Makeup class this Friday (May 25, tentative time: 11:30-2:40pm) in 524 Mudd. The class will be recorded. There will be no class on Tuesday, June 12.

Homework 1 will be out on Friday May 25 and due by 10pm on Monday, June 4. Please note that you must submit your assignment as a pdf file on courseworks.

Lecture videos will be available to every student. Please be patient while I work this out with CVN.

Textbook: Introduction to Algorithms, by Cormen, Leiserson, Rivest and Stein (3rd edition) will be the main text for this course. Other recommended reading: Algorithm Design, by Kleinberg and Tardos; Algorithms, by Dasgupta, Papadimitriou and Vazirani

Grading: Homeworks 40%, Midterm 30%, Final 30%. The midterm and final exams will be in-class. The percentage contributions above are approximate and subject to change.

Prerequisites

COMS 3137/3139: Data Structures and Algorithms and COMS 3203: Discrete Mathematics.

Familiarity with mathematical proofs and how to write one, as well as basic knowledge of probability and linear algebra is required.

You might want to refresh your memory on some of the required math background for this class by going over the following sections in our textbook:

Appendix A: pp. 1145-1147, a basic understanding of 1150-1153

Appendix B: B.1, B.3, B.4 (parts of B4 and B5 will be covered in class)

Appendix C: pp. 1183-1186, C.2, C.3, C.4

Appendix D: D.1, D.2

Course description

This course focuses on the design and analysis of efficient algorithms. We will discuss general design paradigms as well as specific problems.
A preliminary list of topics follows.

Fundamentals: induction, asymptotics, recurrences

Sorting and searching: insertion sort, merge-sort, quicksort, binary search

Greedy algorithms: huffman coding, minimum spanning trees

Network flows

Dynamic programming: data segmentation, sequence alignment

Linear programming

Reductions and NP-completeness

Hashing: Bloom filters, document similarity

Approximation algorithms

Homeworks

General: There will be four homework assignments to help you better assimilate the course material.

Homework assignments will be out on Fridays (late in the evening). They will be due 10 days later by 10pm. For a tentative schedule of homework assignments and/or to download a homework, please go to the end of this section.

You should submit your homework assignment online (on canvas) as a pdf file. If needed, scan your handwritten assignment and submit the resulting pdf---make sure the quality of your scanned document is good and your handwriting very neat!

Late homework policy: No homework extensions will be granted. However you have 5 late days for all assignments: for example, you may submit Homework 1 at 9:59pm on Wed, 6/6, thus using 2 of your 5 late days. Note the following hard deadlines for Homework 2 and Homework 4:

Homework 2: 10am on Wed, 6/13 (if you do so, you will use 0.5 late days)

Homework 4: 7pm on Tue, 6/26 (if you do so, you will use 0.75 late days)

Collaboration policy: You are allowed to brainstorm and think through solutions with a small number (2-3) of your classmates. However you must write up your solutions entirely on your own. If you have used collaborators, you must state their names clearly next to your name on your write-up. Finally, copying solutions from the Internet or other textbooks is strictly prohibited. You should adhere to the department's academic honesty policy

Homework 1: Out: Friday, May 25, Due: 10pm on Monday, June 4

Homework 2: Out: Friday, June 1, Due: 10pm on Tuesday, June 12 (HARD deadline: 10am on Wednesday, June 13)

Homework 3: Out: Friday, June 8, Due: 10pm on Monday, June 18

Homework 4: Out: Friday, June 15, Due: 10pm on Monday, June 25 (HARD deadline: 5pm on Tuesday, June 26)

Tue, 5/29: Analysis of randomized Quicksort; balls-in-bins problems, the coupon collector problem; hashing: hash table, universal hash functions, analysis of chain hashing using balls-and-bins; hashing for saving space: fingerprints, Bloom filters [randQS-ballsinbins, hashing, Sections 7.3, 7.4, 5.1, 5.2 without the hiring problem]

Thu, 5/31: Greedy algorithms: Huffman coding for data compression; cache maintenance, online algorithms [huffman, 16.3, cache, if you need more context on this problem, read Section 4.3 from Tardos-Kleinberg; optional reading: Chapter 6 from your textbook (binary min heap)]

Thu, 6/7: Matrix-chain multiplication; [matrixchainmultiplication, Sections 15.0, 15.2, 15.3 without the discussion on shortest/longest simple path]; single-source shortest paths in weighted graphs (negative edge weights): Bellman-Ford; [shortestpaths-negativeweights, optional reading: Sections 24.0-24.3, Section 25.2]; all-pairs shortest paths: Floyd-Warshall

