javascript-algorithms Documentation

Algorithms and data structures implemented in JavaScript with explanations and links to further readings

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1. Project Overview & Quickstart (trekhleb/javascript-algorithms)

File: README.md

JavaScript Algorithms and Data Structures

🇺🇦 UKRAINE IS BEING ATTACKED BY RUSSIAN ARMY. CIVILIANS ARE GETTING KILLED. RESIDENTIAL AREAS ARE GETTING BOMBED.


This repository contains JavaScript based examples of many
popular algorithms and data structures.

Each algorithm and data structure has its own separate README
with related explanations and links for further reading (including ones
to YouTube videos).

Read this in other languages:
简体中文,
繁體中文,
한국어,
日本語,
Polski,
Français,
Español,
Português,
Русский,
Türkçe,
Italiano,
Bahasa Indonesia,
Українська,
Arabic,
Tiếng Việt,
Deutsch,
Uzbek,
עברית

Data Structures

A data structure is a particular way of organizing and storing data in a computer so that it can
be accessed and modified efficiently. More precisely, a data structure is a collection of data
values, the relationships among them, and the functions or operations that can be applied to
the data.

Remember that each data has its own trade-offs. And you need to pay attention more to why you're choosing a certain data structure than to how to implement it.

B - Beginner, A - Advanced

Algorithms

An algorithm is an unambiguous specification of how to solve a class of problems. It is
a set of rules that precisely define a sequence of operations.

B - Beginner, A - Advanced

Algorithms by Topic

Algorithms by Paradigm

An algorithmic paradigm is a generic method or approach which underlies the design of a class
of algorithms. It is an abstraction higher than the notion of an algorithm, just as an
algorithm is an abstraction higher than a computer program.

How to use this repository

Install all dependencies

text
npm install

Run ESLint

You may want to run it to check code quality.

text
npm run lint

Run all tests

text
npm test

Run tests by name

text
npm test -- 'LinkedList'

Troubleshooting

If linting or testing is failing, try to delete the node_modules folder and re-install npm packages:

text
rm -rf ./node_modules
npm i

Also, make sure that you're using the correct Node version (>=16). If you're using nvm for Node version management you may run nvm use from the root folder of the project and the correct version will be picked up.

Playground

You may play with data-structures and algorithms in ./src/playground/playground.js file and write
tests for it in ./src/playground/__test__/playground.test.js.

Then just, simply run the following command to test if your playground code works as expected:

text
npm test -- 'playground'

Useful Information

References

Big O Notation

Big O notation is used to classify algorithms according to how their running time or space requirements grow as the input size grows.
On the chart below, you may find the most common orders of growth of algorithms specified in Big O notation.

Source: Big O Cheat Sheet.

Below is the list of some of the most used Big O notations and their performance comparisons against different sizes of the input data.

Big O Notation Type Computations for 10 elements Computations for 100 elements Computations for 1000 elements
O(1) Constant 1 1 1
O(log N) Logarithmic 3 6 9
O(N) Linear 10 100 1000
O(N log N) n log(n) 30 600 9000
O(N^2) Quadratic 100 10000 1000000
O(2^N) Exponential 1024 1.26e+29 1.07e+301
O(N!) Factorial 3628800 9.3e+157 4.02e+2567

Data Structure Operations Complexity

Data Structure Access Search Insertion Deletion Comments
Array 1 n n n
Stack n n 1 1
Queue n n 1 1
Linked List n n 1 n
Hash Table - n n n In case of perfect hash function costs would be O(1)
Binary Search Tree n n n n In case of balanced tree costs would be O(log(n))
B-Tree log(n) log(n) log(n) log(n)
Red-Black Tree log(n) log(n) log(n) log(n)
AVL Tree log(n) log(n) log(n) log(n)
Bloom Filter - 1 1 - False positives are possible while searching

Array Sorting Algorithms Complexity

Name Best Average Worst Memory Stable Comments
Bubble sort n n2 n2 1 Yes
Insertion sort n n2 n2 1 Yes
Selection sort n2 n2 n2 1 No
Heap sort n log(n) n log(n) n log(n) 1 No
Merge sort n log(n) n log(n) n log(n) n Yes
Quick sort n log(n) n log(n) n2 log(n) No Quicksort is usually done in-place with O(log(n)) stack space
Shell sort n log(n) depends on gap sequence n (log(n))2 1 No
Counting sort n + r n + r n + r n + r Yes r - biggest number in array
Radix sort n * k n * k n * k n + k Yes k - length of longest key

