# Red Black Tree in C++¶

## Invariants maintained by red-black trees¶

1. Each node has a flag indicating that its red or black
2. The root is always black
3. There can be no two consecutive red nodes, i.e., if a node is red, then its children must be black
4. Every path from the root to every null pointer, i.e. every unsuccessful search, passes through the same number of black nodes

## Implications of invariants¶

The four invariants above guarantee that a red-black tree’s height is always O(log(n)), in particular:

height <= 2 * log2(n + 1).

### Proof:¶

First note that any binary tree in which every root-NULL path has >= k nodes includes a perfectly balanced search tree of height k - 1, which imples the size of the tree must be at least >= 2k- 1 and thus k <= log2(n + 1).

By fourth invariant, thus a red black tree every root-null path has <= log2(n + 1) black nodes. By the third invariant, every root-null path has <= 2 * log2(n + 1) total nodes.

## Implementation of Left-Leaning Red Black Trees¶

The code is based on Robert Sedgwich’s talk Left Leaning Red Black Trees , in which he discusses how both 2 3 trees and 2 3 4 trees can be represented as left leaning (or right leaning) red-black trees. The C++ code below is a port of his java implementation. The invariant–checking methods in the java, used to verify that the tree is always as balanced as possible following insertions or deletions, have been removed to improve performance.

The relationship between 2 3 4 trees and red black trees, and how 2 3 4 trees map to red black tree, is also explained at http://cw.felk.cvut.cz/lib/exe/fetch.php/courses/a4m33pal/paska12x.pdf and http://www.cs.princeton.edu/courses/archive/spr07/cos226/lectures/balanced.pdf.

#ifndef REDBLACK_SDFSEWRSDPGSCP
#define REDBLACK_SDFSEWRSDPGSCP

#include <exception>
#include <memory>

class key_not_found :  public std::exception {
public:
virtual const char* what() const throw()
{
return "Key does not exist in tree";
}
};

template<typename Key, typename Value>  class rbtree {

private:
enum { BLACK = false, RED = true};

class  Node {
public:
Key   key;            // key
Value value;          // its associated data
std::unique_ptr<Node> left;           // left...
std::unique_ptr<Node> right;          // ...and right subtrees
bool color;           // color of parent link

Node(Key key, Value value)
{
this->key   = key;
this->value = value;
this->color = RED;
}
};

std::unique_ptr<Node> root;           // root of the BST

Value get(Node *p, Key key);

Node *getInOrderSuccessorNode(Node *p);

void DestroyTree(std::unique_ptr<Node>& root);

/*
* Returns minimum key of subtree rooted at p
*/
Key min(Node *p)
{
return (p->left == nullptr) ? p->key : min(p->left);
}

Key max(Node *p)
{
return (p->right == nullptr) ? p->key : max(p->right);
}

Node *insert(Node *p, Key key, Value value);

bool isRed(Node *p)
{
return (p == nullptr) ? false : (p->color == RED);
}

void colorFlip(Node *p)
{
p->color        = !p->color;
p->left->color  = !p->left->color;
p->right->color = !p->right->color;
}

Node *rotateLeft(Node *p);
Node *rotateRight(Node *p);

Node *moveRedLeft(Node *p);
Node *moveRedRight(Node *p);

Node *deleteMax(Node *p);
Node *deleteMin(Node *p);

Node *fixUp(Node *p);

Node *remove(std::unique_ptr<Node> &p, Key key);

template<typename Functor> void traverse(Functor f, Node *root);

public:
// The default unique_ptr constructor sets the uderlying pointer to nullptr

explicit rbtree() {  }

~rbtree() { DestroyTree(root); }

bool contains(Key key)
{  return get(key) != nullptr;  }

Value get(Key key)
{  return get(root, key);  }

void put(Key key, Value value)
{
root = insert(root, key, value);
root->color = BLACK;
}

template<typename Functor> void traverse(Functor f);

Key min()
{
return (root == nullptr) ? nullptr : min(root);
}

Key max()
{
return (root == nullptr) ? nullptr : max(root);
}

void deleteMin()
{
root = deleteMin(root);
root->color = BLACK;
}

void deleteMax()
{
root = deleteMax(root);
root->color = BLACK;
}

void remove(Key key)
{
if (root == nullptr) return;

root = remove(root, key);

if (root != nullptr) {

root->color = BLACK;
}
}
};

/*
*  Do post order traversal deleting underlying pointer.
*/
//--template<typename Key, typename Value> void rbtree<Key, Value>::DestroyTree(Node *current)
template<typename Key, typename Value> void rbtree<Key, Value>::DestroyTree(std::unique_ptr<Node>& current)
{
if (current.get() == nullptr) return;

