A Blog about Programming,Algorithms,Coding Competitions, and Discrete Mathematics!
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. 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
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|) |
Notation for asymptotic growth
| letter | bound | growth |
|---|---|---|
| (theta) Θ | upper and lower, tight | equal |
| (big-oh) O | upper, tightness unknown | less than or equal |
| (small-oh) o | upper, not tight | less than |
| (big omega) Ω | lower, tightness unknown | greater than or equal |
| (small omega) ω | lower, not tight | greater than |
[1] Big O is the upper bound, while Omega is the
lower bound. Theta requires both Big O and Omega, so that's why it's
referred to as a tight bound (it must be both the upper and lower
bound). For example, an algorithm taking Omega(n log n) takes at least n
log n time but has no upper limit. An algorithm taking Theta(n log n)
is far preferential since it takes AT LEAST n log n (Omega n log n) and
NO MORE THAN n log n (Big O n log n). SO
[2] f(x)=Θ(g(n)) means f (the running time of the
algorithm) grows exactly like g when n (input size) gets larger. In
other words, the growth rate of f(x) is asymptotically proportional to
g(n).
[3] Same thing. Here the growth rate is no faster
than g(n). big-oh is the most useful because represents the worst-case
behavior.
In short, if algorithm is __ then its performance is __ | algorithm | performance |
|---|---|
| o(n) | < n |
| O(n) | ≤ n |
| Θ(n) | = n |
| Ω(n) | ≥ n |
| ω(n) | > n |
Will edit this post soon to make it more easily readable...
Blog Post Credits - Pawan Tiwari(Microsoft SDET US)
