Prerequisites:

COP 4531: Analysis of Data Structures and Algorithms (this pre-requisite will not be waived). You are also expected to have the mathematical maturity to write formal proofs and algorithms.

Lecture hours:

MW 3:35 pm - 4:50 pm, LOV 103.

Contact information:

Instructor:	Ashok Srinivasan
Office hours:	MW 1:30 pm - 2:30 pm. I am also usually available in my office, and you can feel free to meet me in the afternoons. Alternatively, you may schedule an appointment, either by email or by phone.
Office:	169, Love Building
Phone:	644-0559
Email:	asriniva@cs.fsu.edu
Course web site:	Access through blackboard http://campus.fsu.edu.

Text books:

• Required:
  1. Introduction to algorithms, T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, second edition, MIT press, 2001. (Referred to as CLR). Note: You will be given handouts for all material from sources other than CLR.
• Recommended:
  1. Approximation algorithms, V. V. Vazirani, Springer, 2003 (Referred to as AA).

• Other references:
  1. Computational Geometry: Algorithms and Applications, M. De Berg , M. van Kreveld, M. Overmars, and O. Schwarzkopf, Springer-Verlag, 1997 (Referred to as CG).
  2. Randomized Algorithms, R. Motwani and P. Raghavan, Cambridge University Press, 1995 (Referred to as RA).
  3. A compendium of NP optimization problems, www.nada.kth.se/~viggo/problemlist/compendium.html. (Referred to as VK).

Course rationale:

In your undergraduate algorithms course, you learned efficient algorithms for many common problems in computer science. In this course, you will learn algorithms that use more sophisticated techniques than the ones in the undergraduate course. Many of the algorithms relate to some type of optimization, and arise in many important applications, which you will typically encounter in other graduate courses, your research, and in the real-world. While the applications are important, the techniques used to solve them are well beyond the scope of an undergraduate course.

As an example, there is a class of problems called NP hard, for which it is believed that efficient algorithms cannot be developed. Many important optimization problems belong to this class. While showing a problem to be NP hard relieves you from the responsibility of designing an efficient algorithm to solve it, the problem itself does not disappear, and we still need to be able to do something about it. In this course, you will learn techniques to find reasonably good approximate solutions efficiently, for many NP hard problems.

Thus, techniques you learn in this course will enable you to solve important problems that you would not be able to solve otherwise. Furthermore, you will also learn more powerful techniques for analysis of the algorithms, and learn to write formal proofs, which will help you in your research.

Course description:

Problems that you have studied so far were somewhat simple, in the sense that the algorithms used to solve them were rather straightforward. From example, it was easy to see that an algorithm was correct and to determine its time complexity. Topics we discuss in this course can be broadly classified into two categories: (i) difficult problems, for which we think it is unlikely that efficient algorithms exist, and (ii) difficult algorithms, for which proof of correctness or analysis of time complexity are difficult. The discussion will sometimes span both categories, when we discuss approximation algorithms for the difficult problems

To be more specific, we will discuss the following topics. (i) We will first study network flow problems, which arise in several optimization applications. (ii) We will then discuss techniques to approximate NP hard optimization problems. The emphasis will be on techniques based on linear programming, which are applicable to a wide variety of problems, rather than on ad-hoc solutions to specific problems. (iii) We will next study parallel algorithms, where several processors work together to solve a given problem. We will see that efficient parallelization often requires solving an NP hard optimization problem too. (iv) Typical analyses of algorithms use the computational effort of an algorithm to measure their efficiency. However, in modern processors, efficient use of the memory hierarchy is the major bottleneck, rather than operations in the CPU. We will discuss cache-aware and cache-oblivious algorithms, which exploit the memory hierarchy efficiently. (v) We will next discuss randomized algorithms. Fortunately, these algorithms do not give random answers. The answers are always correct for the algorithms we discuss, but the time taken for a given input can vary. Randomized algorithms can often improve the expected time complexity significantly. (vi) We will introduce computational geometry, by discussing a few algorithms for range searching, which involves searching for data satisfying certain conditions, from a large set of data. (vii) We will finally discuss algorithms for string matching, which has important applications in computational biology.

