of the whole tree.
*
* @public
* @param {Function} callback Callback which will
* be called for each traversed node.
*/
exports.AVLTree.prototype.preorder = function (callback) {
return this._preorder(this._root, callback);
};
/**
* Finds a node by it's value.<br><br>
* Average time complexity: O(log N).
*
* @public
* @param {Number|String} value of the node which should be found.
*/
exports.AVLTree.prototype.find = function (value) {
return this._find(value, this._root);
};
/**
* Finds a node by it's value in a given sub-tree.
* Average time complexity: O(log N).
*
* @private
* @param {Number|String} value of the node which should be found.
* @param {Node} current node to be checked.
*/
exports.AVLTree.prototype._find = function (value, current) {
if (!current) {
return null;
}
if (current.value === value) {
return current;
}
if (current.value > value) {
return this._find(value, current._left);
}
if (current.value < value) {
return this._find(value, current._right);
}
};
/**
* Replaces given child with new one, for given parent.
*
* @private
* @param {Node} parent Parent node.
* @param {Node} oldChild Child to be replaced.
* @param {Node} newChild Child replacement.
*/
exports.AVLTree.prototype._replaceChild = function (parent, oldChild, newChild) {
if (parent === null) {
this._root = newChild;
if (this._root !== null){
this._root._parent = null;
}
} else {
if (parent._left === oldChild) {
parent._left = newChild;
} else {
parent._right = newChild;
}
if (newChild) {
newChild._parent = parent;
}
}
};
/**
* Removes node from the tree. <br><br>
* Average runtime complexity: O(log N).
*
* @public
* @param {Number|String} value of node to be removed
* @returns {Boolean} True/false depending
* on whether the given node is removed.
*/
exports.AVLTree.prototype.remove = function (value) {
var node = this.find(value);
if (!node) {
return false;
}
if (node._left && node._right) {
var min = this._findMin(node._right);
var temp = node.value;
node.value = min.value;
min.value = temp;
return this.remove(temp);
} else {
if (node._left) {
this._replaceChild(node._parent, node, node._left);
this._maintainHeightBalanceProperty(node._left, true);
} else if (node._right) {
this._replaceChild(node._parent, node, node._right);
this._maintainHeightBalanceProperty(node._right, true);
} else {
this._replaceChild(node._parent, node, null);
this._maintainHeightBalanceProperty(node._parent, true);
}
return true;
}
};
/**
* Finds the node with minimum value in given sub-tree.
*
* @private
* @param {Node} node Root of the sub-tree.
* @param {Number|String} current Current minimum value of the sub-tree.
* @returns {Node} Node with the minimum value in the sub-tree.
*/
exports.AVLTree.prototype._findMin = function (node, current) {
current = current || { value: Infinity };
if (!node) {
return current;
}
if (current.value > node.value) {
current = node;
}
return this._findMin(node._left, current);
};
/**
* Finds the node with maximum value in given sub-tree.
*
* @private
* @param {Node} node Root of the sub-tree.
* @param {Number|String} current Current maximum value of the sub-tree.
* @returns {Node} Node with the maximum value in the sub-tree.
*/
exports.AVLTree.prototype._findMax = function (node, current) {
current = current || { value: -Infinity };
if (!node) {
return current;
}
if (current.value < node.value) {
current = node;
}
return this._findMax(node._right, current);
};
/**
* Finds the node with minimum value in the whole tree.
*
* @public
* @returns {Node} The minimum node of the tree.
*/
exports.AVLTree.prototype.findMin = function () {
return this._findMin(this._root);
};
/**
* Finds the node with maximum value in the whole tree.
*
* @public
* @returns {Node} The maximum node of the tree.
*
*/
exports.AVLTree.prototype.findMax = function () {
return this._findMax(this._root);
};
exports.AVLTree.prototype._isBalanced = function (current) {
if (!current) {
return true;
}
return this._isBalanced(current._left) &&
this._isBalanced(current._right) &&
Math.abs(this._getHeight(current._left) -
this._getHeight(current._right)) <= 1;
};
/**
* Returns whether the AVL Tree is balanced.
*
* @public
* @returns {Boolean} Whether the tree is balanced or not.
*/
exports.AVLTree.prototype.isBalanced = function () {
return this._isBalanced(this._root);
};
/**
* Finds the diameter of the AVL tree.
*
* @public
* @returns {Number} The longest path in the AVL Tree.
*/
exports.AVLTree.prototype.getDiameter = function () {
var getDiameter = function (root) {
if (!root) {
return 0;
}
var leftHeight = this._getHeight(root._left);
var rightHeight = this._getHeight(root._right);
var path = leftHeight + rightHeight + 1;
return Math.max(path, getDiameter(root._left), getDiameter(root._right));
}.bind(this);
return getDiameter(this._root);
};
/**
* Returns the height of the tree.
*
* @public
* @returns {Number} The height of the tree.
*/
exports.AVLTree.prototype.getTreeHeight = function () {
return this._getHeight(this._root);
};
exports.AVLTree.prototype._getHeight = function (node) {
if (!node) {
return 0;
}
return 1 + Math.max(this._getHeight(node._left),
this._getHeight(node._right));
};
/**
* Finds the lowest common ancestor of two nodes.
*
* @public
* @returns {Node} The lowest common ancestor of the two nodes or null.
*/
exports.AVLTree.prototype.lowestCommonAncestor = function (firstNode, secondNode) {
return this._lowestCommonAncestor(firstNode, secondNode, this._root);
};
exports.AVLTree.prototype._lowestCommonAncestor = function (firstNode, secondNode, current) {
var firstNodeInLeft = this._existsInSubtree(firstNode, current._left);
var secondNodeInLeft = this._existsInSubtree(secondNode, current._left);
var firstNodeInRight = this._existsInSubtree(firstNode, current._right);
var secondNodeInRight = this._existsInSubtree(secondNode, current._right);
if ((firstNodeInLeft && secondNodeInRight) ||
(firstNodeInRight && secondNodeInLeft)) {
return current;
}
if (secondNodeInLeft && firstNodeInLeft) {
return this._lowestCommonAncestor(firstNode, secondNode, current._left);
}
if (secondNodeInRight && secondNodeInLeft) {
return this._lowestCommonAncestor(firstNode, secondNode, current._right);
}
return null;
};
exports.AVLTree.prototype._existsInSubtree = function (node, root) {
if (!root) {
return false;
}
if (node === root.value) {
return true;
}
return this._existsInSubtree(node, root._left) ||
this._existsInSubtree(node, root._right);
};
})(typeof window === 'undefined' ? module.exports : window);