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66 lines
2 KiB
Markdown
66 lines
2 KiB
Markdown
---
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id: avl
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title: AVL tree
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description: |
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Looking at a tree that's very similar to the red-black tree.
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tags:
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- AVL tree
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- balanced trees
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- zig
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last_update:
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date: 2024-01-22
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---
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## Introduction
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AVL tree is the first self-balanced tree to be ever invented. However if we look
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at the commonly used data structures to represent the set or a map, we will find
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out that red-black trees (or even its relaxation LLRB[^1]) or hash tables are
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the most used.
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Let's look into the AVL tree and how it is related to its “successor” red-black
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tree, they are closer than you can imagine.
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## Properties
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AVL tree is height-balanced same as the red-black tree. The key difference lies
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in the invariants that need to be satisfied. In this case we require that
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subtrees of any node differ in their heights at most by 1. If we were to write
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it as a condition for node $n$, we can write it like:
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$$
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\vert Height(Right(n)) - Height(Left(n)) \vert \leq 1
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$$
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Because the only allowed values here are $\left\{ -1, 0, 1 \right\}$, it is
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possible to simplify the structure of the nodes in memory by using a one byte of
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memory to store the “state” of the subtrees in a node.
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:::tip
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Some representations use a _sign_ notation of `+`, `0` and `-` which represent
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where is the tree leaning to:
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- `-` means the left subtree has bigger height ⇒ tree is leaning to left
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- `0` means both subtrees have the same height ⇒ they're equal
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- `+` means the right subtree has bigger height ⇒ tree is leaning to right
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:::
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It is also possible to find representations that allow leaning only to one side,
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by enforcing this requirement you can use just boolean to represent the state,
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i.e. the tree _is_ leaning to one side, or _is not_.
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## Implementation
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Let's have a look at the implementation of the AVL tree in some language.
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## Insertion
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## Deletion
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## Comparison with the red-black tree
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## Bonus: Implementation of the BST in Java
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[^1]: [Left-leaning Red-Black trees](https://sedgewick.io/wp-content/themes/sedgewick/papers/2008LLRB.pdf)
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