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Chapter2 Algorithm Analysis

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Chapter1(1)

Chapter1(1)

Properties:
- Finite (easy) - Correct (hard) - Efficient (complexity) - Input - Output
Classifications: Some differences:
- Numerical - Nonnumerical - Serial - Parallel -……
Analysis Basics
- Algorithm vs. Problem - Algorithm vs. Procedure - Algorithm vs. Program
What is a problem
Definition:
- A mapping/relation between a set of input instances (domain) and an output set (range)
Analysis Basics
Initializatio n is not very important!
Why and What to Analysis
Purposes:
- choose between algorithms of solving a given problem - count the number of basic operations with respect to N input values - analysis is done without regard to any specific computer
N characters in the file The number of assignment (=) is 257 The total time: f(N) as the time of a certain basic operation. E.g., f(N)=N+257+1. N∞, f(N) ≈ N

【Verilog硬件描述】Verilog HDL-chapter2-3

【Verilog硬件描述】Verilog HDL-chapter2-3
Behavioral HDL model code that can not be synthesized Only for functional verification
Microelectronic center
11
Top-Down and Bottom-Up
2.2 Design methodology
Microelectronic center
3
Review
Spec.
Behavior HDL Top_down Flow
High-Level Synthesis
RTL HDL
Area
Synthesis
Timing
Netlist
Power DFT
FPGA Compiler Placement&Routing
▪ For the final physical implementation ▪ VHDL：very high speed integrated circuits
(VHSIC) HDL-IEEE-1076(1987)
▪ Verilog HDL
Microelectronic center
15
2.3 Framework of Verilog HDL
9
2.1 System on Chip
16-State Viterbi
Technology: 0.18 mm (C10N)
Transfer Rate: 850 Mbit/s
Package:
P-TQFP-100
Analog Area: 20 % (relative)
Digital Area: 80 % (relative)
Bottom-up

数理逻辑 第二章 算法、整数和矩阵 整数和除法

数理逻辑 第二章 算法、整数和矩阵 整数和除法

三、素数
如果整数不能被小于或等于其平方根的 素数整除，它就是素数。
例5：证明101是素数。
解：不超过101的平方根的素数有2，3，5， 7。因为101不能被这些数整除，所以101是 素数。
三、素数
由于每个整数都有素因子分解，如何求 解整数n的素因子分解？
从最小的素数2开始，从小到大用一个个素 数去除n；
最常用的产生伪随机数的过程称为线性同 余法
xn+1=(axn+c) mod m P120
九、密码学
最重要的同余应用之一涉及研究信息保 密的密码学
解为 a p1a1 p2a2 pnan
b p1b1 p2b2 pnbn
每个指数都是非负整数，出现在a和b分解中的所有素数都包 含在两个分解之中，必要时以0为指数出现
gcd(a,b)
p p min(a1,b1) min(a2,b2)
1
2
p min( an ,bn ) n
五、最大公约数
证明：P116 例14：已知120和500的素因子分解分别
定理7：令m为正整数，若a≡b(mod m)， c≡d(mod m)，那么a+c≡b+d(mod m)以及 ac≡bd(mod m)。
证明：P118
例18：由于7≡2(mod 5)和11≡1(mod 5)， 从定理7知： 18≡3(mod 5) 77≡2(mod 5)
八、同余应用
可以用同余为计算机分配内存地址 例19：散列（哈希）函数 散列就是无需查找，直接用元素的查找
数理逻辑
Mathematical Logic
第二章 算法、整数和矩阵
Chapter 2 Algorithm、Integer and Matrix

