大O表示法算法复杂度速查表(Big-O Algorithm Complexity Cheat Sheet)
原文网址：http://bigocheatsheet.com/
Word文档下载：http://download.csdn.net/detail/anshan1984/5583399
Searching（搜索算法）
| Algorithm(算法) |
Data Structure (数据结构) |
Time Complexity (时间复杂度) |
Space Complexity (空间复杂度) |
|
|
|
|
Average(平均) |
Worst(最差) |
Worst(最差) |
| Depth First Search (DFS)(深度优先搜索) |
Graph of |V| vertices and |E| edges |
- |
O(|E| + |V|) |
O(|V|) |
| Breadth First Search (BFS)(广度优先搜索) |
Graph of |V| vertices and |E| edges |
- |
O(|E| + |V|) |
O(|V|) |
| Binary search(二分查找) |
Sorted array of n elements |
O(log(n)) |
O(log(n)) |
O(1) |
| Linear (Brute Force)(线性查找-蛮力法) |
Array |
O(n) |
O(n) |
O(1) |
| Shortest path by Dijkstra, using a Min-heap as priority queue(Dijkstra最短路径，使用最小堆作为优先队列) |
Graph with |V| vertices and |E| edges |
O((|V| + |E|) log |V|) |
O((|V| + |E|) log |V|) |
O(|V|) |
| Shortest path by Dijkstra, |
Graph with |V| vertices and |E| edges |
O(|V|^2) |
O(|V|^2) |
O(|V|) |
| Shortest path by Bellman-Ford(Bellman-Ford最短路径法) |
Graph with |V| vertices and |E| edges |
O(|V||E|) |
O(|V||E|) |
O(|V|) |
Sorting（排序算法）
| Algorithm（算法） |
Data Structure（数据结构） |
Time Complexity(时间复杂度) |
Worst Case Auxiliary Space Complexity （最差额外消耗空间复杂度） |
||
|
|
|
Best |
Average |
Worst |
Worst |
| Quicksort (快速排序) |
Array（数组） |
O(n log(n)) |
O(n log(n)) |
O(n^2) |
O(n) |
| Mergesort (归并排序) |
Array |
O(n log(n)) |
O(n log(n)) |
O(n log(n)) |
O(n) |
| Heapsort (堆排序) |
Array |
O(n log(n)) |
O(n log(n)) |
O(n log(n)) |
O(1) |
| Bubble Sort (冒泡排序) |
Array |
O(n) |
O(n^2) |
O(n^2) |
O(1) |
| Insertion Sort (插入排序) |
Array |
O(n) |
O(n^2) |
O(n^2) |
O(1) |
| Select Sort (选择排序) |
Array |
O(n^2) |
O(n^2) |
O(n^2) |
O(1) |
| Bucket Sort (桶排序) |
Array |
O(n+k) |
O(n+k) |
O(n^2) |
O(nk) |
| Radix Sort (基数排序) |
Array |
O(nk) |
O(nk) |
O(nk) |
O(n+k) |
Heaps（堆）
| Heaps |
Time Complexity（时间复杂度） |
|||||||
|
|
Heapify |
Find Max |
Extract Max |
Increase Key |
Insert |
Delete |
Merge |
|
| Linked List (sorted) (有序链表) |
- |
O(1) |
O(1) |
O(n) |
O(n) |
O(1) |
O(m+n) |
|
| Linked List (unsorted) (无序链表) |
- |
O(n) |
O(n) |
O(1) |
O(1) |
O(1) |
O(1) |
|
| Binary Heap (二叉堆) |
O(n) |
O(1) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(m+n) |
|
| Binomial Heap (多项式堆) |
- |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
|
| Fibonacci Heap (斐波那契堆) |
- |
O(1) |
O(log(n)) |
O(1) |
O(1) |
O(log(n)) |
O(1) |
|
Graphs（图）
| Node / Edge Management |
Storage |
Add Vertex |
Add Edge |
Remove Vertex |
Remove Edge |
Query |
| Adjacency list （邻接表） |
O(|V|+|E|) |
O(1) |
O(1) |
O(|V| + |E|) |
O(|E|) |
O(|V|) |
| Incidence list （关联表） |
O(|V|+|E|) |
O(1) |
O(1) |
O(|E|) |
O(|E|) |
O(|E|) |
| Adjacency matrix （邻接矩阵） |
O(|V|^2) |
O(|V|^2) |
O(1) |
O(|V|^2) |
O(1) |
O(1) |
| Incidence matrix （关联矩阵） |
O(|V|⋅|E|) |
O(|V|⋅|E|) |
O(|V|⋅|E|) |
O(|V|⋅|E|) |
O(|V|⋅|E|) |
O(|E|) |
Data Structures（数据结构）
| Data Structure （数据结构） |
Time Complexity （时间复杂度） |
Space Complexity （空间复杂度） |
|||||||
|
|
Average（平均） |
Worst（最差） |
Worst（最差） |
||||||
|
|
Indexing |
Search |
Insertion |
Deletion |
Indexing |
Search |
Insertion |
Deletion |
|
| Basic Array （基本数组） |
O(1) |
O(n) |
- |
- |
O(1) |
O(n) |
- |
- |
O(n) |
| Dynamic Array （动态数组） |
O(1) |
O(n) |
O(n) |
O(n) |
O(1) |
O(n) |
O(n) |
O(n) |
O(n) |
| Singly-Linked List （单链表） |
O(n) |
O(n) |
O(1) |
O(1) |
O(n) |
O(n) |
O(1) |
O(1) |
O(n) |
| Doubly-Linked List （双链表） |
O(n) |
O(n) |
O(1) |
O(1) |
O(n) |
O(n) |
O(1) |
O(1) |
O(n) |
| Skip List （跳跃表） |
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 （哈希表） |
- |
O(1) |
O(1) |
O(1) |
- |
O(n) |
O(n) |
O(n) |
O(n) |
| Binary Search Tree （二叉查找树） |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(n) |
O(n) |
O(n) |
O(n) |
O(n) |
| Cartesian Tree （笛卡尔树） |
- |
O(log(n)) |
O(log(n)) |
O(log(n)) |
- |
O(n) |
O(n) |
O(n) |
O(n) |
| B-Tree （B树） |
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 （红黑树） |
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 （伸展树） |
- |
O(log(n)) |
O(log(n)) |
O(log(n)) |
- |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(n) |
| AVL Tree (AVL平衡树) |
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) |
Notation for asymptotic growth（渐进增长表示法）
| Letter（字母） |
Bound（限制） |
Growth（增长） |
| (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'swhy it's referred to as atight bound (it must be boththe 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(nlog n) is far preferential since it takes AT LEAST n log n (Omega n log n) andNO MORE THAN n log n (Big O n log n).SO
大O是渐进上界，Ω是渐进下界。Θ需同时满足大O和Ω，故称为确界（必须同时符合上界和下界）。如，算法Ω（nlogn）消耗至少nlogn时间，但是没有上限。优先选择算法Θ（nlogn），因为它消耗至少nlogn（Ω（nlogn）），且不超过nlogn（O（nlogn））。
[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).
f(x)=Θ(g(n))表示当n变大时，f（算法运行时间）的增长与g严格相同。即，f(x)增长率渐进正比于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.
同样，这里增长率不超过g(n)。O极其有用，因为它表示了最差性能。
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
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