This course does not require any programming. It does involve deep thinking and rigorous theoretical analyses. Therefore, much mathematical maturity is required of you. (However, specific mathematical skills, beyond those learned in undergraduate Discrete Mathematics courses, are not required.) For example, you should be able to understand and write formal proofs. This course will also require you to be creative; you will need to solve problems that differ from the ones discussed in class, though the solution strategies will often be related to those discussed.

Learning objectives:

By the end of the semester, you should develop the following skills: (i) understand and apply the algorithms discussed in class, prove their correctness, and analyze their time complexity in a mathematically rigorous manner, (ii) understand the basic idea behind the techniques, so that you are able to develop algorithms for new problems where these techniques can be applied, and (iii) given a practical application, identify the computational issues and apply suitable algorithms to solve it effectively. We give more details below (objectives shown in bold are particularly important). You will also later be given more specific objectives for each lecture.

Network flow problems

• Describe the notion of network flows and define the maximum flow problem.
• Prove the max-flow min-cut theorem.
• Given a flow network, apply the Ford-Fulkerson method, in particular, the Edmonds-Karp implementation, to it, to determine a max-flow.
• Prove the correctness of the Ford-Fulkerson algorithm, and derive the time complexities for the basic Ford-Fulkerson method and for the Edmonds-Karp implementation.
• Given alternate schemes for determining max-flow, prove their correctness and determine their time complexity, or prove that the algorithm is incorrect.
• Given a practical application, show how it can be formulated as a max-flow problem.

Approximation algorithms

• Define the following terms: (i) combinatorial optimization, (ii)approximation factor of an approximation algorithm, (iii) polynomial time approximation scheme, and (iv) fully polynomial time approximation scheme.
• Given a problem instance, show steps in the approximation algorithms for the following problems: vertex cover (two algorithms), set cover (three algorithms), knapsack, and maximum satisfiability.
• Analyze the time complexity and determine the approximation factor for each of the above algorithms and its variants.
• Given an NP hard combinatorial optimization problem, give an approximation algorithm for it and determine its approximation factor.
• Given an application, abstract its computational structure, give an approximation algorithm to solve it, and derive its approximation factor.
• Explain the use of linear programming, with rounding, randomized rounding, derandomization of the rounding, and primal-dual schemes, to approximately solve NP hard problems.

Parallel algorithms

• Give justifications for the model we used to analyze the time complexity of parallel algorithms.
• Define the terms speedup and efficiency.
• Given a parallel algorithm, analyze its time complexity.
• Give efficient algorithms to perform common operation such as reduction, broadcast, gather, scatter, all-gather, and parallel prefix, and derive their time complexity.
• Explain the steps in the algorithms we discuss for solving linear recurrences and matrix operations, apply these algorithms to instances of problems, and also derive the time complexity of these parallel algorithms.
• Given a problem, develop a parallel algorithm for it and analyze its time complexity.
• Explain the use of graph partitioning in parallel computation, give the limitations of the edge cut metric to measure communication cost, and propose improvements to the edge-cut metric.
• Given a graph, show how the spectral method partitions the graph.
• Formally justify the use of the spectral method as a heuristic for graph partitioning.

Cache-aware and cache-oblivious algorithms

• Describe the ideal cache model and derive the number of cache misses for the simple matrix multiplication algorithm.
• Show how blocking can be used to reduce the number of cache misses in matrix multiplication.
• Show how recursion can be used to reduce the number of cache misses in a "cache-oblivious" manner.

Randomized algorithms

• Given a sequence of integers and a sequence of random numbers, demonstrate the steps in the randomized quicksort algorithm RandQS.
• Analyze the expected time complexity of randomized quicksort.
• Given a sequence of integers and a sequence of random numbers, demonstrate the steps in the randomized selection algorithm Find.
• Analyze the expected time complexity of the above selection algorithm.
• Analyze the time complexity of modifications to the randomized sorting and selection algorithms discussed above.

