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 Google, Facebook, Yahoo, LinkedIn, and eBay, 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

If you're trying to catch them all, you might also check out the Pokemon Go Evolution Chart.

Big-O Complexity Chart

Excellent Good Fair Bad Horrible
O(1), O(log n) O(n) O(n log n) O(n^2) O(2^n) O(n!) Operations Elements

Common 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)
Queue 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 N/A O(1) O(1) O(1) N/A 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 N/A O(log(n)) O(log(n)) O(log(n)) N/A 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 N/A O(log(n)) O(log(n)) O(log(n)) N/A 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)
KD Tree O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(n) O(n) O(n) O(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)
Tree Sort O(n log(n)) O(n log(n)) O(n^2) O(n)
Shell Sort O(n log(n)) O(n(log(n))^2) O(n(log(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)
Counting Sort O(n+k) O(n+k) O(n+k) O(k)
Cubesort O(n) O(n log(n)) O(n log(n)) O(n)

Graph Data Structure Operations

Data Structure Time Complexity
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 Data Structure Operations

Data Structure Time Complexity
Find Max Extract Max Increase Key Insert Delete Merge
Binary Heap O(1) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(m+n)
Pairing Heap O(1) O(log(n)) O(log(n)) O(1) O(log(n)) O(1)
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)

Graph Algorithms

Algorithm Time Complexity Space Complexity
Average Worst Worst
Dijkstra's algorithm O(|E| log |V|) O(|V|^2) O(|V| + |E|)
A* search algorithm O(|E|) O(b^d) O(b^d)
Prim's algorithm O(|E| log |V|) O(|V|^2) O(|V| + |E|)
Bellman–Ford algorithm O(|E| ⋅ |V|) O(|E| ⋅ |V|) O(|V|)
Floyd-Warshall algorithm O(|V|^3) O(|V|^3) O(|V|^2)
Topological sort O(|V| + |E|) O(|V| + |E|) O(|V| + |E|)

Learn More

Contributors

  1. Eric Rowell
  2. Quentin Pleple
  3. Michael Abed
  4. Nick Dizazzo
  5. Adam Forsyth
  6. Felix Zhu
  7. Jay Engineer
  8. Josh Davis
  9. Nodir Turakulov
  10. Jennifer Hamon
  11. David Dorfman
  12. Bart Massey
  13. Ray Pereda
  14. Si Pham
  15. Mike Davis
  16. mcverry
  17. Max Hoffmann
  18. Bahador Saket
  19. Damon Davison
  20. Alvin Wan
  21. Alan Briolat
  22. Drew Hannay
  23. Andrew Rasmussen
  24. Dennis Tsang
  25. Vinnie Magro
  26. Adam Arold
  27. Alejandro Ramirez
  28. Aneel Nazareth
  29. Rahul Chowdhury
  30. Jonathan McElroy
  31. steven41292
  32. Brandon Amos
  33. Joel Friedly
  34. Casper Van Gheluwe
  35. Eric Lefevre-Ardant
  36. Oleg
  37. Renfred Harper
  38. Piper Chester
  39. Miguel Amigot
  40. Apurva K
  41. Matthew Daronco
  42. Yun-Cheng Lin
  43. Clay Tyler
  44. Orhan Can Ozalp
  45. Ayman Singh
  46. David Morton
  47. Aurelien Ooms
  48. Sebastian Paaske Torholm
  49. Koushik Krishnan
  50. Drew Bailey
  51. Robert Burke

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