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97 lines
4.1 KiB
Markdown
97 lines
4.1 KiB
Markdown
# Red–Black Tree
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_Read this in other languages:_
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[_Português_](README.pt-BR.md)
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A **red–black tree** is a kind of self-balancing binary search
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tree in computer science. Each node of the binary tree has
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an extra bit, and that bit is often interpreted as the
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color (red or black) of the node. These color bits are used
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to ensure the tree remains approximately balanced during
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insertions and deletions.
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Balance is preserved by painting each node of the tree with
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one of two colors in a way that satisfies certain properties,
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which collectively constrain how unbalanced the tree can
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become in the worst case. When the tree is modified, the
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new tree is subsequently rearranged and repainted to
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restore the coloring properties. The properties are
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designed in such a way that this rearranging and recoloring
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can be performed efficiently.
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The balancing of the tree is not perfect, but it is good
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enough to allow it to guarantee searching in `O(log n)` time,
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where `n` is the total number of elements in the tree.
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The insertion and deletion operations, along with the tree
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rearrangement and recoloring, are also performed
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in `O(log n)` time.
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An example of a red–black tree:
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## Properties
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In addition to the requirements imposed on a binary search
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tree the following must be satisfied by a red–black tree:
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- Each node is either red or black.
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- The root is black. This rule is sometimes omitted.
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Since the root can always be changed from red to black,
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but not necessarily vice versa, this rule has little
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effect on analysis.
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- All leaves (NIL) are black.
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- If a node is red, then both its children are black.
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- Every path from a given node to any of its descendant
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NIL nodes contains the same number of black nodes.
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Some definitions: the number of black nodes from the root
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to a node is the node's **black depth**; the uniform
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number of black nodes in all paths from root to the leaves
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is called the **black-height** of the red–black tree.
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These constraints enforce a critical property of red–black
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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_.
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The result is that the tree is roughly height-balanced.
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Since operations such as inserting, deleting, and finding
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values require worst-case time proportional to the height
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of the tree, this theoretical upper bound on the height
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allows red–black trees to be efficient in the worst case,
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unlike ordinary binary search trees.
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## Balancing during insertion
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### If uncle is RED
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### If uncle is BLACK
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- Left Left Case (`p` is left child of `g` and `x` is left child of `p`)
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- Left Right Case (`p` is left child of `g` and `x` is right child of `p`)
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- Right Right Case (`p` is right child of `g` and `x` is right child of `p`)
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- Right Left Case (`p` is right child of `g` and `x` is left child of `p`)
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#### Left Left Case (See g, p and x)
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#### Left Right Case (See g, p and x)
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#### Right Right Case (See g, p and x)
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#### Right Left Case (See g, p and x)
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## References
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- [Wikipedia](https://en.wikipedia.org/wiki/Red%E2%80%93black_tree)
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- [Red Black Tree Insertion by Tushar Roy (YouTube)](https://www.youtube.com/watch?v=UaLIHuR1t8Q&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8&index=63)
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- [Red Black Tree Deletion by Tushar Roy (YouTube)](https://www.youtube.com/watch?v=CTvfzU_uNKE&t=0s&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8&index=64)
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- [Red Black Tree Insertion on GeeksForGeeks](https://www.geeksforgeeks.org/red-black-tree-set-2-insert/)
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- [Red Black Tree Interactive Visualisations](https://www.cs.usfca.edu/~galles/visualization/RedBlack.html)
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