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 oh 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!

Good Fair Poor

Searching

Algorithm Data Structure Time Complexity Space Complexity
Average Worst Worst
Depth First Search (DFS) Graph of |V| vertices and |E| edges - O(|E| + |V|) O(|V|)
Breadth First Search (BFS) Graph of |V| vertices and |E| edges - O(|E| + |V|) O(|V|)
Binary search Sorted array of n elements O(log(n)) O(log(n)) O(1)
Linear (Brute Force) Array O(n) O(n) O(1)
Shortest path by Dijkstra,
using a Min-heap as priority queue
Graph with |V| vertices and |E| edges O((|V| + |E|) log |V|) O((|V| + |E|) log |V|) O(|V|)
Shortest path by Dijkstra,
using an unsorted array as priority queue
Graph with |V| vertices and |E| edges O(|V|^2) O(|V|^2) O(|V|)
Shortest path by Bellman-Ford Graph with |V| vertices and |E| edges O(|V||E|) O(|V||E|) O(|V|)

Sorting

Algorithm Data Structure Time Complexity Worst Case Auxiliary Space Complexity
Best Average Worst Worst
Quicksort Array O(n log(n)) O(n log(n)) O(n^2) O(n)
Mergesort Array O(n log(n)) O(n log(n)) O(n log(n)) O(n)
Heapsort Array O(n log(n)) O(n log(n)) O(n log(n)) O(1)
Bubble Sort Array O(n) O(n^2) O(n^2) O(1)
Insertion Sort Array O(n) O(n^2) O(n^2) O(1)
Select Sort Array O(n^2) O(n^2) O(n^2) O(1)
Bucket Sort Array O(n+k) O(n+k) O(n^2) O(nk)
Radix Sort Array O(nk) O(nk) O(nk) O(n+k)

Data Structures

Data Structure Time Complexity Space Complexity
Average Worst Worst
Indexing Search Insertion Deletion Indexing Search Insertion Deletion
Basic Array O(1) O(n) - - O(1) O(n) - - O(n)
Dynamic Array O(1) O(n) O(n) O(n) O(1) O(n) O(n) O(n) 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)
Cartresian 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)

Heaps

Heaps 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(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n))
Fibonacci Heap - O(1) O(log(n))* O(1)* O(1) O(log(n))* O(1)

Graphs

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|)

Big-O Complexity Chart

Big O Complexity Graph
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