大O表示法演算法複雜度速查表(Big-O Algorithm Complexity Cheat Sheet)
原文網址：http://bigocheatsheet.com/
Word文件下載：http://download.csdn.net/detail/anshan1984/5583399
Searching（搜尋演算法）
| Algorithm(演算法) |
Data Structure (資料結構) |
Time Complexity (時間複雜度) |
Space Complexity (空間複雜度) |
|
|
|
|
Average(平均) | Worst(最差) |
Worst(最差) |
| Graph of |V| vertices and |E| edges |
- |
O(|E| + |V|) |
O(|V|) |
|
| Graph of |V| vertices and |E| edges |
- |
O(|E| + |V|) |
O(|V|) |
|
| Sorted array of n elements |
O(log(n)) |
O(log(n)) |
O(1) |
|
| Array |
O(n) |
O(n) |
O(1) |
|
| using a Min-heap as priority queue(Dijkstra最短路徑，使用最小堆作為優先佇列) |
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|) |
| Graph with |V| vertices and |E| edges |
O(|V||E|) |
O(|V||E|) |
O(|V|) |
|
Sorting（排序演算法）
| Algorithm（演算法） |
Data Structure（資料結構） |
Time Complexity(時間複雜度) |
Worst Case Auxiliary Space Complexity （最差額外消耗空間複雜度） |
||
|
|
|
Best |
Average |
Worst |
Worst |
| (快速排序) |
Array（陣列） |
O(n log(n)) |
O(n log(n)) |
O(n^2) |
O(n) |
| (歸併排序) |
Array |
O(n log(n)) |
O(n log(n)) |
O(n log(n)) |
O(n) |
| (堆排序) |
Array |
O(n log(n)) |
O(n log(n)) |
O(n log(n)) |
O(1) |
| (氣泡排序) |
Array |
O(n) |
O(n^2) |
O(n^2) |
O(1) |
| (插入排序) |
Array |
O(n) |
O(n^2) |
O(n^2) |
O(1) |
| (選擇排序) |
Array |
O(n^2) |
O(n^2) |
O(n^2) |
O(1) |
| (桶排序) |
Array |
O(n+k) |
O(n+k) |
O(n^2) |
O(nk) |
| (基數排序) |
Array |
O(nk) |
O(nk) |
O(nk) |
O(n+k) |
Heaps（堆）
| Heaps |
Time Complexity（時間複雜度） |
|||||||
|
|
Heapify |
Find Max |
Extract Max |
Increase Key |
Insert |
Delete |
Merge |
|
| (有序連結串列) |
- |
O(1) |
O(1) |
O(n) |
O(n) |
O(1) |
O(m+n) |
|
| (無序連結串列) |
- |
O(n) |
O(n) |
O(1) |
O(1) |
O(1) |
O(1) |
|
| (二叉堆) |
O(n) |
O(1) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(m+n) |
|
| (多項式堆) |
- |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
|
| (斐波那契堆) |
- |
O(1) |
O(log(n)) |
O(1) |
O(1) |
O(log(n)) |
O(1) |
|
Graphs（圖）
| Node / Edge Management |
Storage |
Add Vertex |
Add Edge |
Remove Vertex |
Remove Edge |
Query |
| （鄰接表） |
O(|V|+|E|) |
O(1) |
O(1) |
O(|V| + |E|) |
O(|E|) |
O(|V|) |
| （關聯表） |
O(|V|+|E|) |
O(1) |
O(1) |
O(|E|) |
O(|E|) |
O(|E|) |
| （鄰接矩陣） |
O(|V|^2) |
O(|V|^2) |
O(1) |
O(|V|^2) |
O(1) |
O(1) |
| （關聯矩陣） |
O(|V|⋅|E|) |
O(|V|⋅|E|) |
O(|V|⋅|E|) |
O(|V|⋅|E|) |
O(|V|⋅|E|) |
O(|E|) |
Data Structures（資料結構）
| Data Structure （資料結構） |
Time Complexity （時間複雜度） |
Space Complexity （空間複雜度） |
|||||||
|
|
Average（平均） |
Worst（最差） |
Worst（最差） |
||||||
|
|
Indexing |
Search |
Insertion |
Deletion |
Indexing |
Search |
Insertion |
Deletion |
|
| （基本陣列） |
O(1) |
O(n) |
- |
- |
O(1) |
O(n) |
- |
- |
O(n) |
| （動態陣列） |
O(1) |
O(n) |
O(n) |
O(n) |
O(1) |
O(n) |
O(n) |
O(n) |
O(n) |
| （單鏈表） |
O(n) |
O(n) |
O(1) |
O(1) |
O(n) |
O(n) |
O(1) |
O(1) |
O(n) |
| （雙鏈表） |
O(n) |
O(n) |
O(1) |
O(1) |
O(n) |
O(n) |
O(1) |
O(1) |
O(n) |
| （跳躍表） |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(n) |
O(n) |
O(n) |
O(n) |
O(n log(n)) |
| （雜湊表） |
- |
O(1) |
O(1) |
O(1) |
- |
O(n) |
O(n) |
O(n) |
O(n) |
| （二叉查詢樹） |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(n) |
O(n) |
O(n) |
O(n) |
O(n) |
| （笛卡爾樹） |
- |
O(log(n)) |
O(log(n)) |
O(log(n)) |
- |
O(n) |
O(n) |
O(n) |
O(n) |
| （B樹） |
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) |
| （紅黑樹） |
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) |
| （伸展樹） |
- |
O(log(n)) |
O(log(n)) |
O(log(n)) |
- |
O(log(n)) |
O(log(n)) |
O(log(n)) |
O(n) |
| (AVL平衡樹) |
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（漸進增長表示法）
| Letter（字母） |
Bound（限制） |
Growth（增長） |
| (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是漸進上界，Ω是漸進下界。Θ需同時滿足大O和Ω，故稱為確界（必須同時符合上界和下界）。如，演算法Ω（nlogn）消耗至少nlogn時間，但是沒有上限。優先選擇演算法Θ（nlogn），因為它消耗至少nlogn（Ω（nlogn）），且不超過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))表示當n變大時，f（演算法執行時間）的增長與g嚴格相同。即，f(x)增長率漸進正比於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.
同樣，這裡增長率不超過g(n)。O極其有用，因為它表示了最差效能。
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