Author

@trekhleb

A few more projects and articles about JavaScript and algorithms on trekhleb.dev:

  • 🧠 yesbrainer.ai – council of AI models for the decisions that aren’t no-brainers (BYOK, private, no account)
  • ✍🏻 okso.app – drawing app to express, grasp, and organize your thoughts and ideas

File: src/playground/README.md

Playground

You may use playground.js file to play with data
structures and algorithms. The code from playground.js may
be tested in ./__test__/playground.test.js file.

To run tests simply run:

text
npm test -- -t 'playground'

File: src/data-structures/trie/README.ko-KO.md

Trie

Read this in other languages:
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Русский,
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한국어

컴퓨터 과학에서 트라이는 디지털 트리라고도 불리며 때로는 기수 트리 또는 접두사 트리(접두사로 검색할 수 있기 때문에)라고도 불리며 일종의 검색 트리입니다. 키가 보통 문자열인 동적 집합 또는 연관 배열을 저장하는 데 사용되는 순서가 지정된 트리 데이터 구조입니다. 이진 검색 트리와 달리 트리의 어떤 노드도 해당 노드와 연결된 키를 저장하지 않으며 대신 트리의 위치가 해당 노드와 연결된 키를 정의합니다. 노드의 모든 하위 항목은 해당 노드와 연결된 문자열의 공통 접두사를 가지며 루트는 빈 문자열과 연결됩니다. 값은 모든 노드와 반드시 연결되지는 않습니다. 오히려 값은 나뭇잎과 관심 있는 키에 해당하는 일부 내부 노드에만 연결되는 경향이 있습니다. 접두사 트리의 공간에 최적화된 표현은 콤팩트 접두사 트리를 참조하십시오.

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참조

File: src/data-structures/trie/README.md

Trie

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Русский,
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한국어

In computer science, a trie, also called digital tree and sometimes
radix tree or prefix tree (as they can be searched by prefixes),
is a kind of search tree—an ordered tree data structure that is
used to store a dynamic set or associative array where the keys
are usually strings. Unlike a binary search tree, no node in the
tree stores the key associated with that node; instead, its
position in the tree defines the key with which it is associated.
All the descendants of a node have a common prefix of the string
associated with that node, and the root is associated with the
empty string. Values are not necessarily associated with every
node. Rather, values tend only to be associated with leaves,
and with some inner nodes that correspond to keys of interest.
For the space-optimized presentation of prefix tree, see compact
prefix tree.

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References

File: src/data-structures/trie/README.pt-BR.md

Árvore de Prefixos (Trie)

Na ciência da computação, uma trie, também chamada de árvore digital (digital tree)
e algumas vezes de radix tree ou prefix tree (tendo em vista que eles
podem ser pesquisados por prefixos), é um tipo de árvore de pesquisa, uma
estrutura de dados de árvore ordenada que é usado para armazenar um
conjunto dinâmico ou matriz associativa onde as chaves são geralmente strings.
Ao contrário de uma árvore de pesquisa binária (binary search tree),
nenhum nó na árvore armazena a chave associada a esse nó; em vez disso,
sua posição na árvore define a chave com a qual ela está associada.
Todos os descendentes de um nó possuem em comum o prefixo de uma string
associada com aquele nó, e a raiz é associada com uma string vazia.
Valores não são necessariamente associados a todos nós. Em vez disso,
os valores tendem a ser associados apenas a folhas e com alguns nós
internos que correspondem a chaves de interesse.

Para a apresentação otimizada do espaço da árvore de prefixo (prefix tree),
veja árvore de prefixo compacto.

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Referências

File: src/data-structures/trie/README.ru-RU.md

Префиксное дерево

Префиксное дерево (также бор, луч, нагруженное или суффиксное дерево) в информатике - упорядоченная древовидная
структура данных, которая используется для хранения динамических множеств или ассоциативных массивов, где
ключом обычно выступают строки. Дерево называется префиксным, потому что поиск осуществляется по префиксам.

