CS111, "Discrete Structures"

Winter 2016

  Schedule
  • Lecture: TR 05:10 p.m. - 06:30 p.m., HMNSS 1501.
    Instructor: Katya Mkrtchyan, email: mkrtchyk@cs.ucr.edu.
    Office hours: Tuesdays and Thursdays in WCH 110, 4:00PM-4:50PM.

  • Discussion 1: R 10:10 a.m. - 11:00 a.m.
    Teaching Assistant: Abbas Roayaei Ardakany, email: aroay001@ucr.edu.
    Office hours: WCH 110, Wednesday 10:00-11:00AM

  • Discussion 2: W 5:10 p.m. - 6:00 p.m.
    Teaching Assistant: Abbas Roayaei Ardakany, email: aroay001@ucr.edu.
    Office hours: WCH 110, Wednesday 10:00-11:00AM

  • Discussion 3: W 8:10 a.m. - 9:00 a.m.
    Teaching Assistant: Abbas Roayaei Ardakany, email: aroay001@ucr.edu.
    Office hours: WCH 110, Wednesday 10:00-11:00AM
  Syllabus

Textbook: E. Lehman, T. Leighton and A. Meyer Mathematics for Computer Science. You only need to print the sections that will be covered in the course.

Prerequisites: CS10, CS/MATH11, MATH 9C (or equivalents). The prerequisites are strictly enforced.

Prerequisites by topic: basic programming, logic (propositional, predicate), sets, operations on sets, sequences, relations (equivalence, partial orderings), functions, combinations, basic counting methods, elementary linear algebra (matrices, determinants), proof methods (induction, contradiction), elementary number theory.

Topics Covered:
  • Asymptotic notation: O(f(n)), Ω(f(n)), Θ(f(n)), asymptotic relations between basic functions: polynomial, exponential, and logarithmic functions
  • Number theory: modular arithmetic, Fermat's Theorem, public-key cryptography, the RSA
  • Advanced counting: inclusion-exclusion, linear recurrence equations, divide-and-conquer recurrences
  • Elements of graph theory: undirected and directed graphs, connectivity, planarity, Euler cycles, Hamiltonian cycles, matchings, trees
  • Other possible topics (if time suffices): elements of game theory, error-correcting codes

Homework Assignments: Five homework assignments, due every Thursday. The first homework assignment will be due January 14. To submit an assignment you will need to upload the pdf file into ilearn and turn-in a paper copy in class.
Homework assignments can be done individually or in groups of two (strongly recommended). Each homework will have three problems. Each group submits one paper with two names on it and both students will receive the same credit (unless requested otherwise).
Homework papers must be prepared with LaTeX. Handwritten assignments or assignments in Word or other word processors will not be accepted. LaTeX templates for homework assignments and other help with LaTeX will be available.
Homework papers must be well written, in grammatical English, self-contained, and aesthetically formatted. During the first week of the quarter you are required to read the homework assignment guidelines, and follow these guidelines throughout the quarter. Sloppy papers will not be graded.

Quizzes: Four 30-minute quizzes. The first "entrance" quiz will cover the prerequisite topics.

Final: Saturday, March 12, 2016, 11:30AM-2:30AM.

Attendance: Regular attendance at lectures and discussions is strongly advised. Some of the presented material may not be covered in the book or in posted lecture notes. Students are also strongly encouraged to take advantage of the office hours. In case of a conflict with regular walk-in office hours, special appointments can be arranged. Students that are at risk of failing the class may be required to attend office hours.

Grading: Quizzes 40%, Final 40%, Homeworks 20%. Course grades are expected to be determined as follows: A = 85-100%, B = 75-84%, C = 65-74%, D = 60-64%. Minor adjustments of this scale can be made at the end of the quarter.

Academic Integrity: Zero-tolerance policy on plagiarism is enforced. Cheating on homework assignments or tests will result in an F grade for the course and a disciplinary action, independently of the extent of plagiarism.

