algorithms(avl): add WIP

Signed-off-by: Matej Focko <me@mfocko.xyz>
This commit is contained in:
Matej Focko 2024-01-22 23:38:58 +01:00
parent 5728170a57
commit f46561653f
Signed by: mfocko
GPG key ID: 7C47D46246790496
10 changed files with 1057 additions and 0 deletions

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

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dotnet_naming_symbols.local_constants.required_modifiers = const
dotnet_naming_symbols.parameters.applicable_kinds = parameter
dotnet_naming_symbols.parameters.applicable_accessibilities = *
dotnet_naming_symbols.parameters.required_modifiers =
dotnet_naming_symbols.public_constant_fields.applicable_kinds = field
dotnet_naming_symbols.public_constant_fields.applicable_accessibilities = public, internal
dotnet_naming_symbols.public_constant_fields.required_modifiers = const
dotnet_naming_symbols.public_static_readonly_fields.applicable_kinds = field
dotnet_naming_symbols.public_static_readonly_fields.applicable_accessibilities = public, internal
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dotnet_naming_symbols.private_static_readonly_fields.applicable_kinds = field
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dotnet_naming_style._camelcase.capitalization = camel_case
dotnet_naming_style.camelcase.required_prefix =
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_UpgradeReport_Files/
Backup*/
UpgradeLog*.XML
UpgradeLog*.htm
ServiceFabricBackup/
*.rptproj.bak
# SQL Server files
*.mdf
*.ldf
*.ndf
# Business Intelligence projects
*.rdl.data
*.bim.layout
*.bim_*.settings
*.rptproj.rsuser
*- [Bb]ackup.rdl
*- [Bb]ackup ([0-9]).rdl
*- [Bb]ackup ([0-9][0-9]).rdl
# Microsoft Fakes
FakesAssemblies/
# GhostDoc plugin setting file
*.GhostDoc.xml
# Node.js Tools for Visual Studio
.ntvs_analysis.dat
node_modules/
# Visual Studio 6 build log
*.plg
# Visual Studio 6 workspace options file
*.opt
# Visual Studio 6 auto-generated workspace file (contains which files were open etc.)
*.vbw
# Visual Studio 6 auto-generated project file (contains which files were open etc.)
*.vbp
# Visual Studio 6 workspace and project file (working project files containing files to include in project)
*.dsw
*.dsp
# Visual Studio 6 technical files
*.ncb
*.aps
# Visual Studio LightSwitch build output
**/*.HTMLClient/GeneratedArtifacts
**/*.DesktopClient/GeneratedArtifacts
**/*.DesktopClient/ModelManifest.xml
**/*.Server/GeneratedArtifacts
**/*.Server/ModelManifest.xml
_Pvt_Extensions
# Paket dependency manager
.paket/paket.exe
paket-files/
# FAKE - F# Make
.fake/
# CodeRush personal settings
.cr/personal
# Python Tools for Visual Studio (PTVS)
__pycache__/
*.pyc
# Cake - Uncomment if you are using it
# tools/**
# !tools/packages.config
# Tabs Studio
*.tss
# Telerik's JustMock configuration file
*.jmconfig
# BizTalk build output
*.btp.cs
*.btm.cs
*.odx.cs
*.xsd.cs
# OpenCover UI analysis results
OpenCover/
# Azure Stream Analytics local run output
ASALocalRun/
# MSBuild Binary and Structured Log
*.binlog
# NVidia Nsight GPU debugger configuration file
*.nvuser
# MFractors (Xamarin productivity tool) working folder
.mfractor/
# Local History for Visual Studio
.localhistory/
# Visual Studio History (VSHistory) files
.vshistory/
# BeatPulse healthcheck temp database
healthchecksdb
# Backup folder for Package Reference Convert tool in Visual Studio 2017
MigrationBackup/
# Ionide (cross platform F# VS Code tools) working folder
.ionide/
# Fody - auto-generated XML schema
FodyWeavers.xsd
# VS Code files for those working on multiple tools
.vscode/*
!.vscode/settings.json
!.vscode/tasks.json
!.vscode/launch.json
!.vscode/extensions.json
*.code-workspace
# Local History for Visual Studio Code
.history/
# Windows Installer files from build outputs
*.cab
*.msi
*.msix
*.msm
*.msp
# JetBrains Rider
*.sln.iml
.idea
##
## Visual studio for Mac
##
# globs
Makefile.in
*.userprefs
*.usertasks
config.make
config.status
aclocal.m4
install-sh
autom4te.cache/
*.tar.gz
tarballs/
test-results/
# Mac bundle stuff
*.dmg
*.app
# content below from: https://github.com/github/gitignore/blob/master/Global/macOS.gitignore
# General
.DS_Store
.AppleDouble
.LSOverride
# Icon must end with two \r
Icon
# Thumbnails
._*
# Files that might appear in the root of a volume
.DocumentRevisions-V100
.fseventsd
.Spotlight-V100
.TemporaryItems
.Trashes
.VolumeIcon.icns
.com.apple.timemachine.donotpresent
# Directories potentially created on remote AFP share
.AppleDB
.AppleDesktop
Network Trash Folder
Temporary Items
.apdisk
# content below from: https://github.com/github/gitignore/blob/master/Global/Windows.gitignore
# Windows thumbnail cache files
Thumbs.db
ehthumbs.db
ehthumbs_vista.db
# Dump file
*.stackdump
# Folder config file
[Dd]esktop.ini
# Recycle Bin used on file shares
$RECYCLE.BIN/
# Windows Installer files
*.cab
*.msi
*.msix
*.msm
*.msp
# Windows shortcuts
*.lnk
# Vim temporary swap files
*.swp

