Red–black tree


In computer science, a red–black tree is a kind of self-balancing binary search tree. Each node stores an extra bit representing color, used to ensure that the tree remains approximately balanced during insertions and deletions.
When the tree is modified, the new tree is rearranged and repainted to restore the coloring properties that constrain how unbalanced the tree can become in the worst case. The properties are designed such that this rearranging and recoloring can be performed efficiently.
The re-balancing is not perfect, but guarantees searching in Big-O notation| time, where n is the number of nodes of the tree. The insertion and deletion operations, along with the tree rearrangement and recoloring, are also performed in time.
Tracking the color of each node requires only 1 bit of information per node because there are only two colors. The tree does not contain any other data specific to its being a red–black tree so its memory footprint is almost identical to a classic binary search tree. In many cases, the additional bit of information can be stored at no additional memory cost.

History

In 1972, Rudolf Bayer invented a data structure that was a special order-4 case of a B-tree. These trees maintained all paths from root to leaf with the same number of nodes, creating perfectly balanced trees. However, they were not binary search trees. Bayer called them a "symmetric binary B-tree" in his paper and later they became popular as 2-3-4 trees or just 2-4 trees.
In a 1978 paper, "A Dichromatic Framework for Balanced Trees", Leonidas J. Guibas and Robert Sedgewick derived the red-black tree from the symmetric binary B-tree. The color "red" was chosen because it was the best-looking color produced by the color laser printer available to the authors while working at Xerox PARC. Another response from Guibas states that it was because of the red and black pens available to them to draw the trees.
In 1993, Arne Andersson introduced the idea of right leaning tree to simplify insert and delete operations.
In 1999, Chris Okasaki showed how to make the insert operation purely functional. Its balance function needed to take care of only 4 unbalanced cases and one default balanced case.
The original algorithm used 8 unbalanced cases, but reduced that to 6 unbalanced cases. Sedgewick showed that the insert operation can be implemented in just 46 lines of Java code.
In 2008, Sedgewick proposed the left-leaning red–black tree, leveraging Andersson's idea that simplified the insert and delete operations. Sedgewick originally allowed nodes whose two children are red, making his trees more like 2-3-4 trees, but later this restriction was added, making new trees more like 2-3 trees. Sedgewick implemented the insert algorithm in just 33 lines, significantly shortening his original 46 lines of code.

Terminology

A red–black tree is a special type of binary tree, used in computer science to organize pieces of comparable data, such as text fragments or numbers.
The leaf nodes of red–black trees do not contain data. These leaves need not be explicit in computer memory—a null child pointer can encode the fact that this child is a leaf. However, in the description of this figure, the leaves are considered to be explicit nodes—a view which may simplify the description and the understanding of some algorithms for operating on red–black trees. Now, in order to save a marginal amount of execution time, these NIL-leaves may be implemented as sentinel nodes. On the other hand, in order to save memory, a single sentinel node may perform the role of all leaf nodes: all references from internal nodes to leaf nodes then point to this unique sentinel node.
Red–black trees, like all binary search trees, allow efficient in-order traversal of their elements. The search-time results from the traversal from root to leaf, and therefore a balanced tree of n nodes, having the least possible tree height, results in search time.

Properties

In addition to the requirements imposed on a binary search tree the following must be satisfied by a
  1. Each node is either red or black.
  2. The root is black. This rule is sometimes omitted. Since the root can always be changed from red to black, but not necessarily vice versa, this rule has little effect on analysis.
  3. All leaves are black.
  4. If a node is red, then both its children are black.
  5. Every path from a given node to any of its descendant NIL nodes goes through the same number of black nodes.
The only constraint on the children of black nodes is. In particular, a black node can have a black parent; for example, every perfect binary tree that consists only of black nodes is a red-black tree.
The black depth of a node is defined as the number of black nodes from the root to that node. The black height of a red–black tree is the number of black nodes in any path from the root to the leaves, which, by property 5, is constant.
These constraints enforce a critical property of red–black trees: the path from the root to the farthest leaf is no more than twice as long as the path from the root to the nearest leaf. The result is that the tree is roughly height-balanced. Since operations such as inserting, deleting, and finding values require worst-case time proportional to the height of the tree, this theoretical upper bound on the height allows red–black trees to be efficient in the worst case, unlike ordinary binary search trees.
To see why this is guaranteed, consider a red–black tree having b black nodes in property 5. The shortest path from the root to any leaf has b black nodes and 0 red nodes. We can insert at most one red node between each two black nodes. Therefore, the longest possible path has of 2*b nodes, alternating black and red.

