精研课程

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课程基本信息课程编号： 21190120

上课时间、地点： 2012 年 夏学期周二 (9 、 10) 曹光彪 西 101 、周四（ 3~4 ）曹光彪 西 101

上机：周日（ 11 、 12 ）软件学院机房考试时间： 6 月 11 日（ hard deadline ）考试形式：提交课程论文—综述学时 / 学分： 4-2/ 周 学时 / 2-1 学分

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Office & Homepage

Office ：工商楼 215

My Homepage ：http://www.cs.zju.edu.cn/people/yedeshi/

Course Home:

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Examination

Grading Polices ：1) Homework: 15%

2) Quiz: 10% (5th Week, May 15)

3) Presentation+ Poster session : 20% (Slides submit to TA),

Videos maybe taken

Poster: in Lab time

4) Programming Project: 20% (2-3programms)

5) Final Survey: 35%

TA information

金淑贤 : rolesGuide to group the students

Collect the homework, program, and survey

Review of homework, programs

Collect the scores of each presentations/posters

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What is the position of algorithms in CS

1. Linguists: what shall we talk to the machines?

2. Algorithms: what is a good method for solving a problem fast on my computer

3. Architects: Can I build a better computer?

4. Sculptors of Machine Intelligence: Can I write a computer program that can find its own solution.

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Algorithms in Computer ScienceA

lgo

rithm

s

Hardware

Compilers, Programming languages

Machine learning, Statistics, Information retrieval, AI

Networking, Distributed systems, Fault tolerance, Security

Bioinformatics .......

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MIT Undergraduate Programs

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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.

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What will CS be?

Computer Science:Past, Present, and Future

Ed Lazowska (Washington)

Computer Science is the new Math

Christos H. Papadimitriou (Berkeley)

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Algorithm

Problem definition 问题

Objective 目标 (very important)

Evaluation 算法评价

Methods 方法

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Algorithm evaluation

Quality: how far away from the optimal solution ?

Cost: Running time Space needed

Our goal is to design algorithm with high quality, but in low cost

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Reasonable times

Poly(|I|), Time polynomial in |I|, where |I| is the size of the problem instanceInput size: size(x) of an instance x with rational data is the total number of bits needed for the binary prepresentation.

Time complexity

logarithmic time if T(n) = O(log n).

sub-linear time if T(n) = o(n)

linear time, or O(n) time

linearithmic function: T(n) = O(n log n), quasilinear time if T(n) = O(n logk n)

polynomial time: T(n) = O(nk) for some constant kstrongly polynomial time:

the number of operations in the arithmetic model of computation is bounded by a polynomial in the number of integers in the input instance; and

the space used by the algorithm is bounded by a polynomial in the size of the input.

weakly polynomial time: P but not strongly P

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Time complexity

Quasi-polynomial time: for some fixed c.

Sub-exponential time if T(n) = 2o(n)

Exponential time, if T(n) is upper bounded by 2poly(n)

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Hardness of problems

Polynomial (e.g. n2, n log n, n3, n1000).

Quasi-polynomial(e.g.:n log n, n log2n, c log7n).

Sub-exponential (e.g.: 2√n, 5(n0.98)).

Exponential (e.g.: 2n, 8n, n!, nn).

Easy

Hard

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Running timeComputer A is 100 times faster than computer B

Sort n numbersComputer A requires instructions

Computer B requires 50nlgn instructions

n = 1,000, 000Computer A: 2(10^6)^2/10^9 = 2000 seconds

Computer B: 50*10^6 lg 10^6/10^7 ~ 100 seconds

22n

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Running time

10 < 1 s < 1s < 1 s < 1 s < 1 s 4 s

100 < 1 s < 1 s < 1 s 1 s 18 min

year

1,000 < 1 s < 1 s 1 s 18 min

Very long

Very long

10,000 < 1 s < 1 s 2 min 12 day

Very long

Very long

1 s 20 s 12 days

31710 year

Very long

Very long

n2n3nlognn 2n n!

