算法设计与分析

1. 算法设计与分析 叶 德 仕 yedeshi@gmail.com

2. 课程基本信息 • 课程编号： 21190120 • 上课时间：2007年 夏学期 • 周一、周三的晚上 6:30 ~ • 上课地点：紫金港 东1B-206 • 考试时间：7月6日 14:00 – 16:00 • 考试形式：书面闭卷考试 • 学时/学分：4-2/周 学时/ 2-1 学分

3. Office • 办公室地点：曹光彪主楼 524 • Tel.： 13957173448

4. $1,000,000 problem • P = NP? http://www.claymath.org/millennium/ • Solved???!!!!

5. Examination • Scores： • 1. Final examination (50%) • 2. Two projects (40% ) • 3. Attendances & Questions (10%)

6. 想成为计算机科学家？

7. Erdős project • Paul Erdős(1913-1996) has an Erdős number of zero. If the lowest Erdős number of a coauthor is X, then the author's Erdős number is X + 1.

8. Nevanlinna Prize winners • NAME YEAR COUNTRY ERDÖS NUMBER • Robert Tarjan 1982 USA 2 • Leslie Valiant 1986 Hungary/Gt Brtn 3 • Alexander Razborov 1990 Russia 2 • Avi Wigderson 1994 Israel 2 • Peter Shor 1998 USA 2 • Madhu Sudan 2002 India/USA 2 • Jon Kleinberg 2006 USA 3

9. Other famous people • Albert Einstein 1921 Physics 2 • Chen Ning Yang 1957 Physics 4 • Tsung-dao Lee 1957 Physics 5 • John F. Nash 1994 Economics 4 • Edmund S. Phelps 2006 Economics 4 • Shing-Tung Yau 1982 China 2 • Shiing Shen Chern 1983-84 China 2 • Alan Turing computer science 5 • John von Neumann mathematics 3 • David Hilbert mathematics 4 • Donald E. Knuth 2

10. 还是想成为一名 技术人员？

11. 领导者或决策者？

12. 老板布置任务了

13. 你的答案：Um? 告诉我写那些代码？

14. 还是随便的应付？

15. 你的答案： 我学过这个算法，但想不起来了 图书馆里有现成

16. 你的答案： 我可以设计一个新的算法

17. A Microsoft interview?

18. 未来是属于 • 谁能掌握基本问题和解决方法 • 有能力解决新出现或者不熟悉的问题

19. What is algorithm？ • （Oxford Dict.）Algorithm: • A set of rules that must be followed when solving a particular problem. • From Math world • A specific set of instructions for carrying out a procedure or solving a problem, usually with the requirement that the procedure terminate at some point. • An algorithm is any well-defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values, as output.

20. Algorithm • Problem definition 问题 • Objective 目标 (very important) • Methods 方法 • Performance 算法评价

21. Perspective • Algorithms we can find everywhere • They have been developed to easy our daily life • Train/Airplane timetable schedule • Routing • We live in the age of information • Text, numbers, images, video, audio

22. Algorithm in daily life • Clothes: strip packing • Cooking: menu scheduling • Accommodation: facility location • Traffic: traffic lights

23. Pricing • Water, electrical power pricing: Step pricing • Promotion: Hangbai et al., buy items with total price >= 300, then 60 bonus, but each invoice uses only once. • Bin covering, Bin packing, Open – End bin packing problem.

24. Selfish routing • Pigou's Example Suburb s, a nearby train station t. Assuming that all drivers aim to minimize the driving time from s to t C(x) = 1 s t C(x) = x, with x in [0, 1]

25. Selfish routing • We have good reason to expect all traffic to follow the lower road • Social optimal? ½ to the long, wide highway, ½ to the lower road. • selfish behavior need not produce a socially optimal outcome

26. Braess's Paradox v C(x) = x C(x) = 1 s t C(x) = 1 C(x) = x w

27. Braess's Paradox v C(x) = x C(x) = 1 C(x) = 0 s t C(x) = 1 C(x) = x w

28. Braess's Paradox • Paradox thus shows that the intuitively helpful action of adding a new zero-cost link can negatively impact all of the traffic! • With selfish routing, network improvements can degrade network performance.

