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, using an unsorted array as priority queue(Dijkstra最短路径，使用无序数组作为优先队列)	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