// ----------------------------------------------------------------------
// dyoo: the following code comes from the Google Closure Library.  I've done
// edits to flatten the namespace from goog.structs to just
// AvlTree, commented out inorderTraverse and reverseOrderTraverse.
//
// ----------------------------------------------------------------------
// Original license follows:
// ----------------------------------------------------------------------


// Copyright 2007 The Closure Library Authors. All Rights Reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//      http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS-IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

/**
 * @fileoverview Datastructure: AvlTree.
 *
 *
 * This file provides the implementation of an AVL-Tree datastructure. The tree
 * maintains a set of unique values in a sorted order. The values can be
 * accessed efficiently in their sorted order since the tree enforces an O(logn)
 * maximum height. See http://en.wikipedia.org/wiki/Avl_tree for more detail.
 *
 * The big-O notation for all operations are below:
 * <pre>
 *   Method                 big-O
 * ----------------------------------------------------------------------------
 * - add                    O(logn)
 * - remove                 O(logn)
 * - clear                  O(1)
 * - contains               O(logn)
 * - getCount               O(1)
 * - getMinimum             O(1), or O(logn) when optional root is specified
 * - getMaximum             O(1), or O(logn) when optional root is specified
 * - getHeight              O(1)
 * - getValues              O(n)
 * - inOrderTraverse        O(logn + k), where k is number of traversed nodes
 * - reverseOrderTraverse   O(logn + k), where k is number of traversed nodes
 * </pre>
 */



/**
 * Constructs an AVL-Tree, which uses the specified comparator to order its
 * values. The values can be accessed efficiently in their sorted order since
 * the tree enforces a O(logn) maximum height.
 *
 * @param {Function=} opt_comparator Function used to order the tree's nodes.
 * @constructor
 * @implements {Collection}
 */
var AvlTree = function(opt_comparator) {
    this.comparator_ = opt_comparator ||
        AvlTree.DEFAULT_COMPARATOR_;
};


/**
 * String comparison function used to compare values in the tree. This function
 * is used by default if no comparator is specified in the tree's constructor.
 *
 * @param {string} a The first string.
 * @param {string} b The second string.
 * @return {number} -1 if a < b, 1 if a > b, 0 if a = b.
 * @private
 */
AvlTree.DEFAULT_COMPARATOR_ = function(a, b) {
    if (String(a) < String(b)) {
        return -1;
    } else if (String(a) > String(b)) {
        return 1;
    }
    return 0;
};


/**
 * Pointer to the root node of the tree.
 *
 * @type {AvlTree.Node}
 * @private
 */
AvlTree.prototype.root_ = null;


/**
 * Comparison function used to compare values in the tree. This function should
 * take two values, a and b, and return x where:
 * <pre>
 *  x < 0 if a < b,
 *  x > 0 if a > b,
 *  x = 0 otherwise
 * </pre>
 *
 * @type {Function}
 * @private
 */
AvlTree.prototype.comparator_ = null;


/**
 * Pointer to the node with the smallest value in the tree.
 *
 * @type {AvlTree.Node}
 * @private
 */
AvlTree.prototype.minNode_ = null;


/**
 * Pointer to the node with the largest value in the tree.
 *
 * @type {AvlTree.Node}
 * @private
 */
AvlTree.prototype.maxNode_ = null;


/**
 * Keeps track of the number of nodes in the tree.
 *
 * @type {number}
 * @private
 */
AvlTree.prototype.count_ = 0;


/**
 * Inserts a node into the tree with the specified value if the tree does
 * not already contain a node with the specified value. If the value is
 * inserted, the tree is balanced to enforce the AVL-Tree height property.
 *
 * @param {*} value Value to insert into the tree.
 * @return {boolean} Whether value was inserted into the tree.
 */
AvlTree.prototype.add = function(value) {
    // If the tree is empty, create a root node with the specified value
    if (this.root_ == null) {
        this.root_ = new AvlTree.Node(value);
        this.minNode_ = this.root_;
        this.maxNode_ = this.root_;
        this.count_ = 1;
        return true;
    }

    // Assume a node is not added and change status when one is
    var retStatus = false;

