算法复杂度总结！

刚看到一个很好的算法复杂度总结的贴，果断转了。

原文链接：http://bigocheatsheet.com/#

坑爹，那些表格没法完全显示，大家可以点上面的链接看原文。

Searching

AlgorithmData StructureTime ComplexitySpace Complexity

AverageWorstWorstDepth 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 searchSorted array of n elementsO(log(n))O(log(n))O(1)Linear (Brute Force)ArrayO(n)O(n)O(1)Shortest path by Dijkstra,
using a Min-heap as priority queueGraph with |V| vertices and |E| edgesO((|V| + |E|) log |V|)O((|V| + |E|) log |V|)O(|V|)Shortest path by Dijkstra,
using an unsorted array as priority queueGraph with |V| vertices and |E| edgesO(|V|^2)O(|V|^2)O(|V|)Shortest path by Bellman-FordGraph with |V| vertices and |E| edgesO(|V||E|)O(|V||E|)O(|V|)

Sorting

AlgorithmData StructureTime ComplexityWorst Case Auxiliary Space Complexity  BestAverageWorstWorstQuicksortArrayO(n log(n))O(n log(n))O(n^2)O(n)MergesortArrayO(n log(n))O(n log(n))O(n log(n))O(n)HeapsortArrayO(n log(n))O(n log(n))O(n log(n))O(1)Bubble SortArrayO(n)O(n^2)O(n^2)O(1)Insertion SortArrayO(n)O(n^2)O(n^2)O(1)Select SortArrayO(n^2)O(n^2)O(n^2)O(1)Bucket SortArrayO(n+k)O(n+k)O(n^2)O(nk)Radix SortArrayO(nk)O(nk)O(nk)O(n+k)

Data Structures

Data StructureTime ComplexitySpace Complexity AverageWorstWorst IndexingSearchInsertionDeletionIndexingSearchInsertionDeletion Basic ArrayO(1)O(n)--O(1)O(n)--O(n)Dynamic ArrayO(1)O(n)O(n)O(n)O(1)O(n)O(n)O(n)O(n)Singly-Linked ListO(n)O(n)O(1)O(1)O(n)O(n)O(1)O(1)O(n)Doubly-Linked ListO(n)O(n)O(1)O(1)O(n)O(n)O(1)O(1)O(n)Skip ListO(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 TreeO(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-TreeO(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 TreeO(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 TreeO(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

HeapsTime Complexity HeapifyFind MaxExtract MaxIncrease KeyInsertDeleteMerge 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 HeapO(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 ManagementStorageAdd VertexAdd EdgeRemove VertexRemove EdgeQueryAdjacency listO(|V|+|E|)O(1)O(1)O(|V| + |E|)O(|E|)O(|V|)Incidence listO(|V|+|E|)O(1)O(1)O(|E|)O(|E|)O(|E|)Adjacency matrixO(|V|^2)O(|V|^2)O(1)O(|V|^2)O(1)O(1)Incidence matrixO(|V| ⋅ |E|)O(|V| ⋅ |E|)O(|V| ⋅ |E|)O(|V| ⋅ |E|)O(|V| ⋅ |E|)O(|E|)

Notation for asymptotic growth

letterboundgrowth(theta) Θupper and lower, tight^[1]equal^[2](big-oh) Oupper, tightness unknownless than or equal^[3](small-oh) oupper, not tightless than(big omega) Ωlower, tightness unknowngreater than or equal(small omega) ωlower, not tightgreater 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 __algorithmperformanceo(n)< nO(n)≤ nΘ(n)= nΩ(n)≥ nω(n)> n

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Big-O Complexity Chart

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Authors:

1. Eric Rowell
2. Quentin Pleple
3. Nick Dizazzo
4. Michael Abed
5. Adam Forsyth
6. Jay Engineer
7. Josh Davis
8. makosblade
9. Alejandro Ramirez
10. Joel Friedly
11. Robert Burke
12. David Dorfman
13. Eric Lefevre-Ardant
14. Thomas Dybdahl Ahle