Analogy to B-trees of order 4

A red–black tree is similar in structure to a B-tree of order 4, where each node can contain between 1 and 3 values and between 2 and 4 child pointers. In such a B-tree, each node will contain only one value matching the value in a black node of the red–black tree, with an optional value before and/or after it in the same node, both matching an equivalent red node of the red–black tree.
One way to see this equivalence is to "move up" the red nodes in a graphical representation of the red–black tree, so that they align horizontally with their parent black node, by creating together a horizontal cluster. In the B-tree, or in the modified graphical representation of the red–black tree, all leaf nodes are at the same depth.
The red–black tree is then structurally equivalent to a B-tree of order 4, with a minimum fill factor of 33% of values per cluster with a maximum capacity of 3 values.
This B-tree type is still more general than a red–black tree though, as it allows ambiguity in a red–black tree conversion—multiple red–black trees can be produced from an equivalent B-tree of order 4. If a B-tree cluster contains only 1 value, it is the minimum, black, and has two child pointers. If a cluster contains 3 values, then the central value will be black and each value stored on its sides will be red. If the cluster contains two values, however, either one can become the black node in the red–black tree.
So the order-4 B-tree does not maintain which of the values contained in each cluster is the root black tree for the whole cluster and the parent of the other values in the same cluster. Despite this, the operations on red–black trees are more economical in time because you don't have to maintain the vector of values. It may be costly if values are stored directly in each node rather than being stored by reference. B-tree nodes, however, are more economical in space because you don't need to store the color attribute for each node. Instead, you have to know which slot in the cluster vector is used. If values are stored by reference, e.g. objects, null references can be used and so the cluster can be represented by a vector containing 3 slots for value pointers plus 4 slots for child references in the tree. In that case, the B-tree can be more compact in memory, improving data locality.
The same analogy can be made with B-trees with larger orders that can be structurally equivalent to a colored binary tree: you just need more colors. Suppose that you add blue, then the blue–red–black tree defined like red–black trees but with the additional constraint that no two successive nodes in the hierarchy will be blue and all blue nodes will be children of a red node, then it becomes equivalent to a B-tree whose clusters will have at most 7 values in the following colors: blue, red, blue, black, blue, red, blue.
For moderate volumes of values, insertions and deletions in a colored binary tree are faster compared to B-trees because colored trees don't attempt to maximize the fill factor of each horizontal cluster of nodes. B-trees will be faster for performing rotations. For storing large volumes, however, B-trees will be much faster as they will be more compact by grouping several children in the same cluster where they can be accessed locally.
All optimizations possible in B-trees to increase the average fill factors of clusters are possible in the equivalent multicolored binary tree. Notably, maximizing the average fill factor in a structurally equivalent B-tree is the same as reducing the total height of the multicolored tree, by increasing the number of non-black nodes. The worst case occurs when all nodes in a colored binary tree are black, the best case occurs when only a third of them are black.