Fri, 6/8, 12-3:10pm (makeup class 3, full): Network flows, the Ford-Fulkerson algorithm for max flow; correctness of Ford-Fulkerson, applications of max flow: maximum bipartite matching [flows, pp. 708-720, 724-726, pp. 720-724, Section 26.3]

Fri, 6/8: hw3 out

Tue, 6/12:no class (makeup class on 5/25)

Tue, 6/12: hw2 due by 10pm (HARD deadline: 10am on Wed 6/13)

Thu, 6/14:midterm exam

Midterm exam information

Time: 1-2:30pm (regular time) (no class after the exam, makeup class on 6/1)

Location: 52 Mudd (regular classroom)

Material: all lectures up to and including lecture 6/7, except for the following:

makeup lecture 6/1;

Belman-Ford & Floyd-Warshall from lecture 6/7

Midterm is closed books, no calculators, no cheat sheet, no external aids

Fri, 6/15: hw4 out

Mon, 6/18: hw3 due by 10pm

Tue, 6/19:no class (makeup class on Friday, 6/8)

Thu, 6/21: Reductions; complexity classes P, NP, Satisfiability of boolean functions (formulas, circuits); proving NP-completeness, TSP and more NP-complete problems [reductions, Sections 34.0, 34.1 up to and including p. 1056, 34.3 up to p. 1073, SAT, optional reading: Section 34.4, 34.5.2, more NP-complete problems, optional reading: 34.2 up to and not including Verification Algorithms, 34.5.4, p. 1118]

Mon, 6/25: hw4 due by 10pm (HARD deadline: 5pm on Tue 6/26)

Tue, 6/26: Linear programs, weak and strong duality, dualization; interpreting the dual LP; [LP_part1, optional reading: Sections 29.0, 29.1 (without the slack form), Theorem 29.13 on p. 892 (just the statement, no proof), duality, optional reading: Section 29.4 up to and including Corollary 29.9, Section 29.2 (without min-cost flow/multi-commodity flow)]; Approximation algorithms for set cover, vertex cover [IP-setcover-approximation, Section 11.4]

Thu, 6/28: Final exam

Final exam information:

Day/Time: Thursday, June 28, 1-3pm

Location: 524 Mudd Building

Material: all lectures from 6/8 (Belman-Ford & Floyd-Warshall) up to and including 6/26, WITHOUT linear programming

Final exam is closed books, no calculators, no cheat sheet, no external aids.

## Announcements

May 24: Next Thursday's (5/31) and Friday's (6/1) classes will take place in 1127 Mudd due to construction work next to our regular classroom.May 24: Reminder:makeup class TOMORROW11:30-2:40pmin 524 Mudd. The class will be recorded.May 20: Several important announcements follow:## General course information

Time:TTR, 1-4:10pmLocation:524 Mudd BuildingInstructor:Eleni Drinea, eleni@cs.columbia.eduOffice Hours:Tuesdays, 4:15-6pm (or by appointment), Mudd 414Teaching Assistants and Office Hours:Textbook:Introduction to Algorithms, by Cormen, Leiserson, Rivest and Stein (3rd edition) will be the main text for this course. Other recommended reading: Algorithm Design, by Kleinberg and Tardos; Algorithms, by Dasgupta, Papadimitriou and VaziraniGrading:Homeworks 40%, Midterm 30%, Final 30%. The midterm and final exams will be in-class. The percentage contributions above are approximate and subject to change.## Prerequisites

## Course description

This course focuses on the design and analysis of efficient algorithms. We will discuss general design paradigms as well as specific problems.A preliminary list of topics follows.

## Homeworks

General:There will be four homework assignments to help you better assimilate the course material.Late homework policy: No homework extensions will be granted. However you have 5 late days for all assignments: for example, you may submit Homework 1 at 9:59pm on Wed, 6/6, thus using 2 of your 5 late days. Note the followinghard deadlinesfor Homework 2 and Homework 4:Collaboration policy:You are allowed to brainstorm and think through solutions with a small number (2-3) of your classmates. However you must write up your solutions entirely on your own. If you have used collaborators, you must state their names clearly next to your name on your write-up. Finally, copying solutions from the Internet or other textbooks is strictly prohibited. You should adhere to the department's academic honesty policyTuesday, June 12(HARD deadline: 10am on Wednesday, June 13)HARD deadline: 5pm on Tuesday, June 26)## Preliminary schedule of lectures