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<Project Sdk="Microsoft.NET.Sdk">
<PropertyGroup>
<TargetFramework>net8.0</TargetFramework>
<ImplicitUsings>enable</ImplicitUsings>
<Nullable>enable</Nullable>
<IsPackable>false</IsPackable>
<IsTestProject>true</IsTestProject>
</PropertyGroup>
<ItemGroup>
<PackageReference Include="Microsoft.NET.Test.Sdk" Version="17.6.0" />
<PackageReference Include="xunit" Version="2.4.2" />
<PackageReference Include="xunit.runner.visualstudio" Version="2.4.5">
<IncludeAssets>runtime; build; native; contentfiles; analyzers; buildtransitive</IncludeAssets>
<PrivateAssets>all</PrivateAssets>
</PackageReference>
<PackageReference Include="coverlet.collector" Version="6.0.0">
<IncludeAssets>runtime; build; native; contentfiles; analyzers; buildtransitive</IncludeAssets>
<PrivateAssets>all</PrivateAssets>
</PackageReference>
</ItemGroup>
</Project>

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@ -0,0 +1,25 @@

Microsoft Visual Studio Solution File, Format Version 12.00
# Visual Studio Version 17
VisualStudioVersion = 17.5.002.0
MinimumVisualStudioVersion = 10.0.40219.1
Project("{9A19103F-16F7-4668-BE54-9A1E7A4F7556}") = "avl", "avl.csproj", "{82C655B9-83B6-4AB4-8449-1309913DDC8F}"
EndProject
Global
GlobalSection(SolutionConfigurationPlatforms) = preSolution
Debug|Any CPU = Debug|Any CPU
Release|Any CPU = Release|Any CPU
EndGlobalSection
GlobalSection(ProjectConfigurationPlatforms) = postSolution
{82C655B9-83B6-4AB4-8449-1309913DDC8F}.Debug|Any CPU.ActiveCfg = Debug|Any CPU
{82C655B9-83B6-4AB4-8449-1309913DDC8F}.Debug|Any CPU.Build.0 = Debug|Any CPU
{82C655B9-83B6-4AB4-8449-1309913DDC8F}.Release|Any CPU.ActiveCfg = Release|Any CPU
{82C655B9-83B6-4AB4-8449-1309913DDC8F}.Release|Any CPU.Build.0 = Release|Any CPU
EndGlobalSection
GlobalSection(SolutionProperties) = preSolution
HideSolutionNode = FALSE
EndGlobalSection
GlobalSection(ExtensibilityGlobals) = postSolution
SolutionGuid = {C5C6C0D8-FFA4-4D7A-BE93-6CF1A159F1F2}
EndGlobalSection
EndGlobal

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using System.Collections;
namespace avl;
class AVL<T>(IComparer<T> comparator) : IEnumerable<T> {
private Node<T>? _root = null;
public AVL() : this(Comparer<T>.Default) { }
public int Count { get; private set; } = 0;
public IComparer<T> Comparator { get; } = comparator;
public bool Add(T item) {
throw new NotImplementedException();
}
public bool Remove(T item) {
throw new NotImplementedException();
}
public void Clear() => _root = null;
public bool Contains(T item) {
var node = _root;
while (node != null) {
var cmp = Comparator.Compare(node.Value, item);
if (cmp < 0) {
node = node.Right;
} else if (cmp == 0) {
return true;
} else {
node = node.Left;
}
}
return false;
}
public void CopyTo(T[] array, int arrayIndex) {
throw new NotImplementedException();
}
public IEnumerator<T> GetEnumerator() {
throw new NotImplementedException();
}
IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
}

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@ -0,0 +1,12 @@
namespace avl;
class Node<T> {
public T Value { get; init; }
public readonly Node<T>? Parent;
public readonly Node<T>? Left, Right;
public Node(T value) {
Value = value;
}
}

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namespace avl.test;
public class AVL {
[Fact]
public void EmptyTree() {
var t = new AVL<int>();
Assert.Equal(0, t.Count);
for (int i = 0; i < 10; ++i) {
Assert.False(t.Contains(i));
}
}
[Fact]
public void OneNode() {
var t = new AVL<int> {
123
};
Assert.Equal(1, t.Count);
Assert.True(t.Contains(1));
}
}

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@ -0,0 +1 @@
global using Xunit;

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@ -0,0 +1,8 @@
namespace avl.test;
public class Node {
[Fact]
public void Test1() {
}
}