Computational Geometry

• Define the range search problem.
• Derive the time and space complexities of constructing a range tree, and the time complexity of searching for a given range query.
• Derive the time and space complexities of constructing a layered range tree, and the time complexity of searching for a given range query.
• Given a set of data, determine a range tree or layered range tree that represents it, and show the steps involved in searching that tree for a given range query.

String matching

• Define the string-matching problem, describe a naive algorithm for solving it, and derive its time complexity.
• Derive the time complexities of the Rabin-Karp, finite-automaton, and Knuth-Morris-Pratt algorithms for string-matching.
• Given two strings, show the steps taken by the Rabin-Karp algorithm for the given problem instance.
• Given modified versions of the string-matching problem, give modifications to the Rabin-Karp algorithms to solve the modified problems.
• Given two strings, show the steps taken by the finite-automaton based algorithm for the given problem instance.
• Given two strings, show the steps taken by the Knuth-Morris-Pratt algorithm for the given problem instance.

Grading criteria:

Class participation and review questions 20 Three Homework assignments 3 x 10 = 30 Midterm 20 Final Exam 30

Course average Letter grade 85 - 100 A 80 - 85 A- 75-80 B+ 70-75 B 65-70 B- 0-65 F to C+

Your grade will be based on the scores obtained in the above categories, with weights as given above. You should also score at least 60% in the weighted average of the midterm and finals in order to get a grade of B or higher. Note that the thresholds for grades are a little lower than they normally are in other courses. Please don't think this makes it easier to obtain a high grade. They are lower because the exams are more difficult!

Examples

1. If you get 83% total and 93% on the exams, your grade will be A-.
2. If you get 83% total and 33% on the exams, you will not get A-. Your grade will be C+, since the exam grade is below 60%.

Class participation and review questions

Class participation consists of answering questions correctly in class, and other positive contributions, discussed in greater detail at www.cs.fsu.edu/~asriniva/courses/alg04/classparticipation.html.

Assignments

Assignments will all be theoretical; there will be no programming involved. Hardcopies should be turned in to me in my office. If I am not in my office, you can slip it under my door. Assignments are due 5 pm on the due date.

Midterm and final exams

Both exams will be closed book. You will be tested on knowledge, understanding, and application of material discussed in class. You will also have some questions testing your ability to creatively solve new problems, using techniques discussed in class. The final exam will be comprehensive.

Your responsibilities:

Deadlines and instructions

Following the same professional guidelines that you will encounter in business, there are strict deadlines, and instructions that must be followed. Please read instructions carefully, and schedule your activities so that you submit assignments well in time. You should check your garnet email account and the class web page regularly, and note other announcements, on-line and in class.

Participation

I will ask you questions in class: (i) review questions on the previous lecture, and (ii) questions on the material currently being discussed, in order for me to obtain feedback on how well you understand the material. You should be prepared to answer these questions, and should also participate by asking questions, suggesting ideas, and performing in-class assignments that I give. Of course, you cannot participate in class unless you attend it!

Reading assignments

You will have a reading assignment for each lecture, which you should read. New material often builds on the old ones. So, if you have trouble with some material, please get help from me before the next class. You should also peruse the material for the next lecture, and be prepared to answer questions on it, which I will provide in advance. I expect that you will need to spend between one and two hours studying, for each lecture. The assignments and exams will consume additional time. The following learning components are important, and you may want to verify if you do satisfactorily on these, after studying the material.