В отличие от бинарного дерева, узлы не содержат ключи, соответствующие узлу. Представляет собой корневое дерево, каждое
ребро которого помечено каким-то символом так, что для любого узла все рёбра, соединяющие этот узел с его сыновьями,
помечены разными символами. Некоторые узлы префиксного дерева выделены (на рисунке они подписаны цифрами) и считается,
что префиксное дерево содержит данную строку-ключ тогда и только тогда, когда эту строку можно прочитать на пути из
корня до некоторого выделенного узла.

Таким образом, в отличие от бинарных деревьев поиска, ключ, идентифицирующий конкретный узел дерева, не явно хранится в
данном узле, а неявно задаётся положением данного узла в дереве. Получить ключ можно выписыванием подряд символов,
помечающих рёбра на пути от корня до узла. Ключ корня дерева — пустая строка. Часто в выделенных узлах хранят
дополнительную информацию, связанную с ключом, и обычно выделенными являются только листья и, возможно, некоторые
внутренние узлы.

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На рисунке представлено префиксное дерево, содержащее ключи «A», «to», «tea», «ted», «ten», «i», «in», «inn».

Ссылки

File: src/data-structures/trie/README.uk-UA.md

Префіксне дерево

Префіксне дерево (Також промінь, навантажене або суфіксне дерево) в інформатиці - впорядкована деревоподібна
структура даних, яка використовується для зберігання динамічних множин або асоціативних масивів, де
ключем зазвичай виступають рядки. Дерево називається префіксним, тому що пошук здійснюється за префіксами.

На відміну від бінарного дерева, вузли не містять ключів, що відповідають вузлу. Являє собою кореневе дерево, кожне
ребро якого позначено якимось символом так, що для будь-якого вузла всі ребра, що з'єднують цей вузол з його синами,
позначені різними символами. Деякі вузли префіксного дерева виділені (на малюнку вони підписані цифрами) і вважається,
що префіксне дерево містить цей рядок-ключ тоді і тільки тоді, коли цей рядок можна прочитати на шляху з
кореня до певного виділеного вузла.

Таким чином, на відміну від бінарних дерев пошуку, ключ, що ідентифікує конкретний вузол дерева, не явно зберігається в
цьому вузлі, а неявно задається положенням цього вузла в дереві. Отримати ключ можна виписуванням поспіль символів,
помічають ребра по дорозі від кореня до вузла. Ключ кореня дерева - порожній рядок. Часто у виділених вузлах зберігають
додаткову інформацію, пов'язану з ключем, і зазвичай виділеними є тільки листя і, можливо, деякі
внутрішні вузли.

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На малюнку представлено префіксне дерево, що містить ключі. «A», «to», «tea», «ted», «ten», «i», «in», «inn».

Посилання

File: src/data-structures/trie/README.zh-CN.md

字典树

在计算机科学中, 字典树(trie,中文又被称为”单词查找树“或 ”键树“), 也称为数字树,有时候也被称为基数树或前缀树（因为它们可以通过前缀搜索）,它是一种搜索树--一种已排序的数据结构,通常用于存储动态集或键为字符串的关联数组。

与二叉搜索树不同, 树上没有节点存储与该节点关联的键; 相反,节点在树上的位置定义了与之关联的键。一个节点的全部后代节点都有一个与该节点关联的通用的字符串前缀, 与根节点关联的是空字符串。

值对于字典树中关联的节点来说,不是必需的,相反,值往往和相关的叶子相关,以及与一些键相关的内部节点相关。

有关字典树的空间优化示意,请参阅紧凑前缀树

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参考

File: src/data-structures/tree/binary-search-tree/README.md

Binary Search Tree

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Português

In computer science, binary search trees (BST), sometimes called
ordered or sorted binary trees, are a particular type of container:
data structures that store "items" (such as numbers, names etc.)
in memory. They allow fast lookup, addition and removal of
items, and can be used to implement either dynamic sets of
items, or lookup tables that allow finding an item by its key
(e.g., finding the phone number of a person by name).

