FFT慕课视频播放-算法设计与分析-MOOC慕课视频教程-柠檬大学

我们考虑把系数形式转化为点值形式

也就是给定一个n-1次多项式

求它在n个不同的点x_0到x_n-1处的值

关键的想法在于

我们选择x_k=ω^k

这里ω是n次本原单位根

我们可以用矩阵的方式来进行表达

这个关系我们称为离散傅里叶变换

这个矩阵我们称为傅里叶矩阵

我们回顾一下一些基本的数学知识

x^n=1在复数域上有n个根

称为n次单位根

这n个单位根可以写成ω^0, ω^1, …, ω^n-1

这里ω=e^2πi/n 我们把它称为n次本原单位根

容易验证它们的n次方都等于1

令½n次单位根为ν^0,ν^1, …,ν^n/2-1

这里ν=e^4πi/n为½n次本原单位根

也就是ω^2=ν 以及(ω^2)^k=ν^k

我们可以在一个单位圆周上

标示出这些单位根

我们的目标是给定一个n-1次多项式

求它在ω^0, ω^1, …, ω^n-1处的值

我们讨论分治合的操作

还是把多项式分为奇次项和偶次项

我们定义两个小的多项式

分别称为

A偶次多项式A_even(x)和A奇次多项式A_odd(x)

容易验证

A(x)=A偶次多项式和A奇次多项式

在x^2处取值的和

治的过程就是计算A_even(x)和A_odd(x)

在 ½n次单位根ν^0, ν^1, …,ν^n/2-1处的值

合的过程我们分两段来讨论

注意到 ν^k=(ω^k)^2=(ω^k+n)^2

以及ω^k+n=-ω^k

我们有

当0≤k

A(ω^k)=A_even(ν^k)+ω^kA_odd(ν^k)

在0≤k

A(ω^k+n)=A_even(ν^k)-ω^kA_odd(ν^k)

我们可以递归的求解所需的结果

以上的想法可以写成下面的FFT算法

当n=1时

多项式只有常数项

在任何点处的值都是a_0

递归调用FFT算法

求解偶次多项式和奇次多项式

在½n次单位根处的值

然后按照上面所得到的递归表达式

算出原多项式在n次单位根处的值

我们的结论是

FFT算法计算n-1次多项式

在n次单位根处的值

运算时间为O(nlogn)

由上面的分析我们有

T(2n)=2T(n)+O(n)⇒T(n)=O(nlogn)

现在如果给定多项式的系数形式

我们在O(nlogn)的时间

把它转换为点值形式

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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