1025

106

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Sorting输入： A sequence of n number

输出：排列（ permutation ）< a0

1, a02,… ,a0

n >

使得： a01 <=a0

2 <= ... <=a0n

Example:

Input: 8 2 4 9 3 6Output: 2 3 4 6 8 9

< a1,a2, …,an >

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EX. of insertion sort

8 2 4 9 3 6

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8 2 4 9 3 6

2 8 4 9 3 6

EX. of insertion sort

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8 2 4 9 3 6

2 8 4 9 3 6

EX. of insertion sort

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8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

EX. of insertion sort

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8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

EX. of insertion sort

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8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

2 4 8 9 3 6

EX. of insertion sort

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8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

2 4 8 9 3 6

EX. of insertion sort

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8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

2 4 8 9 3 6

2 3 4 8 9 6

EX. of insertion sort

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8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

2 4 8 9 3 6

2 3 4 8 9 6

EX. of insertion sort

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8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

2 4 8 9 3 6

2 3 4 8 9 6

2 3 4 6 8 9 done

EX. of insertion sort

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Insertion sortINSERTION-SORT (A, n) ⊳ A[1 . . n]

for j ← 2 to ndo key ← A[ j]

i ← j – 1while i > 0 and A[i] > key

do A[i+1] ← A[i]i ← i – 1

A[i+1] = key

“pseudocode”

i j

keysortedsorted

AA::1 n

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Analyzing algorithms

Need a computational model

Random-access machine (RAM) modelInstructions are executed one after another. No concurrent operations.

Arithmetic: add, subtract, multiply, divide, remainder, floor, ceiling

Data movement: load, store, copy

Control: conditional/unconditional branch, subroutine call and return.

Each of these instructions takes a constant amount of time.

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Running time

Running time: The running time of an algorithm on a particular input is the number of primitive operations or “steps” executed.line consists only of primitive operations and takes constant time

Input size: number of items the total number of bits.more than one number: Graph

the number of vertices and the number of edges

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The input size of sorting problem is n.Worst-case running time of Insert sort is O(n2).

Example:

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Running time

The running time depends on the input: an already sorted sequence is easier to sort.

Parameterize the running time by the size of the input, since short sequences are easier to sort than long ones.

Generally, we seek upper bounds on the running time, because everybody likes a guarantee.

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Map of Algorithm Design

Off-line problem

Polynomial NP-C problem

Improve cost running time

Exact Algorithm

HeuristicApproximate

Algorithm

Improve cost running time

QualityAppro. ratio

On-line problem

New problem

QualityAppro. ratio

Polynomial

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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

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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)

课程内容

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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 *

课程内容

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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)

课程内容

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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

课程内容

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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)

课程内容

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3.4 随机算法 (Randomized Algorithm) 3.4.1 随机变量与期望 3.4.2 A Randomized MAX-3-SAT 3.4.3 Randomized Divide-and-Conquer

3.5 在线算法（ Online Algorithm ） 3.5.1 Online Skying 3.5.2 Online Hiring

课程内容

* ：备选内容

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课程内容

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教材 Textbook:Textbook: Introduction to algorithms Introduction to algorithms, Second Edition. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein. The MIT Press, 2001. ISBN: 0262032937.

Recommended:Recommended: Algorithm Design.Algorithm Design. Jon Kleinberg, Éva Tardos. Addison Wesley, 2005. ISBN: 0-321-29535-8.

Rolf Nevanlinna Rolf Nevanlinna Prize, 06Prize, 06

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Algorithms.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 MathematicsConcrete Mathematics. Ronald L. Graham, Donald E. Knuth, Oren Patashnik. Addison-Wesley Publishing Company, 2005. ISBN: o-201-14236-8.