    // Depth traverse the tree and insert the value if we reach a null node
    this.traverse_(function(node) {
        var retNode = null;
        if (this.comparator_(node.value, value) > 0) {
            retNode = node.left;
            if (node.left == null) {
                var newNode = new AvlTree.Node(value, node);
                node.left = newNode;
                if (node == this.minNode_) {
                    this.minNode_ = newNode;
                }
                retStatus = true; // Value was added to tree
                this.balance_(node); // Maintain the AVL-tree balance
            }
        } else if (this.comparator_(node.value, value) < 0) {
            retNode = node.right;
            if (node.right == null) {
                var newNode = new AvlTree.Node(value, node);
                node.right = newNode;
                if (node == this.maxNode_) {
                    this.maxNode_ = newNode;
                }
                retStatus = true; // Value was added to tree
                this.balance_(node); // Maintain the AVL-tree balance
            }
        }
        return retNode; // If null, we'll stop traversing the tree
    });

    // If a node was added, increment count
    if (retStatus) {
        this.count_ += 1;
    }

    // Return true if a node was added, false otherwise
    return retStatus;
};


/**
 * Removes a node from the tree with the specified value if the tree contains a
 * node with this value. If a node is removed the tree is balanced to enforce
 * the AVL-Tree height property. The value of the removed node is returned.
 *
 * @param {*} value Value to find and remove from the tree.
 * @return {*} The value of the removed node or null if the value was not in
 *     the tree.
 */
AvlTree.prototype.remove = function(value) {
    // Assume the value is not removed and set the value when it is removed
    var retValue = null;

    // Depth traverse the tree and remove the value if we find it
    this.traverse_(function(node) {
        var retNode = null;
        if (this.comparator_(node.value, value) > 0) {
            retNode = node.left;
        } else if (this.comparator_(node.value, value) < 0) {
            retNode = node.right;
        } else {
            retValue = node.value;
            this.removeNode_(node);
        }
        return retNode; // If null, we'll stop traversing the tree
    });

    // If a node was removed, decrement count.
    if (retValue) {
        // Had traverse_() cleared the tree, set to 0.
        this.count_ = this.root_ ? this.count_ - 1 : 0;
    }

    // Return the value that was removed, null if the value was not in the tree
    return retValue;
};


/**
 * Removes all nodes from the tree.
 */
AvlTree.prototype.clear = function() {
    this.root_ = null;
    this.minNode_ = null;
    this.maxNode_ = null;
    this.count_ = 0;
};


/**
 * Returns true if the tree contains a node with the specified value, false
 * otherwise.
 *
 * @param {*} value Value to find in the tree.
 * @return {boolean} Whether the tree contains a node with the specified value.
 */
AvlTree.prototype.contains = function(value) {
    // Assume the value is not in the tree and set this value if it is found
    var isContained = false;

    // Depth traverse the tree and set isContained if we find the node
    this.traverse_(function(node) {
        var retNode = null;
        if (this.comparator_(node.value, value) > 0) {
            retNode = node.left;
        } else if (this.comparator_(node.value, value) < 0) {
            retNode = node.right;
        } else {
            isContained = true;
        }
        return retNode; // If null, we'll stop traversing the tree
    });

    // Return true if the value is contained in the tree, false otherwise
    return isContained;
};


/**
 * Returns the number of values stored in the tree.
 *
 * @return {number} The number of values stored in the tree.
 */
AvlTree.prototype.getCount = function() {
    return this.count_;
};


/**
 * Returns the value u, such that u is contained in the tree and u < v, for all
 * values v in the tree where v != u.
 *
 * @return {*} The minimum value contained in the tree.
 */
AvlTree.prototype.getMinimum = function() {
    return this.getMinNode_().value;
};


/**
 * Returns the value u, such that u is contained in the tree and u > v, for all
 * values v in the tree where v != u.
 *
 * @return {*} The maximum value contained in the tree.
 */
AvlTree.prototype.getMaximum = function() {
    return this.getMaxNode_().value;
};


/**
 * Returns the height of the tree (the maximum depth). This height should
 * always be <= 1.4405*(Math.log(n+2)/Math.log(2))-1.3277, where n is the
 * number of nodes in the tree.
 *
 * @return {number} The height of the tree.
 */
AvlTree.prototype.getHeight = function() {
    return this.root_ ? this.root_.height : 0;
};


