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I have organized a number of free online resources for learning Java and Data Structures in the tables below. None of these textbooks are a perfect fit for the class, but I have drawn material from them and hope to provide my own readings from time to time. PDF versions of these may be available online, and physical copies are going to be available through the library. These are NOT required. Ask me more in person.
In computer science , a red—black tree is a kind of self-balancing binary search tree. Each node stores an extra bit representing "color" "red" or "black" , used to ensure that the tree remains 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.
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 uncolored binary search tree. In many cases, the additional bit of information can be stored at no additional memory cost. In , 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 trees or just trees. Guibas and Robert Sedgewick derived the red-black tree from the symmetric binary B-tree. In , Arne Andersson introduced the idea of a right leaning tree to simplify insert and delete operations.
In , 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 Cormen et al. Sedgewick originally allowed nodes whose two children are red, making his trees more like trees, but later this restriction was added, making new trees more like trees.
Sedgewick implemented the insert algorithm in just 33 lines, significantly shortening his original 46 lines of code. A red—black tree is a special type of binary search tree , used in computer science to organize pieces of comparable data , such as text fragments or numbers as e. The leaf nodes of red—black trees black rectangles containing NIL in figure 1 do not contain keys or data. These leaves need not be explicit individuals in computer memory—a NULL child pointer can encode the fact that this child is a leaf.
Nevertheless, it is convenient to attribute a color to these end-of-path objects, namely black, so that it is easy to define and count the number of black "nodes" on a path. On the other hand, in order to save a marginal amount of execution time, these possibly many NIL leaves may be implemented as pointers to one unique and black sentinel node instead of pointers of value NULL.
As a conclusion, it is sufficient to encode the fact that a child does not exist is not a true node, does not contain data; is called NIL throughout this article in all occurrences as the very same NULL pointer or as the very same pointer to a sentinel node.
Red—black trees, like all binary search trees , allow quite efficient sequential access e. In addition to the requirements imposed on a binary search tree the following must be satisfied by a red—black tree: .
Some authors, e. This article also omits it, because it slightly disturbs the recursive algorithms and proofs. As an example, every perfect binary tree that consists only of black nodes is a red-black tree.
The read-only operations, such as search or tree traversal, do not affect any of the properties. The same is true for properties 1 and 2 and the modifying operations insert and delete which may, however, introduce a violation of property 3, called a red-violation , or of property 4, called a black-violation. The representation of the NIL leaves as in figure 1 certainly facilitates the explanation of a path and the counting of black nodes on it.
But it may introduce some problem of their representation e. As already explained in section Terminology , the semantics of the NIL leaves can be expressed without any individuality as the same code NIL or address of a sentinel node for all of them. Figure 2 shows the conceptually same red—black tree without these NIL leaves.
In order to arrive at the same notion of a path, one has to observe that e. This way, these ends of the paths could be interpreted as kind of sockets, equivalent to the NIL leaves of figure 1. The black depth of a node is defined as the number of black nodes from the root to that node i. 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 4, is constant alternatively, it could be defined as the black depth of any leaf node.
In this article, the black height of a NIL leaf shall be set to 0, because its subtree is empty as suggested by figure 2, and its tree height is also 0.
The requirements 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 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.
For a mathematical proof see section Proof of bounds. 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 accordingly between 2 and 4 child pointers. 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. 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 see figure 3. 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 and the other one will be red. 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.
Instead, you have to know which slot in the cluster vector is used. If values are stored by reference, e. 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 each cluster, there will be at most 1 black node, 2 red nodes, and 4 blue nodes.
B-trees will be faster for performing rotations because rotations will frequently occur within the same cluster rather than with multiple separate nodes in a colored binary tree. 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 and the other two thirds are red nodes. 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.
AVL trees can be colored red-black, thus are a subset of RB trees. Worst-case height is 0. 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.
For every tree , there are corresponding red—black trees with data elements in the same order. The insertion and deletion operations on trees are also equivalent to color-flipping and rotations in red—black trees. This makes trees an important tool for understanding the logic behind red—black trees, and this is why many introductory algorithm texts introduce trees just before red—black trees, even though trees are not often used in practice. In , 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 trees ,  or trees,  for any sequence of operations. The tree isometry was described in by Sedgewick.
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. As of Java 8, 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.
The read-only operations, such as search or tree traversal, 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, the restoration of which is called rebalancing so that red—black trees become self-balancing.
The example code may call upon the helper functions below to find the parent, grandparent, uncle, sibling and nephew nodes and to rotate a node left or right:. The proposal breaks down both, insertion and removal not mentioning some very simple cases , into six constellations of nodes, edges and colors, which are called cases.
The code repairing a case sometimes uses the code of a subsequent case. The proposal contains for both, insertion and removal, exactly one case which advances one black level closer to the root and loops, the other five cases rebalance the tree of their own. The more complicated cases are pictured in a diagram.
Insertion begins by adding the node in a very similar manner as a standard binary search tree insertion. It is colored red so that all paths contain the same number of black nodes as before. But if its parent is also red then a red-violation is introduced. The loop has the following invariant :.
Property 4 holds as well according to the loop invariant. Insert case 1: If both the parent P and the uncle U are red, then both of them can be repainted black and the grandparent G becomes red for maintaining property 4 all paths from a node to the leaves contain the same number of black nodes. Since any path through the parent or uncle must pass through the grandparent, the number of black nodes on these paths has not changed.
However, the grandparent G may now violate property 3 Both children of every red node are black if it has a red parent. Insert case 2: The current node N is the root of the tree. Then all properties are satisfied with the red root N. Insert case 4: The parent P is red but the uncle U is black.
The ultimate goal is to rotate the parent node P to the grandparent position, but this will not work if N is an "inner" grandchild of G i. A dir -rotation at P switches the roles of the current node N and its parent P.
The rotation adds paths through N those in the sub-tree labelled 2 , see diagram and removes paths through P those in the sub-tree labelled 4. But both P and N are red, so property 4 all paths from a node to its leaves contain the same number of black nodes is preserved. Property 3 both children of every red node are black is restored in case 5.
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In computer science , a red—black tree is a kind of self-balancing binary search tree. Each node stores an extra bit representing "color" "red" or "black" , used to ensure that the tree remains 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. Tracking the color of each node requires only 1 bit of information per node because there are only two colors.
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