Applications and related data structures

Red–black trees offer worst-case guarantees for insertion time, deletion time, and search time. Not only does this make them valuable in time-sensitive applications such as real-time applications, but it makes them valuable building blocks in other data structures which provide worst-case guarantees; for example, many data structures used in computational geometry can be based on red–black trees, and the Completely Fair Scheduler used in current Linux kernels and epoll system call implementation uses red–black trees.
The AVL tree is another structure supporting search, insertion, and removal. AVL trees can be colored red-black, thus are a subset of RB trees. Worst-case height is 0.720 times the worst-case height of RB trees, so AVL trees are more rigidly balanced. The performance measurements of Ben Pfaff with realistic test cases in 79 runs find AVL to RB ratios between 0.677 and 1.077, median at 0.947, and geometric mean 0.910. WAVL trees have a performance in between those two.
Red–black trees are also particularly valuable in functional programming, where they are one of the most common persistent data structures, used to construct associative arrays and sets which can retain previous versions after mutations. The persistent version of red–black trees requires space for each insertion or deletion, in addition to time.
For every 2-4 tree, there are corresponding red–black trees with data elements in the same order. The insertion and deletion operations on 2-4 trees are also equivalent to color-flipping and rotations in red–black trees. This makes 2-4 trees an important tool for understanding the logic behind red–black trees, and this is why many introductory algorithm texts introduce 2-4 trees just before red–black trees, even though 2-4 trees are not often used in practice.
In 2008, Sedgewick introduced a simpler version of the red–black tree called the left-leaning red–black tree by eliminating a previously unspecified degree of freedom in the implementation. The LLRB maintains an additional invariant that all red links must lean left except during inserts and deletes. Red–black trees can be made isometric to either 2-3 trees, or 2-4 trees, for any sequence of operations. The 2-4 tree isometry was described in 1978 by Sedgewick. With 2-4 trees, the isometry is resolved by a "color flip," corresponding to a split, in which the red color of two children nodes leaves the children and moves to the parent node.
The original description of the tango tree, a type of tree optimized for fast searches, specifically uses red–black trees as part of its data structure.
In the version 8 of Java, the HashMap has been modified such that instead of using a LinkedList to store different elements with colliding hashcodes, a red-black tree is used. This results in the improvement of time complexity of searching such an element from to.

Operations

Read-only operations on a red–black tree require no modification from those used for binary search trees, because every red–black tree is a special case of a simple binary search tree. However, the immediate result of an insertion or removal may violate the properties of a red–black tree. Restoring the red–black properties requires a small number of color changes and no more than three tree rotations. Although insert and delete operations are complicated, their times remain.
If the example implementation [|below] is not suitable, other implementations with explanations may be found in Ben Pfaff's annotated C library .
The details of the insert and removal operations will be demonstrated with example C++ code. The example code may call upon the helper functions below to find the parent, sibling, uncle and grandparent nodes and to rotate a node left or right:

// Basic type definitions:
enum color_t ;
struct Node ;
// Helper functions:
Node* GetParent
Node* GetGrandParent
Node* GetSibling
Node* GetUncle
void RotateLeft
void RotateRight

;Diagram notes
  1. The label N will be used to denote the current node in each case. At the beginning, this is the insertion node or the replacement node and a leaf, but the entire procedure may also be applied recursively to other nodes.
  2. P will denote N's parent node, G will denote N's grandparent, S will denote N's sibling, and U will denote N's uncle.
  3. In between some cases, the roles and labels of the nodes are shifted, but within each case, every label continues to represent the same node throughout.
  4. In the diagrams a blue border rings the current node N in the left half and rings the node that will become N in the right half. In the next step, the other nodes will be newly assigned relative to it.
  5. Red or black shown in the diagram is either assumed in its case or implied by those assumptions. White represents either red or black, but is the same in both halves of the diagram.
  6. A numbered triangle represents a subtree of unspecified depth. A black circle atop a triangle means that black-height of that subtree is greater by one compared to a subtree without this circle.

    Insertion

Insertion begins by adding the node in a very similar manner as a standard binary search tree insertion and by coloring it red. The big difference is that in the binary search tree a new node is added as a leaf, whereas leaves contain no information in the red–black tree, so instead the new node replaces an existing leaf and then has two black leaves of its own added.

