Study started since 2011年10月17日 素性测试算法，素性测定(11Y11) primality testing

7.Styles

Use as much as drawings, pictures.animations as possible to accompany or replace texts

Every definition shall be expressed in bold letters

8.Definitions. Terminology, Signs and Symbols

燃烧界和燃烧污染控制领域存在许多混乱的说法，应该避免

任何科学都从概念开始。

术语名辞定语Nomenclature 或List of symbols

Definitions

Assertion

Hypothesis

Terms

Terminology

8.1.Alphabets and Signs

详细规划和说明下列字母的意义和用途：拉丁字母，希腊字母，俄文字母和希伯来字母。

同时要符合国内外学术界的惯例和习惯，又要强调概念清晰，唯一，统一，易于理解和明确。

花体字母

空心体字母

重体字母

斜体字母

8.1.1.Latin

8.1.2.Greek

8.1.3.Hebrew

8.1.4.Russian

8.2.Nomenclature

Nomenclature is a term that applies to either a list of names or terms, or to the system of principles, procedures and terms related to naming—which is the assigning of a word or phrase to a particular object, event, or property. The principles of naming vary from the relatively informal conventions of everyday speech to the internationally-agreed principles, rules and recommendations that govern the formation and use of the specialist terms used in scientific and other disciplines. ... nomenclature concerns itself more with the rules and conventions that are used for the formation of names (wikipedia)

8.3.Notation

We will use _ to denote the base 2 logarithm instead of _ or lgn. Base 10 logarithms and natural logarithms will be denoted as _ n and _ respectively.

The notation ord _r (a) represents the order of a modulo r, which is the smallest positive integer k, such that _{ } 1(modr).

The notation (r) will be used to represent Euler’s totient function, which is defined as the number of positive integers less than or equal to r that are relatively prime to r.

The notation f(x) ≡g(x)mod(h(x),p) is used throughout to mean that f(x)=g(x) in the ring Zp[x]/(h(x)). In some cases, p will be prime and will have degree d and be irreducible in Zp(x), so that Zp[x]/(h(x)).will be a finite field of order _.

Notation: p(x) ≡ q(x) mod (h(x), n) means that h(x)|p(x) − q(x) all coefficients are taken modulo n

Time complexity functions will be written in “big-O” notation. A function f(x) is considered to be O (g(x))(pronounced “big-oh of g”) if given some function g(x) there exists a constant c such that |f(x)|≤c|g (x)|⋅ for all values of x_0, where x is defined to be the binary input length of n.

The function M(n) will be used to represent the time complexity function for multiplication.

8.4.Definitions

Diophantine equation

Euclidean Algorithm Extended Euclid Algorithm

The greatest common divisor GCD (or greatest common factor GCF)

8.5.List of theorems, formulars and equations

8.6.List of abbreviations

8.7.List of people mentioned in the book

8.8.Index

8.9.Explanations

8.10.Fonts

italic: for scalers;

Bold italic: for vectors in D-dimensional vector space, D = 1, 2, 3;

Boldface: for vectors in Rb ;

Sans serif: for operators or second rank tensors (matrices);

Bold sans serif: for matrices with vector/tensor elements;

BLA C K B O A RD BO LD : for vector space.

8.11.List of symbols

8.12.Sub and super scripts

8.13.Marks

@↑↓

≈≠≡⌂∩∑∏ ≈∵∴⊥∥≌∽∈

8.14.词冠???

