Asymptotic Order of Growth慕课视频播放-算法设计与分析-MOOC慕课视频教程-柠檬大学

我们引进一些关于多项式阶数关系的符号

如果存在常数 c>0 和 n_0≥0

使得对于所有的 n ≥ n_0 都有T(n)≤ cf(n)

那么称T(n)是O(f(n))的

f(n)是T(n)的上界

如果存在常数 c>0 和 n_0≥0

使得对于所有的 n ≥ n_0 , T(n)≥ cf(n)

那么称T(n)是Ω(f(n))的

f(n)是T(n)的下界

如果T(n)是O(f(n))的又是Ω(f(n))的

那么T(n)是θ(f(n))的

f(n)是T(n)的紧界

例如 T(n) = 32n^2 + 17n + 32

T(n) 是 O(n^2), O(n^3), Ω(n^2), Ω(n^3), 以及θ(n^2)的

但T(n) 不是O(n), Ω(n^3), θ(n), 或θ(n^3)的

O(f(n))是一个函数的集合

但通常习惯上我们说 T(n) = O(f(n))

而不是 T(n)∈O(f(n))

对于上面介绍的符号

容易验证具有下面的性质

传递性

如果f = O(g) 且 g = O(h)

则 f = O(h)

如果f = Ω(g) 且 g = Ω(h)

则 f = Ω(h)

如果f = θ(g) 且 g = θ(h)

则 f = θ(h)

可加性

如果f = O(h) 且 g = O(h)

则 f+g = O(h)

如果f = Ω(h) 且 g = Ω(h)

则 f+g = Ω(h)

如果f = θ(h) 且 g = θ(h)

则 f+g = θ(h)

下面我们给出一些常见函数的界

多项式 a_0 + a_1n + … + a_dn^d 是 θ(n^d) 的

这里 a_d > 0

我们说一个算法是多项式时间的

如果存在常数d

对于任意输入规模n

运行时间是O(n^d)

对数

对于任意的常数 a,b >1

O(log_a n) = O(log_b n)

对于任意的 x>0, log n = O(n^x)

即对数函数比任何的多项式函数都增涨的慢

指数

对于任意的 r>0 和任意的 d>0

n^d = O(r^n)

任意的指数函数比任意的多项式函数

增长得快

算法设计与分析课程列表：

1 Introduction of Algorithm

-1.1 Introduction

--Introduction

-1.2 A First Problem: Stable Matching

--A First Problem: Stable Matching

-1.3 Gale-Shapley Algorithm

--Gale-Shapley Algorithm

-1.4 Understanding Gale-Shapley Algorithm

--Understanding Gale-Shapley Algorithm

-Homework1

-Lecture note 1

--Lecture note 1 Introduction of Algorithm

2 Basics of Algorithm Analysis

-2.1 Computational Tractability

--Computational Tractability

-2.2 Asymptotic Order of Growth

--Asymptotic Order of Growth

-2.3 A Survey of Common Running Times

--A Survey of Common Running Times

-Homework2

-Lecture note 2

--Lecture note 2 Basics of Algorithm Analysis

3 Graph

-3.1 Basic Definitions and Applications

--Basic Definitions and Applications

-3.2 Graph Traversal

--Graph Traversal

-3.3 Testing Bipartiteness

--Testing Bipartiteness

-3.4 Connectivity in Directed Graphs

--Connectivity in Directed Graphs

-3.5 DAG and Topological Ordering

--DAG and Topological Ordering

-Homework3

-Lecture note 3

--Lecture note 3 Graph

4 Greedy Algorithms

-4.1 Coin Changing

--Coin Changing

-4.2 Interval Scheduling

--Interval Scheduling

-4.3 Interval Partitioning

--Interval Partitioning

-4.4 Scheduling to Minimize Lateness

--Scheduling to Minimize Lateness

-4.5 Optimal Caching

--Optimal Caching

-4.6 Shortest Paths in a Graph

--Shortest Paths in a Graph

-4.7 Minimum Spanning Tree

--Minimum Spanning Tree

-4.8 Correctness of Algorithms

--Correctness of Algorithms

-4.9 Clustering

--Clustering

-Homework4

-Lecture note 4

--Lecture note 4 Greedy Algorithms

5 Divide and Conquer

-5.1 Mergesort

--Mergesort

-5.2 Counting Inversions

--Counting Inversions

-5.3 Closest Pair of Points

--Closest Pair of Points

-5.4 Integer Multiplication

--Integer Multiplication

-5.5 Matrix Multiplication

--Video

-5.6 Convolution and FFT

--Convolution and FFT

-5.7 FFT

--FFT

-5.8 Inverse DFT

--Inverse DFT

-Homework5

-Lecture note 5

--Lecture note 5 Divide and Conquer

6 Dynamic Programming

-6.1 Weighted Interval Scheduling

--Weighted Interval Scheduling

-6.2 Segmented Least Squares

--Segmented Least Squares

-6.3 Knapsack Problem

--Knapsack Problem

-6.4 RNA Secondary Structure

--RNA Secondary Structure

-6.5 Sequence Alignment

--Sequence Alignment

-6.6 Shortest Paths

--Shortest Paths

-Homework6

-Lecture note 6

--Lecture note 6 Dynamic Programming

7 Network Flow

-7.1 Flows and Cuts

--Flows and Cuts

-7.2 Minimum Cut and Maximum Flow

--Minimum Cut and Maximum Flow

-7.3 Ford-Fulkerson Algorithm

--Ford-Fulkerson Algorithm

-7.4 Choosing Good Augmenting Paths

--Choosing Good Augmenting Paths

-7.5 Bipartite Matching

--Bipartite Matching

-Homework7

-Lecture note 7

--Lecture note 7 Network Flow

8 NP and Computational Intractability

-8.1 Polynomial-Time Reductions

--Polynomial-Time Reductions

-8.2 Basic Reduction Strategies I

--Basic Reduction Strategies I

-8.3 Basic Reduction Strategies II

--Basic Reduction Strategies II

-8.4 Definition of NP

--Definition of NP

-8.5 Problems in NP

--Problems in NP

-8.6 NP-Completeness

--NP-Completeness

-8.7 Sequencing Problems

--Sequencing Problems

-8.8 Numerical Problems

--Numerical Problems

-8.9 co-NP and the Asymmetry of NP

--co-NP and the Asymmetry of NP

-Homework8

-Lecture note 8

--Lecture note 8 NP and Computational Intractability

9 Approximation Algorithms

-9.1 Load Balancing

--Load Balancing

-9.2 Center Selection

--Center Selection

-9.3 The Pricing Method: Vertex Cover

--The Pricing Method: Vertex Cover

-9.4 LP Rounding: Vertex Cover

--LP Rounding: Vertex Cover

-9.5 Knapsack Problem

--Knapsack Problem

-Homework9

-Lecture note 9

--Lecture note 9 Approximation Algorithms

10 Local Search

-10.1 Landscape of an Optimization Problem

--Landscape of an Optimization Problem

-10.2 Maximum Cut

--Maximum Cut

-10.3 Nash Equilibria

--Nash Equilibria

-10.4 Price of Stability

--Price of Stability

-Homework10

-Lecture note 10

--Lecture note 10 Local Search

11 Randomized Algorithms

-11.1 Contention Resolution

--Contention Resolution

-11.2 Linearity of Expectation

--Linearity of Expectation

-11.3 MAX 3-SAT

--MAX 3-SAT

-11.4 Chernoff Bounds

--Chernoff Bounds

-Homework11

-Lecture note 11

--Lecture note 11 Randomized Algorithms

Exam

-Exam

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