算法设计与分析基础课后习题答案solu2

算法设计与分析基础课后习题答案solu2
offered the inventor any reward he wanted. Sashi asked for some grain to be obtained as follows: just a single grain of wheat was to be placed on the first square of the chess board, two on the second, four on the third, eight on the fourth, and so on, until all 64 squares had been filled. What would the ultimate result of this algorithm have been?
3
Hints to Exercises 2.1
1. The questions are indeed as straightforward as they appear, though some of them may have alternative answers. Also, keep in mind the caveat about measuring an integer’s size. 2. a. The sum of two matrices is defined as the matrix whose elements are the sums of the corresponding elements of the matrices given. b. Matrix multiplication requires two operations: multiplication and addition. Which of the two would you consider basic and why? 3. Will the algorithm’s efficiency vary on different inputs of the same size? 4. a. Gloves are not socks: they can be right-handed and left-handed. b. You have only two qualitatively different outcomes possible. the number of ways to get each of the two. Count

《算法导论(第二版)》(中文版)课后答案

《算法导论(第二版)》(中文版)课后答案

5
《算法导论（第二版） 》参考答案 do z←y 调用之前保存结果 y←INTERVAL-SEARCH-SUBTREE(y, i) 如果循环是由于y没有左子树，那我们返回y 否则我们返回z，这时意味着没有在z的左子树找到重叠区间 7 if y≠ nil[T] and i overlap int[y] 8 then return y 9 else return z 5 6 15.1-5 由 FASTEST-WAY 算法知：
15
lg n
2 lg n1 1 2cn 2 cn (n 2 ) 2 1
4.3-1 a) n2 b) n2lgn c) n3 4.3-4
2
《算法导论（第二版） 》参考答案 n2lg2n 7.1-2 (1)使用 P146 的 PARTION 函数可以得到 q=r 注意每循环一次 i 加 1，i 的初始值为 p 1 ，循环总共运行 (r 1) p 1次，最 终返回的 i 1 p 1 (r 1) p 1 1 r (2)由题目要求 q=(p+r)/2 可知，PARTITION 函数中的 i，j 变量应该在循环中同 时变化。 Partition(A, p, r) x = A[p]; i = p - 1; j = r + 1; while (TRUE) repeat j--; until A[j] <= x; repeat i++; until A[i] >= x; if (i < j) Swap(A, i, j); else return j; 7.3-2 （1）由 QuickSort 算法最坏情况分析得知：n 个元素每次都划 n-1 和 1 个，因 为是 p<r 的时候才调用，所以为Θ （n） （2）最好情况是每次都在最中间的位置分，所以递推式是： N(n)= 1+ 2*N(n/2) 不难得到：N(n) =Θ （n） 7.4-2 T(n)=2*T(n/2)+ Θ （n） 可以得到 T(n) =Θ （n lgn） 由 P46 Theorem3.1 可得：Ω (n lgn)

算法设计与分析(第2版)

算法设计与分析(第2版)
该教材在编写过程中参考了很多同行的教材和络博客，特别是“牛客”中企业面试、笔试题和资源，河南工 程学院张天伍老师和使用该教材第1版的多位老师指正多处问题和错误。
出版工作
2018年8月1日，该教材由清华大学出版社出版。
内容简介
内容简介
全书由12章构成，各章的内容如下。
第1章概论：介绍算法的概念、算法分析方法和STL在算法设计中的应用。
教材目录
教材目录
（注：目录排版顺序为从左列至右列 ）
教学资源
教学资源
该教材配有配套教材——《算法设计与分析（第2版）学习与实验指导》，该配套教材涵盖所有练习题、上 机实验题和在线编程题的参考答案。
该教材每个知识点都配套了视频讲解，提供PPT课件、源码、答案、教学大纲、题库、书中全部源程序代码 （在VC++6.0中调试通过）等教学资源。
算法设计与分析（第2版）
20xx年清华大学出版社出版的图书
01 成书过程
03 教材目录源 06 作者简介
基本信息
《算法设计与分析（第2版）》是由李春葆主编，2018年清华大学出版社出版的高等学校数据结构课程系列 教材。该教材适合作为高等院校“算法设计与分析”课程的教材，也可供ACM和各类程序设计竞赛者参考。
第5章回溯法：介绍解空间概念和回溯法算法框架，讨论采用回溯法求解0/1背包问题、装载问题、子集和问 题、n皇后问题、图的m着色问题、任务分配问题、活动安排问题和流水作业调度问题的典型算法。
第6章分枝限界法：介绍分枝限界法的特点和算法框架、队列式分枝限界法和优先队列式分枝限界法，讨论 采用分枝限界法求解0/1背包问题、图的单源最短路径、任务分配问题和流水作业调度问题的典型算法。
该教材介绍了各种常用的算法设计策略，包括递归、分治法、蛮力法、回溯法、分枝限界法、贪心法、动态 规划、概率算法和近似算法等，并讨论了各种图算法和计算几何设计算法。书中配有图表、练习题、上机实验题 和在线编程题。