10 12 Τ Τερα 核 10 9 Giga 京 10 -9 Nano 钎 10 -12 Pico

8.15.UNITS

8.16.Units designation

Atm BYU bar etc

8.17.Constants

9.Comparisons of Algorithm

Available software such as Maple, Mathmatica in computing primality tests

Different high level computer languages such as C/C++, Java

Low-level languages such as assembly

Machine languages

Different Oss such as Windows, Linux, BSD
Software available

Plarforms available

10.Intro

10.1.TOC for Introduction to Algorithms

Table of Contents

Preface

I Foundations

1 The Role of Algorithms in Computing

1.1 Algorithms

1.2 Algorithms as a technology

2 Getting Started

2.1 Insertion sort

2.2 Analyzing algorithms

2.3 Designing Algorithms

3 Growth of Functions

3.1 Asymptotic notation

3.2 Standard notations and common functions

4 Recurrences

4.1 The substitution method

4.2 The recursion-tree method

4.3 The master method

4.4 Proof of the master theorem

5 Probabilistic Analysis and Randomized Algorithms

5.1 The hiring problem

5.2 Indicator random variables

5.3 Randomized algorithms

5.4 Probabilistic analysis and further uses of indicator random variables

II Sorting and Order Statistics

6 Heapsort

6.1 Heaps

6.2 Maintaining the heap property

6.3 Building a heap

6.4 The heapsort algorithm

6.5 Priority queues

7 Quicksort

7.1 Description of quicksort

7.2 Performance of quicksort

7.3 Randomized versions of quicksort

7.4 Analysis of quicksort

8 Sorting in Linear Time

8.1 Lower bounds for sorting

8.2 Counting sort

8.3 Radix sort

8.4 Bucket sort

9 Medians and Order Statistics

9.1 Minimum and maximum

9.2 Selection in expected linear time

9.3 Selection in worst-case linear time

III Data Structures

10 Elementary Data Structures

10.1 Stacks and queues

10.2 Linked lists

10.3 Implementing pointers and objects

10.4 Representing rooted trees

11 Hash Tables

11.1 Direct-address tables

11.2 Hash tables

11.3 Hash functions

11.4 Open addressing

11.5 Perfect hashing

12 Binary Search Trees

12.1 What is a binary search tree?

12.2 Querying a binary search tree

12.3 Insertion and deletion

12.4 Randomly built binary search trees

13 Red-Black Trees

13.1 Properties of red-black trees

13.2 Rotations

13.3 Insertion

13.4 Deletion

14 Augmenting Data Structures

14.1 Dynamic order statistics

14.2 How to augment a data structure

14.3 Interval trees

IV Advanced Design and Analysis Technique

15 Dynamic Programming

15.1 Assembly-line scheduling

15.2 Matrix-chain multiplication

15.3 Elements of dynamic programming

15.4 Longest common subsequence

15.5 Optimal binary search trees

16 Greedy Algorithms

16.1 An activity-selection problem

16.2 Elements of the greedy strategy

16.3 Huffman codes

16.4 Theoretical foundations for greedy methods

16.5 A task-scheduling problem

17 Amortized Analysis

17.1 Aggregate analysis

17.2 The accounting method

17.3 The potential method

17.4 Dynamic tables

V Advanced Data Structures

18 B-Trees

18.1 Definition of B-trees

18.2 Basic operations on B-trees

18.3 Deleting a key from a B-tree

19 Binomial Heaps

19.1 Binomial trees and binomial heaps

19.2 Operations on binomial heaps

20 Fibonacci Heaps

20.1 Structure of Fibonacci heaps

20.2 Mergeable-heap operations

20.3 Decreasing a key and deleting a node

20.4 Bounding the maximum degree

21 Data Structures for Disjoint Sets

21.1 Disjoint-set operations

21.2 Linked-list representation of disjoint sets

21.3 Disjoint-set forests

21.4 Analysis of union by rank with path compression

VI Graph Algorithms

22 Elementary Graph Algorithms

22.1 Representations of graphs

22.2 Breadth-first search

22.3 Depth-first search

22.4 Topological sort

22.5 Strongly connected components

23 Minimum Spanning Trees

23.1 Growing a minimum spanning tree

23.2 The algorithms of Kruskal and Prim

24 Single-Source Shortest Paths

24.1 The Bellman-Ford algorithm

24.2 Single-source shortest paths in directed acyclic graphs

24.3 Dijkstra's algorithm

24.4 Difference constraints and shortest paths

24.5 Proofs of shortest-paths properties

25 All-Pairs Shortest Paths

25.1 Shortest paths and matrix multiplication

25.2 The Floyd-Warshall algorithm

25.3 Johnson's algorithm for sparse graphs

26 Maximum Flow

26.1 Flow networks

26.2 The Ford-Fulkerson method

26.3 Maximum bipartite matching

26.4 Push-relabel algorithms

26.5 The relabel-to-front algorithm

VII Selected Topics

27 Sorting Networks

27.1 Comparison networks

27.2 The zero-one principle

27.3 A bitonic sorting network

27.4 A merging network

27.5 A sorting network

28 Matrix Operations

28.1 Properties of matrices

28.2 Strassen's algorithm for matrix multiplication

28.3 Solving systems of linear equations

28.4 Inverting matrices

28.5 Symmetric positive-definite matrices and least-squares approximation

29 Linear Programming

29.1 Standard and slack forms

29.2 Formulating problems as linear programs

29.3 The simplex algorithm

29.4 Duality

29.5 The initial basic feasible solution

30 Polynomials and the FFT

30.1 Representation of polynomials

30.2 The DFT and FFT

30.3 Efficient FFT implementations

31 Number-Theoretic Algorithms

31.1 Elementary number-theoretic notions

31.2 Greatest common divisor

31.3 Modular arithmetic

31.4 Solving modular linear equations

31.5 The Chinese remainder theorem

31.6 Powers of an element

31.7 The RSA public-key cryptosystem

31.8 Primality testing

31.9 Integer factorization

32 String Matching

32.1 The naive string-matching algorithm

32.2 The Rabin-Karp algorithm

32.3 String matching with finite automata

32.4 The Knuth-Morris-Pratt algorithm

33 Computational Geometry

33.1 Line-segment properties

33.2 Determining whether any pair of segments intersects

33.3 Finding the convex hull

33.4 Finding the closest pair of points

34 NP-Completeness

34.1 Polynomial time

34.2 Polynomial-time verification

34.3 NP-completeness and reducibility

34.4 NP-completeness proofs

34.5 NP-complete problems

35 Approximation Algorithms

35.1 The vertex-cover problem

35.2 The traveling-salesman problem

35.3 The set-covering problem

35.4 Randomization and linear programming

35.4 The subset-sum problem

VIII Appendix: Mathematical Background

A Summations

A.1 Summation formulas and properties

A.2 Bounding summations

B Sets, Etc.

B.1 Sets

B.2 Relations

B.3 Functions

B.4 Graphs

B.5 Trees

C Counting and Probability

C.1 Counting

C.2 Probability

C.3 Discrete random variables

C.4 The geometric and binomial distributions

C.5 The tails of the binomial distribution

Bibliography

Index (created by the authors)