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46 lines
1.9 KiB
Markdown
46 lines
1.9 KiB
Markdown
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id: index
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slug: /rank-balanced-trees/
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title: Rank-Balanced Trees
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description: |
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Explaining the rank-balanced trees. The web version of my bachelor thesis.
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tags:
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- balanced trees
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- red-black trees
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- avl tree
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- wavl tree
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last_update:
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date: 2024-06-08
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---
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Data structures have become a regular part of the essential toolbox for
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problem-solving. In many cases, they also help to improve the existing
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algorithms’ performance, e.g., using a priority queue in _Dijkstra’s algorithm_
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_for the shortest path_. This thesis will mainly discuss the implementation of
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a set.
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Currently, the most commonly used implementations of sets use hash tables, but
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we will talk about another common alternative, implementation via self-balancing
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search trees. Compared to a hash table, they provide consistent time complexity,
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but at the cost of a requirement for ordering on the elements. The most
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implemented self-balancing binary tree is a _red-black tree_, as described by
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_Guibas and Sedgewick_[^1]. Among other alternatives, we can find (non-binary)
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_B-tree_[^2] and _AVL tree_[^3].
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We will focus on the _Weak AVL_ (_WAVL_) _tree_[^4] that is a relaxed variant of
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the AVL tree, but still provides better balancing than a red-black tree.
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We will start by describing the AVL tree, then we will introduce the idea
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of a _rank-balanced tree_. Given this insight we will be able to explore various
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implementations of aforementioned trees using the rank-balanced tree. At the end
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we will focus on the _Weak AVL tree_
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[^1]: [A dichromatic framework for balanced trees.](https://doi.org/10.1109/SFCS.1978.3)
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[^2]: [Organization and Maintenance of Large Ordered Indices.](https://doi.org/10.1145/1734663.1734671)
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[^3]:
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ADELSON-VELSKIJ, Georgij; LANDIS, Evgenij. An algorithm for the
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organization of information. _Doklady Akad. Nauk SSSR._ 1962, vol. 146,
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pp. 263–266.
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[^4]: [Rank-Balanced Trees](https://doi.org/10.1145/2689412)
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