Binary search trees keep their keys in sorted order, so that lookup
and other operations can use the principle of binary search:
when looking for a key in a tree (or a place to insert a new key),
they traverse the tree from root to leaf, making comparisons to
keys stored in the nodes of the tree and deciding, on the basis
of the comparison, to continue searching in the left or right
subtrees. On average, this means that each comparison allows
the operations to skip about half of the tree, so that each
lookup, insertion or deletion takes time proportional to the
logarithm of the number of items stored in the tree. This is
much better than the linear time required to find items by key
in an (unsorted) array, but slower than the corresponding
operations on hash tables.

A binary search tree of size 9 and depth 3, with 8 at the root.
The leaves are not drawn.

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Pseudocode for Basic Operations

Insertion

text
insert(value)
  Pre: value has passed custom type checks for type T
  Post: value has been placed in the correct location in the tree
  if root = ø
    root ← node(value)
  else
    insertNode(root, value)
  end if
end insert
text
insertNode(current, value)
  Pre: current is the node to start from
  Post: value has been placed in the correct location in the tree
  if value < current.value
    if current.left = ø
      current.left ← node(value)
    else
      InsertNode(current.left, value)
    end if
  else
    if current.right = ø
      current.right ← node(value)
    else
      InsertNode(current.right, value)
    end if
  end if
end insertNode

Searching

text
contains(root, value)
  Pre: root is the root node of the tree, value is what we would like to locate
  Post: value is either located or not
  if root = ø
    return false
  end if
  if root.value = value
    return true
  else if value < root.value
    return contains(root.left, value)
  else
    return contains(root.right, value)
  end if
end contains

Deletion

text
remove(value)
  Pre: value is the value of the node to remove, root is the node of the BST
      count is the number of items in the BST
  Post: node with value is removed if found in which case yields true, otherwise false
  nodeToRemove ← findNode(value)
  if nodeToRemove = ø
    return false
  end if
  parent ← findParent(value)
  if count = 1
    root ← ø
  else if nodeToRemove.left = ø and nodeToRemove.right = ø
    if nodeToRemove.value < parent.value
      parent.left ←  nodeToRemove.right
    else
      parent.right ← nodeToRemove.right
    end if
  else if nodeToRemove.left != ø and nodeToRemove.right != ø
    next ← nodeToRemove.right
    while next.left != ø
      next ← next.left
    end while
    if next != nodeToRemove.right
      remove(next.value)
      nodeToRemove.value ← next.value
    else
      nodeToRemove.value ← next.value
      nodeToRemove.right ← nodeToRemove.right.right
    end if
  else
    if nodeToRemove.left = ø
      next ← nodeToRemove.right
    else
      next ← nodeToRemove.left
    end if
    if root = nodeToRemove
      root = next
    else if parent.left = nodeToRemove
      parent.left = next
    else if parent.right = nodeToRemove
      parent.right = next
    end if
  end if
  count ← count - 1
  return true
end remove

Find Parent of Node

text
findParent(value, root)
  Pre: value is the value of the node we want to find the parent of
       root is the root node of the BST and is != ø
  Post: a reference to the prent node of value if found; otherwise ø
  if value = root.value
    return ø
  end if
  if value < root.value
    if root.left = ø
      return ø
    else if root.left.value = value
      return root
    else
      return findParent(value, root.left)
    end if
  else
    if root.right = ø
      return ø
    else if root.right.value = value
      return root
    else
      return findParent(value, root.right)
    end if
  end if
end findParent

Find Node

text
findNode(root, value)
  Pre: value is the value of the node we want to find the parent of
       root is the root node of the BST
  Post: a reference to the node of value if found; otherwise ø
  if root = ø
    return ø
  end if
  if root.value = value
    return root
  else if value < root.value
    return findNode(root.left, value)
  else
    return findNode(root.right, value)
  end if
end findNode

Find Minimum

text
findMin(root)
  Pre: root is the root node of the BST
    root = ø
  Post: the smallest value in the BST is located
  if root.left = ø
    return root.value
  end if
  findMin(root.left)
end findMin

Find Maximum

text
findMax(root)
  Pre: root is the root node of the BST
    root = ø
  Post: the largest value in the BST is located
  if root.right = ø
    return root.value
  end if
  findMax(root.right)
end findMax