29. History of Algorithm • The word algorithmcomes from the name of the 9th century Persian mathematician Abu Abdullah Muhammad ibn Musa al-Khwarizmi whose works introduced Arabic numerals and algebraic concepts. • The word algorism originally referred only to the rules of performing arithmetic using Arabic numerals but evolved into algorithm by the 18th century. The word has now evolved to include all definite procedures for solving problems or performing tasks.

30. History – con. • The first case of an algorithm written for a computer was Ada Byron's notes on the analytical engine written in 1842, for which she is considered by many to be the world's first programmer. However, since Charles Babbage never completed his analytical engine the algorithm was never implemented on it. • This problem was largely solved with the description of the Turing machine, an abstract model of a computer formulated by Alan Turing, and the demonstration that every method yet found for describing "well-defined procedures" advanced by other mathematicians could be emulated on a Turing machine (a statement known as the Church-Turing thesis).

31. 课程内容 • 1.数学基础 • 1.1 算法基础 • 1.2 和 (SUMS) 集合运算 (Sets) • 1.3 特殊数 （Stirling numbers, Harmonic numbers, Eulerian numbers et al.） • 2.基本算法 • 2.1分治（Divide-and-Conquer）* • 2.1.1 Mergesort * • 2.1.2 自然数相乘（Multiplication）* • 2.1.3 矩阵相乘（Matrix multiplication） • 2.1.4 Discrete Fourier transform and Fast Fourier transform

32. 课程内容 • 2.2 动态规划 (Dynamic Programming) • 2.2.1 背包问题（Knapsack problem） • 2.2.2 最长递增子序列（Longest increasing subsequence） • 2.2.3 Sequence alignment • 2.2.4 最长相同子序列（Longest common subsequence） • 2.3.5 Matrix-chain multiplication • 2.3.6 树上的独立集 (Max Independent set in tree)

33. 课程内容 • 2.3 贪婪算法 （Greedy） • 2.3.1 区间规划（Interval scheduling） • 2.3.2 集合覆盖（Set cover） • 2.3.3 拟阵（Matroids） • 2.4 NP 问题 （NP-completeness） • 2.4.1 The classes P and NP • 2.4.2 NP-completeness and reducibility • 2.4.3 NP-complete problems *

34. 课程内容 • 2.5 近似算法 (Approximate Algorithm) • 2.5.1 顶点覆盖问题 （Vertex cover） • 2.5.2 负载平衡问题 (Load balancing) • 2.5.3 旅行商问题 (Traveling salesman problem) • 2.5.4 子集和问题 (Subset sum problem)

35. 课程内容 • 3. 算法的应用 • 3.1 局部搜索 (Local Search) • 3.1.1 The Metropolis Algorithm and Simulated Annealing • 3.1.2 Local Search to Hopfield Neural Networks（Nash Equilibria） • 3.1.3 Maximum Cut Approximation via Local Search

36. 课程内容 • 3.2 图论 (Graph Theorem) • 3.2.1 图论的基本知识 （Fundamental） • 3.2.2 线性规划 (Linear Programming) • 网络流（Network Flow），二分图，完全图的匹配 • 3.3计算几何学 (Computational Geometry) • 3.3.1 基本概念与折线段的性质 (Line-segment ) • 3.3.2 线段的一些性质 (Segments intersects ) • 3.3.3 凸包问题 (Convex Hull ) • 3.3.4 最近点对问题 (The closet pair of points) • 3.3.5 多边形三角剖分 (Polygon Triangulation)