• Knowledge: Examples -- Do you understand the terminology used? Given an algorithm and problem instance, can you describe the steps carried out by the algorithm and the output?
• Understanding: Examples -- If some aspect of an algorithm, were modified, can you analyze the time complexity, prove its correctness, or prove that it is incorrect? If some aspect of the problem where changed, can you suggest modifications to the algorithm used to solve it? In order to answer such questions, you need to understand how each component of an algorithm problem affects the time complexity, and why each component of an algorithm is important for its correctness. After you learn about what an algorithm does (and have, thus, acquired "knowledge"), it will be useful for you to think of different things that can be changed, and see how that will affect the time complexity or correctness. Similarly, you have to understand how different aspects of the problem are dealt with by different components of the algorithm.
• Theoretical analysis: You can often intuitively understand why an algorithm works, or why its time complexity is what we claim it to be. However, you should also learn to formally prove this in a mathematically rigorous manner. You will also need to write algorithms precisely. Students often have trouble with this. I suggest that, until you attain some proficiency in these skills, you practice writing proofs and algorithms for review questions that I post after each lecture, and get feedback from me.
• Creativity: Can you modify algorithms that we have studied, to make them more efficient for special situations? Given a problem for which our algorithm is not valid as designed, can you modify the algorithm to solve the problem, and then prove the correctness of your solution, and analyze its time complexity? Can you prove that a new problem which you are presented is NP complete, or that it is hard to approximate within a given factor? Questions related to creativity will form the major part of the grade.

Assignments

You will have three assignments in this course, and you will have around ten days to work on each one.

Course policies:

Attendance Policy:

The university requires attendance in all classes, and it is also important to your learning. The attendance record may be provided to deans who request it. If your grade is just a little below the cutoff for a higher grade, your attendance will be one of the factors that we consider, in deciding whether to "bump" you up to the higher grade. Missing three or fewer lectures will be considered good attendance. In rare cases, such as medical needs or jury duty, absences may be excused with appropriate documentation. You should let me know in advance, when possible, and submit the documentation I seek. You should make up for any materials missed due to absences.

Missed exam Policy:

A missed exam will be recorded as a grade of zero. We will follow the university rules regarding missed final exams (see http://registrar.fsu.edu/dir_class/fall/exam_schedule.htm), for all the exams, including the final exam.

Late Assignment Policy:

In order to enable us to provide timely solutions to assignments, we have the following policy regarding submission of late assignments.

• An assignment that is turned in no more than 48 hours late will be scored with a 20% penalty.
• An assignment that is turned in more than 48 hours late will receive the score of zero, though we will review it and comment on it.

Grade of 'I' Policy:

The grade of 'I' will be assigned only under the following exceptional circumstances:

• The final exam is missed with an accepted excuse for the absence. In this case, the final exam must be made up during the first two weeks of the following semester.

Professional ethics:

You will gain confidence in your ability only when you do the work independently. On the other hand, one does learn a lot through discussions with ones peers. In order to balance these two goals, I give below a list of things that you may, and may not, do.

Things you may not do: You should not copy from the assignments of other students. This includes directly copying or making modifications to others' assignments. If you happen to find a solution to an assignment problem from source, you may not copy it. Furthermore, you should take steps to ensure that others cannot copy your assignment.

Things you may do: You may ask others about (i) terminology that you do not understand, (ii) clarifications on what is required of you in an assignment problem (though you may be much better of asking me), and (iii) how to submit your assignment. After the assignment has been graded, you are encouraged to discuss each others' solutions.

Honor Code: Students are expected to uphold the academic honor code published in "The Florida State University Bulletin" and the "Student Handbook". Please read the provisions of the Academic Honor Code: http://www.fsu.edu/Books/Student-Handbook/codes/honor.html. Also read the section on "Honor code" below.

Plagiarism:

Plagiarism is "representing another's work or any part thereof, be it published or unpublished, as ones own. For example, plagiarism includes failure to use quotation marks or other conventional markings around material quoted from any source" (Florida State University General Bulletin 1998-1999, p. 69). Failure to document material properly, that is, to indicate that the material came from another source, is also considered a form of plagiarism. Copying someone else's program, and turning it in as if it were your own work, is also considered plagiarism.