算法设计与分析 课件 一


Assessment
Marks from these components and from your final examination will be combined to produce a single mark as follows: –30% (3 x 10%) for home works –30% (3 x 10%) for assignments –40% for the final examination (written, close book, close note) Attendance – 0 to -10
Getting to 1st base
If your program fails to compile you will receive zero credit. Please also ensure that it runs as per the specification given. If you are not sure as to the exact requirements, please ask.
Assignments: general advice
For all assignments, use at least the given test data. Your program will be tested with this and some other data and it must behave at least as well as my version, which is by no means perfect. Examine the sample output for each program. From this it should be clear that you need to check for various error conditions.

Chapter_3__Sequence_Analysis_of_Nucleic_Acid


（二） pairwise alignment
Alignment between two sequences.
1 Simple alignment
1）unconsidering gap，determine match score and mismatch score in advance，and then judge the similarity according to the scores.
该模型不考虑核苷酸之间的关联。双核苷酸的全部 16种组合中，两个碱基相邻的频率等于序列中两碱 基的频率的乘积。 2. Markov Model（马尔可夫模型）
该模型认为，由4种碱基组成的一条DNA序列中，如 果完全是随机的，那么任何一个字母后出现其它字 母的频率都相同，如AA、AC、AG、AT出现的频率都 相同（1/4）。
直系同源物ortholog：在进化上起源于同一祖先并垂直遗传
（vertical descent）的同源基因，在结构和功能上高度保守。
旁系同源物paralog是指同一基因组中由于祖先基因的加倍而
横向传递（horizontal transfer）产生的几个同源基因，即一 个基因组中既有一定同源关系而又不十分相同的某些基因。
There are versions of BLAST for
searching nucleic acid and protein databases, which can be used to translate DNA sequences prior to comparing them to protein sequence databases.
3） Doolitter经验显著性检验 Doolitter针对蛋白质序列提出经验法则 （1）若两序列的长度都大于100，在适当加入空位后， 其配对的相同率达25%以上，则认为这两序列相关； 若小于15%，不可能相关；若15%～25%，可能是相 关的。

算法分析基础


6
程序(Program)
• 程序是算法用某种程序设计语言的具体实现。 • 程序可以不满足算法的有限性的性质。例如操作系统，是一个
在无限循环中执行的程序，因而不是一个算法。 • 操作系统的各种任务可看成是单独的问题，每一个问题由操作
系统中的一个子程序通过特定的算法来实现。该子程序得到输 出结果后便终止。
《算法设计技巧与分析》
第1章 算法分析基本概念
曹霑懋 caozhanmao@
Chapter 1 Basic Concepts in Algorithmic Analysis 内容
1.1 Introduction l.2 Historical Background 1.3 Binary Search 1.3.1 Analysis of the binary search algorithm 1.4 Merging Two Sorted Lists 1.5 Selectinn Sort 1.6 Insertion Sort 1.7 Bottom-Up Merge Sorting 1.7.1 Analysis of bottom-up merge sorting
最大搜索次数：满足Floor(n/2j-1)=1 时的ｊ 值
– 即：１n/2j-1 <2 – 也即： 2j-1n <2j
– j-1 log n <j
j=Floor(log n)+1
2020/6/10
华南师范大学 计算
16
设随有堂序数练组习：
试搜索x=20, 以及 X =22. 1）拟用什么法？为什么？ 2）试给出用你想要得算法求解的过程。
2020/6/10
华南师范大学 计算
14
1. 3 二分搜索及其时间复杂