参考教材

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Algorithms in Computer ScienceP = NPP = 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

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$1,000,000 problem

P = NPP = NP ? http://www.claymath.org/millennium/Solved???!!!!

Algorithms in Computer Science

Selfish Routing

Privacy preserve in database

TSP

Ad auction

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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

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Selfish routingPigou's Example

Suburb s, a nearby train station t.

Assuming that all drivers aim to minimize the driving time from s to t

s t

C(x) = 1

C(x) = x, with x in [0, 1]

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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

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Braess's Paradox

s

v

w

t

C(x) = 1

C(x) = xC(x) = 1

C(x) = x

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Braess's Paradox

s

v

w

t

C(x) = 1

C(x) = xC(x) = 1

C(x) = x

C(x) = 0

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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.

Link attack example

Re-identify the medical record of the governor of Massachussetts

MA collects and publishes sanitized medical data for state employees (microdata) left circle

voter registration list of MA (publicly available data) right circle

• looking for governor’s record• join the tables:

– 6 people had his birth date

– 3 were men

– 1 in his zipcode

• regarding the US 1990 census data– 87% of the population are unique based on (zipcode,

gender, dob)

Privacy in microdatathe role of attributes in microdata

explicit identifiers are removed

quasi identifiers can be used to re-identify individuals

sensitive attributes (may not exist!) carry sensitive information

Name Birthdate

Sex Zipcode Disease

Andre 21/1/79 male 53715 Flu

Beth 10/1/81 female 55410 Hepatitis

Carol 1/10/44 female 90210 Brochitis

Dan 21/2/84 male 02174 Sprained Ankle

Ellen 19/4/72 female 02237 AIDS

identifier

quasi identifiers sensitive

Name Birthdate

Sex Zipcode Disease

Andre 21/1/79 male 53715 FluBeth 10/1/81 female 55410 HepatitisCarol 1/10/44 female 90210 BrochitisDan 21/2/84 male 02174 Sprained

AnkleEllen 19/4/72 female 02237 AIDS

k-anonymityk-anonymity: intuitively, hide each individual among k-1 others

each QI set of values should appear at least k times in the released microdata

linking cannot be performed with confidence > 1/k

sensitive attributes are not considered (going to revisit this...)

how to achieve this?generalization and suppression

value perturbation is not considered (we should remain truthful to original values )

privacy vs utility tradeoffdo not anonymize more than necessary

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Advertisement Auction

Auction Dutch auction

Vickrey auction

Ad placement

k-anonymity exampletools for anonymization

generalizationpublish more general values, i.e., given a domain hierarchy, roll-up

suppressionremove tuples, i.e., do not publish outliers

often the number of suppressed tuples is bounded

Birthdate

Sex Zipcode

21/1/79 male 53715

10/1/79 female 55410

1/10/44 female 90210

21/2/83 male 02274

19/4/82 male 02237

Birthdate

SexZipcode

group 1*/1/79 person 5*****/1/79 person 5****

suppressed

1/10/44 female 90210

group 2*/*/8* male 022***/*/8* male 022**

original microdata 2-anonymous data

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TSPTrucking 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.

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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?

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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?

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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 ?

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Erdős project – shortest path

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.

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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

Daniel Alan Spielman 2010 USA

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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

Extensions of shortest path

On k-skip Shortest Paths (SIGMOD 2011)

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History of Algorithm

The word algorithm comes 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.

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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).

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Why you come here?

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Requirement

Come to the class (*)

Ask questions

Thinking ：Why it is ok now ？How about other methods ？

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Kinds of analyses

Worst-case: (usually)T(n) = maximum time of algorithm on any input of size n.

Average-case: (sometimes)T(n) = expected time of algorithm over all inputs of size n.Need assumption of statistical distribution of inputs.

Best-case: (bogus) Cheat with a slow algorithm that works fast on

some input.

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Uniform distribution

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Performance Measures for On-line Algorithms

Competitive ratio

Max/Max ratio

Smoothed Competitiveness