Traversal

InOrder Traversal

text
inorder(root)
  Pre: root is the root node of the BST
  Post: the nodes in the BST have been visited in inorder
  if root != ø
    inorder(root.left)
    yield root.value
    inorder(root.right)
  end if
end inorder

PreOrder Traversal

text
preorder(root)
  Pre: root is the root node of the BST
  Post: the nodes in the BST have been visited in preorder
  if root != ø
    yield root.value
    preorder(root.left)
    preorder(root.right)
  end if
end preorder

PostOrder Traversal

text
postorder(root)
  Pre: root is the root node of the BST
  Post: the nodes in the BST have been visited in postorder
  if root != ø
    postorder(root.left)
    postorder(root.right)
    yield root.value
  end if
end postorder

Complexities

Time Complexity

Access Search Insertion Deletion
O(log(n)) O(log(n)) O(log(n)) O(log(n))

Space Complexity

O(n)

References

File: src/data-structures/tree/binary-search-tree/README.pt-BR.md

Árvore de Pesquisa Binária (Binary Search Tree)

Na ciência da computação binary search trees (BST), algumas vezes
chamadas de árvores binárias ordenadas (ordered or sorted binary trees),
é um tipo particular de container: estruturas de dados que armazenam
"itens" (como números, nomes, etc.) na memória. Permite pesquisa rápida,
adição e remoção de itens além de poder ser utilizado para implementar
tanto conjuntos dinâmicos de itens ou, consultar tabelas que permitem
encontrar um item por seu valor chave. E.g. encontrar o número de
telefone de uma pessoa pelo seu nome.

Árvore de Pesquisa Binária mantem seus valores chaves ordenados, para
que uma pesquisa e outras operações possam usar o princípio da pesquisa
binária: quando pesquisando por um valor chave na árvore (ou um lugar
para inserir uma nova chave), eles atravessam a árvore da raiz para a folha,
fazendo comparações com chaves armazenadas nos nós da árvore e decidindo então,
com base nas comparações, continuar pesquisando nas sub-árvores a direita ou
a esquerda. Em média isto significa que cara comparação permite as operações
pular metade da árvore, para que então, cada pesquisa, inserção ou remoção
consuma tempo proporcional ao logaritmo do número de itens armazenados na
árvore. Isto é muito melhor do que um tempo linear necessário para encontrar
itens por seu valor chave em um array (desorndenado - unsorted), mas muito
mais lento do que operações similares em tableas de hash (hash tables).

Uma pesquisa de árvore binária de tamanho 9 e profundidade 3, com valor 8
na raíz.
As folhas não foram desenhadas.

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Pseudocódigo para Operações Básicas

Inserção

text
insert(value)
  Pre: value has passed custom type checks for type T
  Post: value has been placed in the correct location in the tree
  if root = ø
    root ← node(value)
  else
    insertNode(root, value)
  end if
end insert
text
insertNode(current, value)
  Pre: current is the node to start from
  Post: value has been placed in the correct location in the tree
  if value < current.value
    if current.left = ø
      current.left ← node(value)
    else
      InsertNode(current.left, value)
    end if
  else
    if current.right = ø
      current.right ← node(value)
    else
      InsertNode(current.right, value)
    end if
  end if
end insertNode

Pesquisa

text
contains(root, value)
  Pre: root is the root node of the tree, value is what we would like to locate
  Post: value is either located or not
  if root = ø
    return false
  end if
  if root.value = value
    return true
  else if value < root.value
    return contains(root.left, value)
  else
    return contains(root.right, value)
  end if
end contains