/**
 * Inserts the values stored in the tree into a new Array and returns the Array.
 *
 * @return {Array} An array containing all of the trees values in sorted order.
 */
AvlTree.prototype.getValues = function() {
    var ret = [];
    this.inOrderTraverse(function(value) {
        ret.push(value);
    });
    return ret;
};


/**
 * Performs an in-order traversal of the tree and calls {@code func} with each
 * traversed node, optionally starting from the smallest node with a value >= to
 * the specified start value. The traversal ends after traversing the tree's
 * maximum node or when {@code func} returns a value that evaluates to true.
 *
 * @param {Function} func Function to call on each traversed node.
 * @param {Object=} opt_startValue If specified, traversal will begin on the
 *    node with the smallest value >= opt_startValue.
 */
AvlTree.prototype.inOrderTraverse =
    function(func, opt_startValue) {
        // If our tree is empty, return immediately
        if (!this.root_) {
            return;
        }

        // Depth traverse the tree to find node to begin in-order traversal from
        var startNode;
        if (opt_startValue) {
            this.traverse_(function(node) {
                var retNode = null;
                if (this.comparator_(node.value, opt_startValue) > 0) {
                    retNode = node.left;
                    startNode = node;
                } else if (this.comparator_(node.value, opt_startValue) < 0) {
                    retNode = node.right;
                } else {
                    startNode = node;
                }
                return retNode; // If null, we'll stop traversing the tree
            });
        } else {
            startNode = this.getMinNode_();
        }

        // Traverse the tree and call func on each traversed node's value
        var node = startNode, prev = startNode.left ? startNode.left : startNode;
        while (node != null) {
            if (node.left != null && node.left != prev && node.right != prev) {
                node = node.left;
            } else {
                if (node.right != prev) {
                    if (func(node.value)) {
                        return;
                    }
                }
                var temp = node;
                node = node.right != null && node.right != prev ?
                    node.right :
                    node.parent;
                prev = temp;
            }
        }
    };


// /**
//  * Performs a reverse-order traversal of the tree and calls {@code func} with
//  * each traversed node, optionally starting from the largest node with a value
//  * <= to the specified start value. The traversal ends after traversing the
//  * tree's minimum node or when func returns a value that evaluates to true.
//  *
//  * @param {Function} func Function to call on each traversed node.
//  * @param {Object=} opt_startValue If specified, traversal will begin on the
//  *    node with the largest value <= opt_startValue.
//  */
// AvlTree.prototype.reverseOrderTraverse =
//     function(func, opt_startValue) {
//   // If our tree is empty, return immediately
//   if (!this.root_) {
//     return;
//   }

//   // Depth traverse the tree to find node to begin reverse-order traversal from
//   var startNode;
//   if (opt_startValue) {
//     this.traverse_(goog.bind(function(node) {
//       var retNode = null;
//       if (this.comparator_(node.value, opt_startValue) > 0) {
//         retNode = node.left;
//       } else if (this.comparator_(node.value, opt_startValue) < 0) {
//         retNode = node.right;
//         startNode = node;
//       } else {
//         startNode = node;
//       }
//       return retNode; // If null, we'll stop traversing the tree
//     }, this));
//   } else {
//     startNode = this.getMaxNode_();
//   }

//   // Traverse the tree and call func on each traversed node's value
//   var node = startNode, prev = startNode.right ? startNode.right : startNode;
//   while (node != null) {
//     if (node.right != null && node.right != prev && node.left != prev) {
//       node = node.right;
//     } else {
//       if (node.left != prev) {
//         if (func(node.value)) {
//           return;
//         }
//       }
//       var temp = node;
//       node = node.left != null && node.left != prev ?
//              node.left :
//              node.parent;
//       prev = temp;
//     }
//   }
// };


/**
 * Performs a traversal defined by the supplied {@code traversalFunc}. The first
 * call to {@code traversalFunc} is passed the root or the optionally specified
 * startNode. After that, calls {@code traversalFunc} with the node returned
 * by the previous call to {@code traversalFunc} until {@code traversalFunc}
 * returns null or the optionally specified endNode. The first call to
 * traversalFunc is passed the root or the optionally specified startNode.
 *
 * @param {Function} traversalFunc Function used to traverse the tree. Takes a
 *     node as a parameter and returns a node.
 * @param {AvlTree.Node=} opt_startNode The node at which the
 *     traversal begins.
 * @param {AvlTree.Node=} opt_endNode The node at which the
 *     traversal ends.
 * @private
 */
AvlTree.prototype.traverse_ =
    function(traversalFunc, opt_startNode, opt_endNode) {
        var node = opt_startNode ? opt_startNode : this.root_;
        var endNode = opt_endNode ? opt_endNode : null;
        while (node && node != endNode) {
            node = traversalFunc.call(this, node);
        }
    };