Node* Insert
void InsertRecurse

What happens next depends on the color of other nearby nodes. There are several cases of red–black tree insertion to handle:
  1. N is the root node, i.e., first node of red–black tree
  2. N's parent is black
  3. P is red and N's uncle is red
  4. P is red and U is black

void InsertRepairTree

Note that:
Case 1: The current node N is at the root of the tree. In this case, it is repainted black to satisfy property 2. Since this adds one black node to every path at once, property 5 is not violated.

void InsertCase1

Case 2: The current node's parent P is black, so property 4 holds. Property 5 is not threatened, because the new node N has two black leaf children, but because N is red, the paths through each of its children have the same number of black nodes as the path through the leaf it replaced, which was black, and so this property remains satisfied. So the tree remains valid.

void InsertCase2


void InsertCase3


void InsertCase4


void InsertCase4Step2

Note that inserting is actually in-place, since all the calls above use tail recursion.
In the algorithm above, all cases are called only once, except in Case 3 where it can recurse back to Case 1 with the grandparent node, which is the only case where an iterative implementation will effectively loop. Because the problem of repair in that case is escalated two levels higher each time, it takes maximally iterations to repair the tree. Because the probability for escalation decreases exponentially with each iteration the average insertion cost is practically constant.

Removal

In a regular binary search tree when deleting a node with two non-leaf children, we find either the maximum element in its left subtree or the minimum element in its right subtree and move its value into the node being deleted. We then delete the node we copied the value from, which must have fewer than two non-leaf children. Because merely copying a value does not violate any red–black properties, this reduces to the problem of deleting a node with at most one non-leaf child. Once we have solved that problem, the solution applies equally to the case where the node we originally want to delete has at most one non-leaf child as to the case just considered where it has two non-leaf children.
Therefore, for the remainder of this discussion we address the deletion of a node with at most one non-leaf child. We use the label M to denote the node to be deleted; C will denote a selected child of M, which we will also call "its child". If M does have a non-leaf child, call that its child, C; otherwise, choose either leaf as its child, C.
If M is a red node, we simply replace it with its child C, which must be black by property 4. All paths through the deleted node will simply pass through one fewer red node, and both the deleted node's parent and child must be black, so property 3 and property 4 still hold.
Another simple case is when M is black and C is red. Simply removing a black node could break Properties 4 and 5, but if we repaint C black, both of these properties are preserved.
The complex case is when both M and C are black. We begin by replacing M with its child C – we recall that in this case "its child C" is either child of M, both being leaves. We will relabel this child C N, and its sibling S.
In the diagrams below, we will also use P for N's new parent, SL for S's left child, and SR for S's right child.
We can perform the steps outlined above with the following code, where the function ReplaceNode substitutes child into n's place in the tree. For convenience, code in this section will assume that null leaves are represented by actual node objects rather than NULL.

void ReplaceNode
void DeleteOneChild

If both N and its original parent are black, then deleting this original parent causes paths which proceed through N to have one fewer black node than paths that do not. As this violates property 5, the tree must be rebalanced. There are several cases to consider:
Case 1: N is the new root. In this case, we are done. We removed one black node from every path, and the new root is black, so the properties are preserved.

void DeleteCase1


void DeleteCase2


void DeleteCase3


void DeleteCase4


void DeleteCase5


void DeleteCase6

Again, the function calls all use tail recursion, so the algorithm is in-place.
In the algorithm above, all cases are chained in order, except in delete case 3 where it can recurse to case 1 back to the parent node: this is the only case where an iterative implementation will effectively loop. No more than loops back to case 1 will occur. And because the probability for escalation decreases exponentially with each iteration the average removal cost is constant.
Additionally, no tail recursion ever occurs on a child node, so the tail recursion loop can only move from a child back to its successive ancestors. If a rotation occurs in case 2, then the parent of the node N becomes red after the rotation and we will exit the loop. Therefore, at most one rotation will occur within this loop. Since no more than two additional rotations will occur after exiting the loop, at most three rotations occur in total.
point out: "AVL trees do not support constant amortized deletion costs", but red-black trees do.

Proof of asymptotic bounds

A red black tree which contains n internal nodes has a height of.
Definitions:
  • h = height of subtree rooted at node v
  • bh = the number of black nodes from v to any leaf in the subtree, not counting v if it is black - called the black-height
Lemma: A subtree rooted at node v has at least internal nodes.
Proof of Lemma :
Basis: h = 0
If v has a height of zero then it must be null, therefore bh = 0. So:
Inductive Step: v such that h = k, has at least internal nodes implies that such that h = k+1 has at least internal nodes.
Since has h > 0 it is an internal node. As such it has two children each of which have a black-height of either bh or bh-1. By the inductive hypothesis each child has at least internal nodes, so has at least:
internal nodes.
Using this lemma we can now show that the height of the tree is logarithmic. Since at least half of the nodes on any path from the root to a leaf are black, the black-height of the root is at least h/2. By the lemma we get:
Therefore, the height of the root is.

