Sorting algorithms are a fundamental part of computer science. Being able to sort through a large data set quickly and efficiently is a problem you will be likely to encounter on nearly a daily basis.
Time Cheat Sheet
Time complexity Cheat Sheet. BigO Graph.Correction:- Best time complexity for TIM SORT is O(nlogn) Tweet. Software Development Engineer at Amazon. AlgorithmicComplexity.com Cheat Sheet Algorithms Sorting Algorithms: 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) Merge Sort Array O(n log(n)) O(n log(n)) O(n log(n)) O(n) Heapsort. Big-O Cheat Sheet Generated December 10, 2013. Algorithm Data Structure Time Complexity Space Complexity Average Worst Depth First Search (DFS) Graph of jVj. Time Complexity of Algorithm De nition Time Complexity of Algorithmis the number of dominating operations executed by the algorithm as the function of data size. Time complexity measures the amount of work done by the algorithm during solving the problem in the way which is independent on the implementation and particular input data.
Here are the main sorting algorithms:
| Algorithm | Data Structure | Time Complexity - Best | Time Complexity - Average | Time Complexity - Worst | Worst Case Auxiliary Space Complexity |
|---|---|---|---|---|---|
| Quicksort | Array | O(n log(n)) | O(n log(n)) | O(n^2) | O(n) |
| Merge Sort | 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) |
Another crucial skill to master in the field of computer science is how to search for an item in a collection of data quickly. Here are the most common searching algorithms, their corresponding data structures, and time complexities.
Here are the main searching algorithms:
| Algorithm | Data Structure | Time Complexity - Average | Time Complexity - Worst | Space Complexity - Worst |
|---|---|---|---|---|
| Depth First Search | Graph of |V| vertices and |E| edges | - | O(|E|+|V|) | O(|V|) |
| Breadth First Search | 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) |
| Brute Force | Array | O(n) | O(n) | O(1) |
| Bellman-Ford | Graph of |V| vertices and |E| edges | O(|V||E|) | O(|V||E|) | O(|V|) |
Graphs are an integral part of computer science. Mastering them is necessary to become an accomplished software developer. Here is the data structure analysis of 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|) |
Storing information in a way that is quick to retrieve, add, and search on, is a very important technique to master. Here is what you need to know about heap data structures:
| Heaps | Heapify | Find Max | Extract Max | Increase Key | Insert | Delete | Merge |
|---|---|---|---|---|---|---|---|
| Sorted Linked List | - | O(1) | O(1) | O(n) | O(n) | O(1) | O(m+n) |
| Unsorted Linked List | - | 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) |
| Sorting Algorithms | Space complexity | Time complexity | ||
|---|---|---|---|---|
| Worst case | Best case | Average case | Worst case | |
| Insertion Sort | O(1) | O(n) | O(n2) | O(n2) |
| Selection Sort | O(1) | O(n2) | O(n2) | O(n2) |
| Smooth Sort | O(1) | O(n) | O(n log n) | O(n log n) |
| Bubble Sort | O(1) | O(n) | O(n2) | O(n2) |
| Shell Sort | O(1) | O(n) | O(n log n2) | O(n log n2) |
| Mergesort | O(n) | O(n log n) | O(n log n) | O(n log n) |
| Quicksort | O(log n) | O(n log n) | O(n log n) | O(n log n) |
| Heapsort | O(1) | O(n log n) | O(n log n) | O(n log n) |
| Data Structures | Average Case | Worst Case | ||||
|---|---|---|---|---|---|---|
| Search | Insert | Delete | Search | Insert | Delete | |
| Array | O(n) | N/A | N/A | O(n) | N/A | N/A |
| Sorted Array | O(log n) | O(n) | O(n) | O(log n) | O(n) | O(n) |
| Linked List | O(n) | O(1) | O(1) | O(n) | O(1) | O(1) |
| Doubly Linked List | O(n) | O(1) | O(1) | O(n) | O(1) | O(1) |
| Stack | O(n) | O(1) | O(1) | O(n) | O(1) | O(1) |
| Hash table | O(1) | O(1) | O(1) | O(n) | O(n) | O(n) |
| Binary Search Tree | O(log n) | O(log n) | O(log 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) |
| Red-Black tree | O(log n) | O(log n) | O(log n) | O(log n) | O(log n) | O(log n) |
| AVL Tree | O(log n) | O(log n) | O(log n) | O(log n) | O(log n) | O(log n) |
Time Space Complexity Cheat Sheet Pdf
| n f(n) | log n | n | n log n | n2 | 2n | n! |
|---|---|---|---|---|---|---|
| 10 | 0.003ns | 0.01ns | 0.033ns | 0.1ns | 1ns | 3.65ms |
| 20 | 0.004ns | 0.02ns | 0.086ns | 0.4ns | 1ms | 77years |
| 30 | 0.005ns | 0.03ns | 0.147ns | 0.9ns | 1sec | 8.4x1015yrs |
| 40 | 0.005ns | 0.04ns | 0.213ns | 1.6ns | 18.3min | -- |
| 50 | 0.006ns | 0.05ns | 0.282ns | 2.5ns | 13days | -- |
| 100 | 0.07 | 0.1ns | 0.644ns | 0.10ns | 4x1013yrs | -- |
| 1,000 | 0.010ns | 1.00ns | 9.966ns | 1ms | -- | -- |
| 10,000 | 0.013ns | 10ns | 130ns | 100ms | -- | -- |
| 100,000 | 0.017ns | 0.10ms | 1.67ms | 10sec | -- | -- |
| 1'000,000 | 0.020ns | 1ms | 19.93ms | 16.7min | -- | -- |
| 10'000,000 | 0.023ns | 0.01sec | 0.23ms | 1.16days | -- | -- |
| 100'000,000 | 0.027ns | 0.10sec | 2.66sec | 115.7days | -- | -- |
| 1,000'000,000 | 0.030ns | 1sec | 29.90sec | 31.7 years | -- | -- |