Algorithm complexity
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 Minheap 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 BellmanFord  Graph with V vertices and E edges  O(VE) 
O(VE) 
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) 
SinglyLinked List  O(n) 
O(n) 
O(1) 
O(1) 
O(n) 
O(n) 
O(1) 
O(1) 
O(n) 
DoublyLinked 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) 
BTree  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) 
RedBlack 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 
(bigoh) O  upper, tightness unknown  less than or equal 
(smalloh) 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). bigoh is the most useful because represents the worstcase 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 
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