Set operations and bulk operations

In addition to the single-element insert, delete and lookup operations, several set operations have been defined on red-black trees: union, intersection and set difference. Then fast bulk operations on insertions or deletions can be implemented based on these set functions. These set operations rely on two helper operations, Split and Join. With the new operations, the implementation of red-black trees can be more efficient and highly-parallelizable. This implementation allows a red root.
  • Join: The function Join is on two red-black trees and and a key and will return a tree containing all elements in, as well as. It requires to be greater than all keys in and smaller than all keys in. If the two trees have the same black height, Join simply create a new node with left subtree, root and right subtree. If both and have black root, set to be red. Otherwise is set black. Suppose that has larger black height than . Join follows the right spine of until a black node which is balanced with. At this point a new node with left child, root and right child is created to replace c. The new node may invalidate the red-black invariant because at most three red nodes can appear in a row. This can be fixed with a double rotation. If double red issue propagates to the root, the root is then set to be black, restoring the properties. The cost of this function is the difference of the black heights between the two input trees.
  • Split: To split a red-black tree into two smaller trees, those smaller than key x, and those larger than key x, first draw a path from the root by inserting x into the red-black tree. After this insertion, all values less than x will be found on the left of the path, and all values greater than x will be found on the right. By applying Join, all the subtrees on the left side are merged bottom-up using keys on the path as intermediate nodes from bottom to top to form the left tree, and the right part is asymmetric. For some applications, Split also returns a boolean value denoting if x appears in the tree. The cost of Split is, order of the height of the tree. This algorithm actually has nothing to do with any special properties of a red-black tree, and thus is generic to other balancing schemes such as AVL trees.
The join algorithm is as follows:
function joinRightRB
if r=⌊r/2⌋×2:
return Node
else
=expose
T'=Node
if and :
T'.right.right.color=black;
return rotateLeft
else return T'
function joinLeftRB
/* symmetric to joinRightRB */
function join
if ⌊r/2⌋>⌊r/2⌋×2:
T'=joinRightRB
if and :
T'.color=black
return T'
else if ⌊r/2⌋>⌊r/2⌋×2
/* symmetric */
else if and
Node
else
Node
Here of a node means twice the black height of a black node, and the twice the black height of a red node. expose= means to extract a tree node 's left child, the key of the node, the color of the node and the right child. Node means to create a node of left child, key, color and right child.
The split algorithm is as follows:
function split
if return
=expose
if return
if
=split
return
if
=split
return )
The union of two red-black trees and representing sets and, is a red-black tree that represents. The following recursive function computes this union:
function union:
if t1 = nil:
return t2
if t2 = nil:
return t1
t<, t> ← split t2 on t1.root
return join
Here, Split is presumed to return two trees: one holding the keys less its input key, one holding the greater keys.
The algorithm for intersection or difference is similar, but requires the Join2 helper routine that is the same as Join but without the middle key. Based on the new functions for union, intersection or difference, either one key or multiple keys can be inserted to or deleted from the red-black tree. Since Split calls Join but does not deal with the balancing criteria of red-black trees directly, such an implementation is usually called the "join-based" implementation.
The complexity of each of union, intersection and difference is for two red-black trees of sizes and. This complexity is optimal in terms of the number of comparisons. More importantly, since the recursive calls to union, intersection or difference are independent of each other, they can be executed in parallel with a parallel depth. When, the join-based implementation has the same computational directed acyclic graph as single-element insertion and deletion if the root of the larger tree is used to split the smaller tree.

Parallel algorithms

Parallel algorithms for constructing red–black trees from sorted lists of items can run in constant time or time, depending on the computer model, if the number of processors available is asymptotically proportional to the number of items where. Fast search, insertion, and deletion parallel algorithms are also known.
The join-based algorithms for red-black trees are parallel for bulk operations, including union, intersection, construction, filter, map-reduce, and so on.

