Know Thy Complexities!

Hi there! This webpage covers the space and time Big-O complexities of common algorithms used in Computer Science. When preparing for technical interviews in the past, I found myself spending hours crawling the internet putting together the best, average, and worst case complexities for search and sorting algorithms so that I wouldn't be stumped when asked about them. Over the last few years, I've interviewed at several Silicon Valley startups, and also some bigger companies, like Yahoo, eBay, LinkedIn, and Google, and each time that I prepared for an interview, I thought to myself "Why hasn't someone created a nice Big-O cheat sheet?". So, to save all of you fine folks a ton of time, I went ahead and created one. Enjoy! - Eric

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Excellent Good Fair Bad Horrible

Data Structure Operations

Data Structure	Time Complexity								Space Complexity
	Average				Worst				Worst
	Access	Search	Insertion	Deletion	Access	Search	Insertion	Deletion
Array	`O(1)`	`O(n)`	`O(n)`	`O(n)`	`O(1)`	`O(n)`	`O(n)`	`O(n)`	`O(n)`
Stack	`O(n)`	`O(n)`	`O(1)`	`O(1)`	`O(n)`	`O(n)`	`O(1)`	`O(1)`	`O(n)`
Singly-Linked List	`O(n)`	`O(n)`	`O(1)`	`O(1)`	`O(n)`	`O(n)`	`O(1)`	`O(1)`	`O(n)`
Doubly-Linked List	`O(n)`	`O(n)`	`O(1)`	`O(1)`	`O(n)`	`O(n)`	`O(1)`	`O(1)`	`O(n)`
Skip List	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(n)`	`O(n)`	`O(n)`	`O(n)`	`O(n log(n))`
Hash Table	`-`	`O(1)`	`O(1)`	`O(1)`	`-`	`O(n)`	`O(n)`	`O(n)`	`O(n)`
Binary Search Tree	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(n)`	`O(n)`	`O(n)`	`O(n)`	`O(n)`
Cartesian Tree	`-`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`-`	`O(n)`	`O(n)`	`O(n)`	`O(n)`
B-Tree	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(n)`
Red-Black Tree	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(n)`
Splay Tree	`-`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`-`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(n)`
AVL Tree	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(n)`

Array Sorting Algorithms

Algorithm	Time Complexity			Space Complexity
	Best	Average	Worst	Worst
Quicksort	`O(n log(n))`	`O(n log(n))`	`O(n^2)`	`O(log(n))`
Mergesort	`O(n log(n))`	`O(n log(n))`	`O(n log(n))`	`O(n)`
Timsort	`O(n)`	`O(n log(n))`	`O(n log(n))`	`O(n)`
Heapsort	`O(n log(n))`	`O(n log(n))`	`O(n log(n))`	`O(1)`
Bubble Sort	`O(n)`	`O(n^2)`	`O(n^2)`	`O(1)`
Insertion Sort	`O(n)`	`O(n^2)`	`O(n^2)`	`O(1)`
Selection Sort	`O(n^2)`	`O(n^2)`	`O(n^2)`	`O(1)`
Shell Sort	`O(n)`	`O((nlog(n))^2)`	`O((nlog(n))^2)`	`O(1)`
Bucket Sort	`O(n+k)`	`O(n+k)`	`O(n^2)`	`O(n)`
Radix Sort	`O(nk)`	`O(nk)`	`O(nk)`	`O(n+k)`

Graph Operations

Node / Edge Management	Storage	Add Vertex	Add Edge	Remove Vertex	Remove Edge	Query
Adjacency list	`O(\|V\|+\|E\|)`	`O(1)`	`O(1)`	`O(\|V\| + \|E\|)`	`O(\|E\|)`	`O(\|V\|)`
Incidence list	`O(\|V\|+\|E\|)`	`O(1)`	`O(1)`	`O(\|E\|)`	`O(\|E\|)`	`O(\|E\|)`
Adjacency matrix	`O(\|V\|^2)`	`O(\|V\|^2)`	`O(1)`	`O(\|V\|^2)`	`O(1)`	`O(1)`
Incidence matrix	`O(\|V\| ⋅ \|E\|)`	`O(\|V\| ⋅ \|E\|)`	`O(\|V\| ⋅ \|E\|)`	`O(\|V\| ⋅ \|E\|)`	`O(\|V\| ⋅ \|E\|)`	`O(\|E\|)`

Heap Operations

Type	Time Complexity
	Heapify	Find Max	Extract Max	Increase Key	Insert	Delete	Merge
Linked List (sorted)	`-`	`O(1)`	`O(1)`	`O(n)`	`O(n)`	`O(1)`	`O(m+n)`
Linked List (unsorted)	`-`	`O(n)`	`O(n)`	`O(1)`	`O(1)`	`O(1)`	`O(1)`
Binary Heap	`O(n)`	`O(1)`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(log(n))`	`O(m+n)`
Binomial Heap	`-`	`O(1)`	`O(log(n))`	`O(log(n))`	`O(1)`	`O(log(n))`	`O(log(n))`
Fibonacci Heap	`-`	`O(1)`	`O(log(n))`	`O(1)`	`O(1)`	`O(log(n))`	`O(1)`

Big-O Complexity Chart

Contributors

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