/**
 * Ensures that the specified node and all its ancestors are balanced. If they
 * are not, performs left and right tree rotations to achieve a balanced
 * tree. This method assumes that at most 2 rotations are necessary to balance
 * the tree (which is true for AVL-trees that are balanced after each node is
 * added or removed).
 *
 * @param {AvlTree.Node} node Node to begin balance from.
 * @private
 */
AvlTree.prototype.balance_ = function(node) {

    this.traverse_(function(node) {
        // Calculate the left and right node's heights
        var lh = node.left ? node.left.height : 0;
        var rh = node.right ? node.right.height : 0;

        // Rotate tree rooted at this node if it is not AVL-tree balanced
        if (lh - rh > 1) {
            if (node.left.right && (!node.left.left ||
                                    node.left.left.height < node.left.right.height)) {
                this.leftRotate_(node.left);
            }
            this.rightRotate_(node);
        } else if (rh - lh > 1) {
            if (node.right.left && (!node.right.right ||
                                    node.right.right.height < node.right.left.height)) {
                this.rightRotate_(node.right);
            }
            this.leftRotate_(node);
        }

        // Recalculate the left and right node's heights
        lh = node.left ? node.left.height : 0;
        rh = node.right ? node.right.height : 0;

        // Set this node's height
        node.height = Math.max(lh, rh) + 1;

        // Traverse up tree and balance parent
        return node.parent;
    }, node);

};


/**
 * Performs a left tree rotation on the specified node.
 *
 * @param {AvlTree.Node} node Pivot node to rotate from.
 * @private
 */
AvlTree.prototype.leftRotate_ = function(node) {
    // Re-assign parent-child references for the parent of the node being removed
    if (node.isLeftChild()) {
        node.parent.left = node.right;
        node.right.parent = node.parent;
    } else if (node.isRightChild()) {
        node.parent.right = node.right;
        node.right.parent = node.parent;
    } else {
        this.root_ = node.right;
        this.root_.parent = null;
    }

    // Re-assign parent-child references for the child of the node being removed
    var temp = node.right;
    node.right = node.right.left;
    if (node.right != null) node.right.parent = node;
    temp.left = node;
    node.parent = temp;
};


/**
 * Performs a right tree rotation on the specified node.
 *
 * @param {AvlTree.Node} node Pivot node to rotate from.
 * @private
 */
AvlTree.prototype.rightRotate_ = function(node) {
    // Re-assign parent-child references for the parent of the node being removed
    if (node.isLeftChild()) {
        node.parent.left = node.left;
        node.left.parent = node.parent;
    } else if (node.isRightChild()) {
        node.parent.right = node.left;
        node.left.parent = node.parent;
    } else {
        this.root_ = node.left;
        this.root_.parent = null;
    }

    // Re-assign parent-child references for the child of the node being removed
    var temp = node.left;
    node.left = node.left.right;
    if (node.left != null) node.left.parent = node;
    temp.right = node;
    node.parent = temp;
};


/**
 * Removes the specified node from the tree and ensures the tree still
 * maintains the AVL-tree balance.
 *
 * @param {AvlTree.Node} node The node to be removed.
 * @private
 */
AvlTree.prototype.removeNode_ = function(node) {
    // Perform normal binary tree node removal, but balance the tree, starting
    // from where we removed the node
    if (node.left != null || node.right != null) {
        var b = null; // Node to begin balance from
        var r;        // Node to replace the node being removed
        if (node.left != null) {
            r = this.getMaxNode_(node.left);
            if (r != node.left) {
                r.parent.right = r.left;
                if (r.left) r.left.parent = r.parent;
                r.left = node.left;
                r.left.parent = r;
                b = r.parent;
            }
            r.parent = node.parent;
            r.right = node.right;
            if (r.right) r.right.parent = r;
            if (node == this.maxNode_) this.maxNode_ = r;
        } else {
            r = this.getMinNode_(node.right);
            if (r != node.right) {
                r.parent.left = r.right;
                if (r.right) r.right.parent = r.parent;
                r.right = node.right;
                r.right.parent = r;
                b = r.parent;
            }
            r.parent = node.parent;
            r.left = node.left;
            if (r.left) r.left.parent = r;
            if (node == this.minNode_) this.minNode_ = r;
        }