Tue, 5/22:Insertion sort, induction, worst-case running time analysis, efficient algorithms, asymptotic notation, divide and conquer principle, mergesort, solving recurrences: recursion trees [insertionsort, Sections 1.1, 2.1, 2.2, mergesort, mastertheorem, Sections 3.1, 2.3, 4.0]Thu, 5/24:Master theorem; more divide & conquer algorithms: binary search, fast integer multiplication, fast matrix multiplication (Strassen's algorithm); quicksort, randomized algorithms, randomized Quicksort, balls-in-bins problems [moredivideandconquer, quicksort, balls in bins, Sections 4.2, 4.3, 4.4, 4.5, 7.1, 7.2]Fri, 5/25, 11:30am-2:40pm (makeup class 1, full):Graphs, graph representation, Breadth-First Search and applications: connected components, testing bipartiteness; Depth-First Search and applications: cycle detection, topological sort [graphsandBFS, optional reading: Appendix B, Sections B.4, B.5, pp. 1168-1179, Sections 22.1, 22.2, dfs-toposort, optional reading: Sections 22.3, 22.4]Fri, 5/25: hw1 outTue, 5/29:Analysis of randomized Quicksort; balls-in-bins problems, the coupon collector problem; hashing: hash table, universal hash functions, analysis of chain hashing using balls-and-bins; hashing for saving space: fingerprints, Bloom filters [randQS-ballsinbins, hashing, Sections 7.3, 7.4, 5.1, 5.2 without the hiring problem]Thu, 5/31:Greedy algorithms: Huffman coding for data compression; cache maintenance, online algorithms [huffman, 16.3, cache, if you need more context on this problem, read Section 4.3 from Tardos-Kleinberg; optional reading: Chapter 6 from your textbook (binary min heap)]Fri, 6/1, 12-1:30pm (makeup class 2, short):strongly connected components in directed graphs [stronglyconnectedcomponents, optional reading: Section 22.5]; single-source shortest paths in weighted graphs (non-negative edge weights): Dijkstra's algorithm [shortestpaths-negativeweights, optional reading: Sections 24.0-24.3]Fri, 6/1: hw2 outMon, 6/4: hw1 due by 10pmTue, 6/5:Dynamic programming principle: segmented least squares, sequence alignment, iterative and memoized implementations [datasegmentation, sequencealignment, recommended reading: Chapter 15, Section 15.4],Thu, 6/7:Matrix-chain multiplication; [matrixchainmultiplication, Sections 15.0, 15.2, 15.3 without the discussion on shortest/longest simple path]; single-source shortest paths in weighted graphs (negative edge weights): Bellman-Ford; [shortestpaths-negativeweights, optional reading: Sections 24.0-24.3, Section 25.2]; all-pairs shortest paths: Floyd-WarshallFri, 6/8, 12-3:10pm (makeup class 3, full):Network flows, the Ford-Fulkerson algorithm for max flow; correctness of Ford-Fulkerson, applications of max flow: maximum bipartite matching [flows, pp. 708-720, 724-726, pp. 720-724, Section 26.3]Fri, 6/8: hw3 outTue, 6/12:no class (makeup class on 5/25)Tue, 6/12: hw2 due by 10pm (HARD deadline: 10am on Wed 6/13)Thu, 6/14:midterm examMidterm exam informationTime: 1-2:30pm (regular time) (no class after the exam, makeup class on 6/1)Location: 52 Mudd (regular classroom)Material: all lectures up to and including lecture6/7,except for the following:closed books, no calculators, no cheat sheet, no external aidsFri, 6/15: hw4 outMon, 6/18: hw3 due by 10pmTue, 6/19:no class (makeup class on Friday, 6/8)Thu, 6/21: Reductions; complexity classes P, NP, Satisfiability of boolean functions (formulas, circuits); proving NP-completeness, TSP and more NP-complete problems [reductions, Sections 34.0, 34.1 up to and including p. 1056, 34.3 up to p. 1073, SAT, optional reading: Section 34.4, 34.5.2, more NP-complete problems, optional reading: 34.2 up to and not including Verification Algorithms, 34.5.4, p. 1118])Tue, 6/26:Linear programs, weak and strong duality, dualization; interpreting the dual LP; [LP_part1, optional reading: Sections 29.0, 29.1 (without the slack form), Theorem 29.13 on p. 892 (just the statement, no proof), duality, optional reading: Section 29.4 up to and including Corollary 29.9, Section 29.2 (without min-cost flow/multi-commodity flow)]; Approximation algorithms for set cover, vertex cover [IP-setcover-approximation, Section 11.4]Thu, 6/28: Final examFinal exam information:Day/Time:Thursday, June 28, 1-3pmLocation:524 Mudd BuildingMaterial:all lectures from 6/8 (Belman-Ford & Floyd-Warshall) up to and including 6/26, WITHOUT linear programmingclosed books, no calculators, no cheat sheet, no external aids.