Remoção

text
remove(value)
  Pre: value is the value of the node to remove, root is the node of the BST
      count is the number of items in the BST
  Post: node with value is removed if found in which case yields true, otherwise false
  nodeToRemove ← findNode(value)
  if nodeToRemove = ø
    return false
  end if
  parent ← findParent(value)
  if count = 1
    root ← ø
  else if nodeToRemove.left = ø and nodeToRemove.right = ø
    if nodeToRemove.value < parent.value
      parent.left ←  nodeToRemove.right
    else
      parent.right ← nodeToRemove.right
    end if
  else if nodeToRemove.left != ø and nodeToRemove.right != ø
    next ← nodeToRemove.right
    while next.left != ø
      next ← next.left
    end while
    if next != nodeToRemove.right
      remove(next.value)
      nodeToRemove.value ← next.value
    else
      nodeToRemove.value ← next.value
      nodeToRemove.right ← nodeToRemove.right.right
    end if
  else
    if nodeToRemove.left = ø
      next ← nodeToRemove.right
    else
      next ← nodeToRemove.left
    end if
    if root = nodeToRemove
      root = next
    else if parent.left = nodeToRemove
      parent.left = next
    else if parent.right = nodeToRemove
      parent.right = next
    end if
  end if
  count ← count - 1
  return true
end remove

Encontrar o Nó Pai

text
findParent(value, root)
  Pre: value is the value of the node we want to find the parent of
       root is the root node of the BST and is != ø
  Post: a reference to the prent node of value if found; otherwise ø
  if value = root.value
    return ø
  end if
  if value < root.value
    if root.left = ø
      return ø
    else if root.left.value = value
      return root
    else
      return findParent(value, root.left)
    end if
  else
    if root.right = ø
      return ø
    else if root.right.value = value
      return root
    else
      return findParent(value, root.right)
    end if
  end if
end findParent

Encontrar um Nó

text
findNode(root, value)
  Pre: value is the value of the node we want to find the parent of
       root is the root node of the BST
  Post: a reference to the node of value if found; otherwise ø
  if root = ø
    return ø
  end if
  if root.value = value
    return root
  else if value < root.value
    return findNode(root.left, value)
  else
    return findNode(root.right, value)
  end if
end findNode

Encontrar Mínimo

text
findMin(root)
  Pre: root is the root node of the BST
    root = ø
  Post: the smallest value in the BST is located
  if root.left = ø
    return root.value
  end if
  findMin(root.left)
end findMin

Encontrar Máximo

text
findMax(root)
  Pre: root is the root node of the BST
    root = ø
  Post: the largest value in the BST is located
  if root.right = ø
    return root.value
  end if
  findMax(root.right)
end findMax

Traversal

Na Ordem Traversal (InOrder Traversal)

text
inorder(root)
  Pre: root is the root node of the BST
  Post: the nodes in the BST have been visited in inorder
  if root = ø
    inorder(root.left)
    yield root.value
    inorder(root.right)
  end if
end inorder

Pré Ordem Traversal (PreOrder Traversal)

text
preorder(root)
  Pre: root is the root node of the BST
  Post: the nodes in the BST have been visited in preorder
  if root = ø
    yield root.value
    preorder(root.left)
    preorder(root.right)
  end if
end preorder

Pós Ordem Traversal (PostOrder Traversal)

text
postorder(root)
  Pre: root is the root node of the BST
  Post: the nodes in the BST have been visited in postorder
  if root = ø
    postorder(root.left)
    postorder(root.right)
    yield root.value
  end if
end postorder

Complexidades

Complexidade de Tempo

Access Search Insertion Deletion
O(log(n)) O(log(n)) O(log(n)) O(log(n))

Complexidade de Espaço

O(n)

Referências

2. Official Technical Reference & Guides (trekhleb/trekhleb.github.io)

trekhleb.dev

My personal website with a list of my projects that help people learn and blog posts about life, web-development, and machine-learning.

Development

The website is built on Gatsby. See the Gatsby cheat-sheet for development hints.

Running the project locally

Install proper version of Node:

bash
nvm use

Install dependencies:

bash
npm i

To run the project locally in development mode on http://localhost:8000:

bash
npm run develop

View GraphiQL, an in-browser IDE, to explore your site's data and schema on http://localhost:8000/___graphql

To build the production version of the project:

bash
npm run build

Serve the production build for testing on http://localhost:9000:

bash
npm run serve

To do ESLint and Type checking:

bash
npm run lint
npm run type

To fix the Lint errors run:

bash
npm run lint -- --fix

Working with icons

Use react-icons.github.io to search for available icons.

Styling components

Use tailwindcss.com to search for available CSS classes.

Custom Domain Support

To serve the trekhleb.github.io web-site on a custom domain trekhleb.dev make sure that the public branch has CNAME file in the root folder. For more details read this.

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