ADA: Students with disabilities needing academic accommodation should (1) register with and provide documentation to the Student Disability Center, and (2) bring a letter to the instructor indicating the need for accommodation and what type. This should be done during the first week of class. For more information about services available to FSU students with disabilities, contact:

Student Disability Resource Center
Dean of Students Department
08 Kellum Hall
Florida State University
Tallahassee, FL 32306-4400
(850) 644-9566 (voice)
(850) 644-8504 (TDD)
SDRC@admin.fsu.edu
http://www.fsu.edu/~staffair/dean/StudentDisability/

Course calender:

Week	Lecture	Chapter	Assignments
1	3 Jan	No class
	5 Jan	Introduction.
2	10 Jan	Maximum flow -- flow networks CLR -- Chapter 26: up to sec 26.1
	12 Jan	Maximum flow -- Ford-Fulkerson method (to be continued) Read: CLR -- Sec 26.2
3	17 Jan	Martin Luther King day -- no class.
	19 Jan	Maximum flow -- Ford-Fulkerson method (completed) Read: CLR -- Sec 26.2
4	24 Jan	Maximum flow -- Maximum bipartite matching Read: CLR -- Sec 26.3	Assignment 1 announced Jan 25.
	26 Jan	Approximation algorithms -- Vertex cover Read: CLR -- Chapter 35: up to sec 35.1, AA (handout) -- Chapter 1: up to sec 1.1
5	31 Jan	Approximation algorithms -- Set cover Read: CLR -- sec 35.3, AA (handout) -- Chapter 2: up to sec 2.1	Assignment 1 due Feb 4.
	2 Feb	Approximation algorithms -- Knapsack (to be continued) Read: AA (handout) -- Chapter 8: up to sec 8.2.
6	7 Feb	Approximation algorithms -- Knapsack (completed) Read: AA (handout) -- Chapter 8: up to sec 8.2.
	9 Feb	Linear programming concepts Read: CLR -- Chapter 29: up to sec 29.2.
7	14 Feb	Linear programming concepts -- Duality, approximation algorithms Read: AA (handout) -- Sec 12.1, sec 12.3 (CLR -- Sec 29.4 has some related material).
	16 Feb	Approximation algorithms -- Rounding applied to set cover Read: AA (handout) -- Chapter 14, up to sec 14.1.
8	21 Feb	Approximation algorithms -- Set cover via primal-dual schema Read: AA (handout) -- Chapter 15.	Assignment 2 announced Feb 21.
	23 Feb	Approximation algorithms -- Maximum satisfiability AA (handout) -- Chapter 16.
9	28 Feb	Midterm review	Assignment 2 due Mar 4.
	2 Mar	Midterm
10	7 Mar	Spring break -- no class
	9 Mar	Spring break -- no class
11	14 Mar	Parallel algorithms -- models and basic operations Read: Class notes.
	16 Mar	Parallel algorithms -- Linear recurrences, matrix operations Read: Class notes.
12	21 Mar	Parallel algorithms -- Graph partitioning Read: Class notes.
	23 Mar	Cache aware and cache oblivious algorithms -- Matrix multiplication Read: Class notes.
13	28 Mar	Randomized algorithms -- randomized quicksort Read: RA (handout) -- Chapter 1, up to sec 1.1.	Assignment 3 announced Mar 28.
	30 Mar	Randomized algorithms -- randomized selection Read: RA (handout) -- Sec 1.4, sec 3.3.
14	4 Apr	Computational Geometry -- Range search (to be continued) Read: CG (handout) -- Chapter 5, sec 5.1, 5.3.	Assignment 3 due Apr 8.
	6 Apr	Computational Geometry -- Range search (completed) Read: CG (handout) -- Chapter 5, sec 5.4, 5,5, and 5.6.
15	11 Apr	String matching (to be continued) Read: CLR -- Chapter 32, up to sec 32.1.
	13 Apr	String matching (to be continued) Read: CLR -- Sec 32.2.
16	18 Apr	String matching (to be continued) Read: CLR -- Sec 32.3..
	20 Apr	String matching (completed) Read: CLR -- Sec 32.4.
17	29 Apr	Final exam: 3 - 5 pm

SYLLABUS CHANGE POLICY:

This syllabus is a guide for the course and is subject to change with advanced notice.