        // Update the parent of the node being removed to point to its replace
        if (node.isLeftChild()) {
            node.parent.left = r;
        } else if (node.isRightChild()) {
            node.parent.right = r;
        } else {
            this.root_ = r;
        }

        // Balance the tree
        this.balance_(b ? b : r);
    } else {
        // If the node is a leaf, remove it and balance starting from its parent
        if (node.isLeftChild()) {
            this.special = 1;
            node.parent.left = null;
            if (node == this.minNode_) this.minNode_ = node.parent;
            this.balance_(node.parent);
        } else if (node.isRightChild()) {
            node.parent.right = null;
            if (node == this.maxNode_) this.maxNode_ = node.parent;
            this.balance_(node.parent);
        } else {
            this.clear();
        }
    }
};


/**
 * Returns the node with the smallest value in tree, optionally rooted at
 * {@code opt_rootNode}.
 *
 * @param {AvlTree.Node=} opt_rootNode Optional root node.
 * @return {AvlTree.Node} The node with the smallest value in
 *     the tree.
 * @private
 */
AvlTree.prototype.getMinNode_ = function(opt_rootNode) {
    if (!opt_rootNode) {
        return this.minNode_;
    }

    var minNode = opt_rootNode;
    this.traverse_(function(node) {
        var retNode = null;
        if (node.left) {
            minNode = node.left;
            retNode = node.left;
        }
        return retNode; // If null, we'll stop traversing the tree
    }, opt_rootNode);

    return minNode;
};


/**
 * Returns the node with the largest value in tree, optionally rooted at
 * opt_rootNode.
 *
 * @param {AvlTree.Node=} opt_rootNode Optional root node.
 * @return {AvlTree.Node} The node with the largest value in
 *     the tree.
 * @private
 */
AvlTree.prototype.getMaxNode_ = function(opt_rootNode) {
    if (!opt_rootNode) {
        return this.maxNode_;
    }

    var maxNode = opt_rootNode;
    this.traverse_(function(node) {
        var retNode = null;
        if (node.right) {
            maxNode = node.right;
            retNode = node.right;
        }
        return retNode; // If null, we'll stop traversing the tree
    }, opt_rootNode);

    return maxNode;
};



/**
 * Constructs an AVL-Tree node with the specified value. If no parent is
 * specified, the node's parent is assumed to be null. The node's height
 * defaults to 1 and its children default to null.
 *
 * @param {*} value Value to store in the node.
 * @param {AvlTree.Node=} opt_parent Optional parent node.
 * @constructor
 */
AvlTree.Node = function(value, opt_parent) {
    /**
     * The value stored by the node.
     *
     * @type {*}
     */
    this.value = value;

    /**
     * The node's parent. Null if the node is the root.
     *
     * @type {AvlTree.Node}
     */
    this.parent = opt_parent ? opt_parent : null;
};


/**
 * The node's left child. Null if the node does not have a left child.
 *
 * @type {AvlTree.Node?}
 */
AvlTree.Node.prototype.left = null;


/**
 * The node's right child. Null if the node does not have a right child.
 *
 * @type {AvlTree.Node?}
 */
AvlTree.Node.prototype.right = null;


/**
 * The height of the tree rooted at this node.
 *
 * @type {number}
 */
AvlTree.Node.prototype.height = 1;


/**
 * Returns true iff the specified node has a parent and is the right child of
 * its parent.
 *
 * @return {boolean} Whether the specified node has a parent and is the right
 *    child of its parent.
 */
AvlTree.Node.prototype.isRightChild = function() {
    return !!this.parent && this.parent.right == this;
};


/**
 * Returns true iff the specified node has a parent and is the left child of
 * its parent.
 *
 * @return {boolean} Whether the specified node has a parent and is the left
 *    child of its parent.
 */
AvlTree.Node.prototype.isLeftChild = function() {
    return !!this.parent && this.parent.left == this;
};