常见的算法的时间和空间复杂度
Big-O Algorithm Complexity Cheat Sheet 列出了常见的算法的时间和空间复杂度，非常直观。
转载到了这里，大家可以看看。
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(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[1] | equal[2] |
| (big-oh) O | upper, tightness unknown | less than or equal[3] |
| (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 |
Big-O Complexity Chart