Parallel bulk operations

Basic operations like insertion, removal or update can be parallelized by defining operations that process bulks of multiple elements.
It is also possible to process bulks with several basic operations, for example bulks may contain elements to insert and also elements to remove from the tree.
The algorithms for bulk operations aren't just applicable to the red-black tree, but can be adapted to other sorted sequence data structures as well, like the 2-3 tree, 2-3-4 tree and -tree.
In the following different algorithms for bulk insert will be explained, but the same algorithms can also be applied to removal and update.
Bulk insert is an operation that inserts each element of a sequence into a tree.

Join-based

This approach can be applied to every sorted sequence data structure that supports efficient join- and split-operations.
The general idea is to split and in multiple parts and perform the insertions on these parts in parallel.
  1. First the bulk of elements to insert has to be sorted.
  2. After that, the algorithm splits into parts of about equal sizes.
  3. Next the tree has to be split into parts in a way, so that for every following constraints hold:
  4. #
  5. #
  6. Now the algorithm inserts each element of into sequentially. This step hast to be performed for every, which can be done by up to processors in parallel.
  7. Finally, the resulting trees will be joined to form the final result of the entire operation.
Note that in Step 3 the constraints for splitting assure that in Step 5 the trees can be joined again and the resulting sequence is sorted.
The pseudo code shows a simple divide-and-conquer implementation of the join-based algorithm for bulk-insert.
Both recursive calls can be executed in parallel.
The join operation used here differs from the version explained in this article, instead join2 is used which misses the second parameter k.
bulkInsert:
I.sort
bulklInsertRec
bulkInsertRec:
if k = 1:
forall e in I: T.insert
else
m := ⌊size / 2⌋
:= split
bulkInsertRec
|| bulkInsertRec
T ← join2
Execution time
Sorting is not considered in this analysis.
#recursion levels
T + T
insertions per thread
T
T with = #processors

This can be improved by using parallel algorithms for splitting and joining.
In this case the execution time is.
Work

Pipelining

Another method of parallelizing bulk operations is to use a pipelining approach.
This can be done by breaking the task of processing a basic operation up into a sequence of subtasks.
For multiple basic operations the subtasks can be processed in parallel by assigning each subtask to a separate processor.
  1. First the bulk of elements to insert has to be sorted.
  2. For each element in the algorithm locates the according insertion position in. This can be done in parallel for each element since won't be mutated in this process. Now has to be divided into subsequences according to the insertion position of each element. For example is the subsequence of which contains the elements whose insertion position would be to the left of node.
  3. The middle element of every subsequence will be inserted into as a new node. This can be done in parallel for each since by definition the insertion position of each is unique. If contains elements to the left or to the right of, those will be contained in a new set of subsequences as or.
  4. Now possibly contains up to two consecutive red nodes at the end of the paths form the root to the leaves, which needs to be repaired. Note that, while repairing, the insertion position of elements have to be updated, if the corresponding nodes are affected by rotations.
If two nodes have different nearest black ancestors, they can be repaired in parallel. Since at most four nodes can have the same nearest black ancestor, the nodes at the lowest level can be repaired in a constant number of parallel steps.
This Step will be applied successively to the black levels above until is fully repaired.
  1. The steps 3 to 5 will be repeated on the new subsequences until is empty. At this point every element has been inserted. Each application of these steps is called a stage. Since the length of the subsequences in is and in every stage the subsequences are being cut in half, the number of stages is.
Since all stages move up the black levels of the tree, they can be parallelized in a pipeline. Once a stage has finished processing one black level, the next stage is able to move up and continue at that level.
Execution time
Sorting is not considered in this analysis.
Also, is assumed to be smaller than, otherwise it would be more sufficient to construct the resulting tree from scratch.
T
#stages
T + T
T with ~ #processors
Work

Popular culture

A red-black-tree was referenced correctly in an episode of Missing as noted by Robert Sedgewick in one of his lectures: