Integer Multiplication慕课视频播放-算法设计与分析-MOOC慕课视频教程-柠檬大学

下面我们来看一个最基本的计算问题

整数乘法

在所有的计算问题中

最基本的运算是加法

给定两个n位的整数a和b

计算a加b

通过按位相加的运算

我们可以用O(n)次运算得到它们的和

这里一次运算指的是一位数的运算

这里我们给的例子是二进制的

但是它与十进制的运算没有本质的区别

我们再来看一下乘法

给定两个n位的整数a和b

计算a乘以b

我们用小学算术的竖式乘法

用O(n^2)次的一位数乘法

和O(n^2)次的一位数加法

可以得到乘积

下面尝试用分治算法来进行求解

我们可以把x和y分别划分为高位和低位

令x的前n/2位为高位值为x_1

后n/2位为低位值为x_0

相应的y的高位为y_1 低位y_0

我们可以把xy的乘积按照高低位展开

得到xy=2nx_1y_1+2n/2(x_1y_0+x_0y_1)+x_0y_0

我们把原来一个乘积

分解为4个乘积的计算

每个乘积中

问题的规模都是原来的一半

我们分析一下算法所需要的运行时间

T(n)=4T(n/2)再加上合并转换

所需要的时间Θ(n)

我们可以得到T(n)=Θ(n^2)

我们可以把刚才的运算稍作改变

在xy的展开式中

我们保持高位和低位不变

分别记为A和C

对于中间项x_1y_0+x_0y_1

可以写为(x_1+x_0)(y_1+y_0)-(x_1y_1+x_0y_0)

我们只需要计算一个新的乘积(x_1+x_0) (y_1+y_0)

我们把它记为B

原来的算法需要计算4个乘积

而现在只需要计算3个乘积

62年发现了这个结果

对于两个n位数的乘法

用O(n^1.585)次方的运算

可以实现两个n位整数的乘法

算法的运算时间T(n)≤3T(n/2)+

加减合并转换所需要的时间Θ(n)

可以得到T(n)=O(n^log 3)=O(n^1.585)

递归运算所需要的时间T(n)

可以用归纳法或者是递归树的方法来得到

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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