DestroyTree(current->left);
DestroyTree(current->right);

current.reset();
}

template<typename Key, typename Value>
typename rbtree<Key, Value>::Node *rbtree<Key, Value>::rotateLeft(Node *p)
{
// Make a right-leaning 3-node lean to the left.
Node  *x = p->right;

p->right = x->left;

x->left  = p;

x->color = x->left->color;

x->left->color = RED;

return x;
}

template<typename Key, typename Value>
typename rbtree<Key, Value>::Node * rbtree<Key, Value>::rotateRight(Node *p)
{
// Make a left-leaning 3-node lean to the right.
Node *x = p->left;

p->left = x->right;

x->right = p;

x->color  = x->right->color;

x->right->color = RED;

return x;
}

template<typename Key, typename Value>
typename rbtree<Key, Value>::Node * rbtree<Key, Value>::moveRedLeft(Node *p)
{
// Assuming that p is red and both p->left and p->left->left
// are black, make p->left or one of its children red
colorFlip(p);

if (isRed(p->right->left)) {

p->right = rotateRight(p->right);

p = rotateLeft(p);

colorFlip(p);
}
return p;
}

template<typename Key, typename Value>
typename rbtree<Key, Value>::Node * rbtree<Key, Value>::moveRedRight(Node *p)
{
// Assuming that p is red and both p->right and p->right->left
// are black, make p->right or one of its children red
colorFlip(p);

if (isRed(p->left->left)) {

p = rotateRight(p);
colorFlip(p);
}
return p;
}

template<typename Key, typename Value>
typename rbtree<Key, Value>::Node *rbtree<Key, Value>::fixUp(Node *p)
{
if (isRed(p->right))
p = rotateLeft(p);

if (isRed(p->left) && isRed(p->left->left))
p = rotateRight(p);

if (isRed(p->left) && isRed(p->right)) // four node
colorFlip(p);

return p;
}

template<typename Key, typename Value>
typename rbtree<Key, Value>::Node *rbtree<Key, Value>::deleteMax(Node *p)
{
if (isRed(p->left))
p = rotateRight(p);

if (p->right == nullptr)
return nullptr;

if (!isRed(p->right) && !isRed(p->right->left))
p = moveRedRight(p);

p->right = deleteMax(p->right);

return fixUp(p);
}

template<typename Key, typename Value>
typename rbtree<Key, Value>::Node *rbtree<Key, Value>::deleteMin(Node *p)
{
if (p->left == nullptr) {

// http://www.teachsolaisgames.com/articles/balanced_left_leaning.html, another C++ implementation, that p's underlying pointer must be deleted.
p.reset();
return nullptr;
}

if (!isRed(p->left) && !isRed(p->left->left))
p = moveRedLeft(p);

p->left = deleteMin(p->left);

return fixUp(p);
}

template<typename Key, typename Value>
//--typename rbtree<Key, Value>::Node *rbtree<Key, Value>::remove(Node *p, Key key)
typename rbtree<Key, Value>::Node *rbtree<Key, Value>::remove(std::unique_ptr<Node>& p, Key key)
{
if (key < p->key) {

if (!isRed(p->left) && !isRed(p->left->left)) {

p = moveRedLeft(p);
}

p->left = remove(p->left, key);

} else {

if (isRed(p->left)) {

p = rotateRight(p);
}

if ((key == p->key) && (p->right == nullptr)) {

/* From code at http://www.teachsolaisgames.com/articles/balanced_left_leaning.html
* Taken from the LeftLeaningRedBlack::DeleteRec method
*/
p.reset();
return nullptr;
}

if (!isRed(p->right) && !isRed(p->right->left)) {

p = moveRedRight(p);
}

if (key == p->key) {

Node *successor = getInOrderSuccessorNode(p);
p->value  = successor->value;  // Assign p in-order successor key and value
p->key    = successor->key;

p->right = deleteMin(p->right);

} else {

p->right = remove(p->right, key);
}
}

return fixUp(p);
}

/*
* Returns key's associated value. The search for key starts in the subtree rooted at p.
*/
template<typename Key, typename Value>  Value rbtree<Key, Value>::get(Node *p, Key key)
{

/* alternate recursive code
if (p == 0) {   ValueNotFound(key);}
if (key == p->key) return p->value;
if (key < p->key)  return get(p->left,  key);
else              return get(p->right, key);
*/
// non-recursive version
while (p != nullptr) {
if      (key < p->key) p = p->left;
else if (key > p->key) p = p->right;
else             return p->value;
}

throw key_not_found();
}

template<typename Key, typename Value> inline
typename rbtree<Key, Value>::Node *rbtree<Key, Value>::getInOrderSuccessorNode(rbtree<Key, Value>::Node *p)
{
p = p->right;

while (p->left != nullptr) {

p = p->left;
}

return p;
}

template<typename Key, typename Value>
typename rbtree<Key, Value>::Node *rbtree<Key, Value>::insert(rbtree<Key, Value>::Node *p, Key key, Value value)
{
if (p == nullptr)
return std::make_unique<Node>(key, value);

/* We view the left-leaning red black tree as a 2 3 4 tree. So first check if p is a
* 4 node and needs to be "split" by flipping colors.  */
if (isRed(p->left) && isRed(p->right))
colorFlip(p);

if (key == p->key)     /* if key already exists, overwrite its value */
p->value = value;
else if (key < p->key) /* otherwise recurse */
p->left = insert(p->left, key, value);
else
p->right = insert(p->right, key, value);

/* rebalance tree */
if (isRed(p->right))
p = rotateLeft(p);

if (isRed(p->left) && isRed(p->left->left))
p = rotateRight(p);

return p;
}
template<typename Key, typename Value> template<typename Functor> inline void rbtree<Key, Value>::traverse(Functor f)
{
return traverse(f, root);
}

/* in order traversal */
template<typename Key, typename Value>  template<typename Functor> void rbtree<Key, Value>::traverse(Functor f, rbtree<Key, Value>::Node *root)
{
if (root == nullptr) {
return;
}

traverse(f, root->left);

f(root->value);

traverse(f, root->right);
}
#endif