37. 课程内容 • 3.4 字符串匹配 (String Matching)* • 3.4.1 字符串匹配的简单算法 • 3.4.2 The Karp-Rabin algorithm • 3.4.3 Finite automata, The Knuth-Morris-Pratt algorithm • 3.5 随机算法 (Randomized Algorithm) • 3.5.1 随机变量与期望 • 3.5.2 A Randomized MAX-3-SAT • 3.5.3 Randomized Divide-and-Conquer

38. 课程内容 • 3.6 组合数学与数论 （Number-Theoretic） • 3.7 高精度 (BigNums) • 3.8 数据结构 （Data Structure）* • 跳跃链表(Skip list), Fibonacci Heaps, Data Structures for Disjoint Sets • 3.9 矩阵运算/计算数学 （Matrix operations）* • 矩阵性质, 矩阵相乘, 解线性方程组, 高斯消去法 *：备选内容

39. 主要参考教材 • Introduction to algorithms, SecondEdition. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein. The MIT Press, 2001. ISBN: 0262032937. • Algorithm Design.Jon Kleinberg, Éva Tardos. Addison Wesley, 2005. ISBN: 0-321-29535-8. Rolf Nevanlinna Prize, 06

41. 参考教材 • Algorithms.S. Dasgupta, C.H. Papadimitriou, and U. V. Vazirani.May 2006. • Combinatorial Algorithms. Jeff Erickson. University of Illinois, Urbana-Champaign. Lecture Notes. Fall 2002. • Concrete Mathematics. Ronald L. Graham, Donald E. Knuth, Oren Patashnik. Addison-Wesley Publishing Company, 2005. ISBN: o-201-14236-8. • 数据结构与算法 – Data structures and algorithms.A. V. Aho J. E. Hopcroft, J. D. Ullman.清华大学出版社, 2003. • 算法艺术与信息学竞赛.刘汝佳 黄亮. ISBN 7-302-07800-9, 清华大学出版社,2004. • 计算几何—算法分析与设计.周培德.清华大学出版社ISBN 7-302-03801-5,2000.

42. 课程目标： • 实践算法设计和分析的技能。 • 培养各种证明能力，如：归纳法证明。 • 培养寻找最坏情况的例子。 • 培养对算法的直觉。能够对陌生问题提出算法，寻找解决问题的途径。 能判断什么情况下已找到最优解？或者能设计更好的算法。

43. Tools？

44. Requirement • Come to the class (*) • Ask questions • Thinking： Why it is ok now？ How about other methods？

45. Algorithms in Computer Science • P = NP? • Can we solve a problem efficiently? • Tradeoff between quality of solution and the running time • Solve a problem with optimal solution, but it might cost long time • Solve a problem approximately in short time

46. Example • Trucking company with a central warehouse • Each day, it loads up the truck at the warehouse and sends it around to several locations to make deliveries. • At the end of the day, the truck must end up back at the warehouse so that it ready to be loaded for the next day. • To reduce the costs, the company wants to select an order of delivery stops that yields the lowest overall distance traveled by the truck.

47. Algorithm as technology • Suppose computers were infinitely fast and computer memory was free. Do we have any reason to study algorithms? • The answer is: Yes

48. Lost cow problem • A short-sighted cow (or assume it’s dark, or foggy, or ...) is standing in front of a fence and does not know in which direction the only gate in the fence might be. How can the cow find the gate without walking too great a detour? • How can two soldiers get together when lost in battlefield ?

49. The Ski problem • The Ski problem [Karp 92]: A skier must decide every day she goes skiing whether to rent or buy skis, unless or until she decides to buy them. The skiier doesn’t know how many days she will go on skiing before she gets tired of this hobbie. The cost to rent skis for a day is 1 unit, while the cost to buy the skis is B units. • How can she save money?

50. Pizza delivery • One can give a call or via internet to order a pizza for dinner • We want the hot,fresh and tasty pizzas • How should they delivery the pizzas upon the reception of orders?? • Immediately or wait some minutes for next orders in the near places?