时间复杂度和空间复杂度——总结
2013-05-05 22:25
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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 msyelf "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(log(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(1) | 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(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(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(n) |
| Red-Black 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(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(log(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 Graph
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