数据结构与算法分析C++版英文版第二版课程设计

Course Design of Data Structures and AlgorithmAnalysis C++ Version (2nd Edition) ObjectiveThe primary objective of this course design is to reinforce the understanding of data structures and algorithms through implementationin C++ programming language. The course also ms to familiarize students with the usage of various C++ libraries while designing and implementing algorithms. By the end of this course design, students should be able to: •Understand the characteristics and properties of basic data structures such as arrays, stacks, queues, trees, graphs, andsearching/sorting algorithms.•Analyze the efficiency and complexity of algorithms using big O notation•Design and implement data structures and algorithms using C++ programming language, including OOP concepts such asinheritance and polymorphism.•Solve real-world problems using data structures and algorithms.SynopsisThe course design focuses on the following topics:1.Introduction to Data Structures and Algorithm Analysis2.Arrays and Vectors3.Linked Lists4.Stacks and Queues5.Trees6.Graphs7.Searching8.Sorting9.Hashing10.Binary Heaps and Priority Queues11.Heapsort12.Balanced Search Trees13.Advanced TopicsThe course design emphasizes practical implementation, therefore, each topic is accompanied by coding exercises using C++ programming language. Additionally, students are required to implement one major project in a team of two or three, which involves the usage of data structures and algorithms studied in class to solve a real-world problem.GradingThe final grade for this course design will be based on thefollowing criteria:•30%: Programming assignments•30%: Final project•20%: Mid-term exam•20%: Final examPrerequisitesIt is expected that students have a good understanding of programming concepts and basic data structures such as arrays, linked lists, and stacks/queues. Additionally, knowledge of object-oriented programming (OOP) concepts such as inheritance, polymorphism, and encapsulation is required.Course MaterialsThe primary course material will be the textbook。

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【Definition】 T (N) = O( f (N) ) if there are positive constants c 】
and n0 such that T (N) ≤ c f (N) for all N ≥ n0.
【Definition】 T (N) = ( g(N) ) if there are positive constants c 】
But it takes more time to compute each step.
float rsum ( float list[ ], int n ) { /* add a list of numbers */ if ( n ) /* count ++ */ return rsum(list, n1) + list[n 1];
Note: A program is written in some programming language, and does not have to be finite (e.g. an operation system). An algorithm can be described by human languages, flow charts, some programming languages, or pseudocode. 〖Example〗 Selection Sort: Sort a set of n ≥ 1 integers in 〗 increasing order.
logk N = O(N) for any constant k. This tells us that
Sort = Find the smallest integer + Interchange it with list[i].
§1 What to Analyze
Machine & compiler-dependent run times. Time & space complexities : machine & compilerindependent. Assumptions:
≥ 2N + 3 = O( N ) = O( Nk≥1 ) = O( 2N ) = We shall always take the smallest f (N). 2N + N2 = ( 2N ) = ( N2 ) = ( N ) = ( 1 ) = We shall always take the largest g(N).
数据结构
Data Structures
主讲人: 主讲人: 李晓红
教材 (Text Book)
Data Structures and Algorithm Analysis in C
(2nd Edition)
Mark Allen Weiss
参考书目 (Reference)
数据结构 (C语言版)严蔚敏等编著 清华大学出版社 数据结构 (C语言版)严蔚敏等编著,清华大学出版社 (C语言版 数据结构 (用面向对象方法与C++描述) 数据结构 (用面向对象方法与C++描述) C++描述 殷人昆等编著, 殷人昆等编著,清华大学出版社
Rules of Asymptotic Notation
If T1 (N) = O( f (N) ) and T2 (N) = O( g(N) ), then (a) T1 (N) + T2 (N) = max( O( f (N)), O( g(N)) ), (b) T1 (N) * T2 (N) = O( f (N) * g(N) ). If T (N) is a polynomial of degree k, then T (N) = Θ( N k ).
From those integers that are currently unsorted, find the smallest and place it next in the sorted list. for ( i = 0; i < n; i++) { list[n Examine list[i] to list[n1] and suppose that the smallest integer is at list[min]; Interchange list[i] and list[min]; }
instructions are executed sequentially each instruction is simple, and takes exactly one time unit integer size is fixed and we have infinite memory
Typically the following two functions are analyzed:
/* count ++ */
Tsum ( n ) = 2n + 3
}
tempsum += list [ i ] ; /* count ++ */
/* count ++ for last execution of for */ return tempsum; /* count ++ */
〖Example〗Recursive 〗 function for summing a list of numbers Trsum ( n ) = 2n + 2
课程评分方法 (Grading Policies)
Lecture Grade (80) = Final Exam (80)
1 5 Laboratory Grade (15) = ∑ Lab ( i ) × 0.15 5 i =1
Other Grade(5)=Attendance + Homework
实验 (Laboratory Projects)
/day;抄袭者0 共 5 次;迟交罚扣为 10% /day;抄袭者0分; 3人一组,各组自行分工(例如:按功能模块,按开 人一组,各组自行分工(例如:按功能模块, 人一组 发过程等),各自工作量应相当,试验报告中必须 发过程等),各自工作量应相当, ),各自工作量应相当 写明各自工作量; 写明各自工作量; 通过e-mail提交,请在邮件标题写明班级学号及试验 提交,请在邮件标题写明班级学号及 通过 提交 邮件标题写明班级学号 ID,并在文档末尾写明分工. 文档末尾写明分工. ,并在文档末尾写明分工
助教( 助教(Teaching Assistant): ): 杜洪伟( 杜洪伟(dhw198426@) 潘东 (pandongshilige@)
CHAPTER 2
ALGORITHM ANALYSIS 【Definition】An algorithm is a finite set of instructions 】 that, if followed, accomplishes a particular task. In addition, all algorithms must satisfy the following criteria:
作业&出勤 作业 出勤
布置作业的下周(Tuesday)交,迟交不交 布置作业的下周( ) 扣分.做则有分,不计对错, 扣分.做则有分,不计对错, 一周后公布 参考答案. 参考答案. 必须按照软件学院规范做作业, 必须按照软件学院规范做作业,否则不计 作业未交者不能参加期末考试. 分,1/3作业未交者不能参加期末考试. 作业未交者不能参加期末考试 点名三次不到者不能参加期末考试. 点名三次不到者不能参加期末考试.
and n0 such that T (N) ≥ c g(N) for all N ≥ n0. T (N) = ( h(N) ) . Θ( p(N) ) .
【Definition】 T (N) = Θ( h(N) ) if and only if T (N) = O( h(N) ) and 】 【Definition】 T (N) = o( p(N) ) if T (N) = O( p(N) ) and T (N) ≠ 】 Note:
(1) Input There are zero or more quantities that are externally supplied. (2) Output At least one quantity is produced. (3) Definiteness Each instruction is clear and unambiguous. (4) Finiteness If we trace out the instructions of an algorithm, then for all cases, the algorithm terminates after finite number of steps. (5) Effectiveness Every instruction must be basic enough to be carried out, in principle, by a person using only pencil and paper. It is not enough that each operation be definite as in(3); it also must be feasible.
Tavg(N) & Tworst(N) -- the average and worst case time complexities, respectively, as functions of input size N.
〖Example〗 Matrix addition 〗
void add ( int int int int { int i, j ; for ( i = 0; i < rows; i++ ) /* rows + 1 */ for ( j = 0; j < cols; j++ ) /* rows(cols+1) */ c[ i ][ j ] = a[ i ][ j ] + b[ i ][ j ]; } T(rows, cols ) = 2 rows cols + 2rows + 1 /* rows cols */ a[ ][ MAX_SIZE ], b[ ][ MAX_SIZE ], c[ ][ MAX_SIZE ], rows, int cols )