  Lectures

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Week 1 Tuesday, January 5
THINGS TO DO during the first week
Review: logic, sets, functions, relations, basic summation formulas, important numbers, sequences
Reading: Chapters 1, 2, 3, 4, 5, Sections 8.1, 8.2, 8.3, 14.1, 14.2.
Recommended exercises: 1.2, 1.5, 1.7, 3.2, 3.8, 3.21, 3.24
Thursday, January 7
Review (cont.): approximations, number theory basics, proofs, proofs by induction
Reading: Chapters 1, 2, 3, 4, 5, Sections 8.1, 8.2, 8.3, 14.1, 14.2.
Recommended exercises: 4.1, 4.7, 15.2, 15.4, 15.12, 15.15
Week 2 Tuesday, January 12
Quiz 1 (30 minutes)
Asymptotic notation
Reading: Section 14.7
Recommended exercises: 14.12, 14.20, 14.30, 14.32
Thursday, January 14
Proofs
Reading: Chapters 1-3
Recommended exercises: 6.1, 6.3, 6.17
Week 3 Tuesday, January 19
Number theory and cryptography
Breaking Turing's code (page 47-51)
Gcd, Euclid's algorithm, computing inverses mod p
Reading: Sections 8.1, 8.2, 8.3, 8.4, 8.5, 8.6
Recommended exercises: 8.2, 8.3, 8.4, 8.17, 8.20, 8.22
Thursday, January 21
HW 1 is due
Fermat's theorem
Turing's code, version 2 (page 47-51)
Computing powers modulo an integer
The RSA cryptosystem
Reading: Chapter 4,5
Recommended exercises: 8.23, 8.25
Week 4 Tuesday, January 26
RSA: correctness, security, efficiency
Famous open (and solved) problems in number theory
Reading:
Recommended exercises: 8.37
Thursday, January 28
Quiz 2 (30 minutes)
Linear recurrence equations (homogeneous)
Reading: Section 8.1 (page 189)
Recommended exercises:
Week 5 Tuesday, February 2
HW 2 is due
Linear recurrence equations (cont.)
Reading: Section 21.1, 21.3
Recommended exercises:
Thursday, February 4
Linear recurrence equations (non-homogeneous)
Reading: Section 21.2, 21.4
Recommended exercises:
Week 6 Tuesday, February 9
Quiz 3 (30 minutes)
Divide-and-conquer recurrences (cont.)

Reading: Section 21.2. 21.4
Recommended exercises:
Thursday, February 11
Inclusion-Exclusion
Integer partitions
Reading: Chapter 15, 6.
Recommended exercises:
Week 7 Tuesday, February 16
HW 3 is due
Graphs
Euler tours
Reading: Chapter 11.
Recommended exercises: 11.3, 11.4, 11.7, 11.30,
Thursday, February 18
Hamiltonian cycles, Dirac's theorem, Ore's theorem
Reading:
Recommended exercises:
Week 8 Tuesday, February 23
Quiz 4 (30 minutes)
Graph coloring, coloring graphs with maximum degree D
Reading:
Recommended exercises:
Thursday, February 25
Bipartite graphs: matchings, Hall's Theorem.
Reading: Chapter 11.
Recommended exercises: 11.21, 11.22, 11.23, 11.25, 11.8, 11.10, 11.11, 11.12
Week 9 Tuesday, March 1
HW 4 is due
Trees
Planar graphs: Kuratowski's theorem.
Reading: Chapter 12.
Recommended exercises: 12.2, 12.3, 12.6, 12.8,
Thursday, March 3
Euler's formula/inequality for planar graphs.
The 4-Color Theorem. Coloring planar graphs with 6 and 5 colors.
Reading: Chapter 12.
Recommended exercises:
Week 10 Tuesday, March 8
Extra credit
Adjacency matrices and matrix multiplication
Trees. Binary trees. Applications (lower bound for comparison sorting).
Reading: Section 9.3
Recommended exercises:
Thursday, March 10
HW 5 is due
TBA
Final Exam: NEW ROOM - exam will be held in HUMN 400, Saturday, March 12, 2016, 11:30am - 2:30pm


  Homework Assignments


LaTeX and Homework help.
Online LaTeX Editor Overleaf

LaTeX guides/manuals.

  Quizzes


  Final


  Other Books
  • V. Shoup, A Computational Introduction to Number Theory and Algebra (free)
  • K. Rosen, Discrete Mathematics and its Applications
  • S. Lipschutz, M. Lipson, Schaum's Outline of Discrete Mathematics
  • K. Bogart, C. Stein, R. Drysdale, Discrete Mathematics for Computer Science
  • B. Kolman, R. Busby, S. Ross, Discrete Mathematical Structures
  • R.C. Penner, Proof techniques and Mathematical Structures
  • F. Preparata, R. Tzu-Yau Yeh, Introduction to Discrete Structures for Computer Science and Engineering
  • S. Ross, Topics in Finite and Discrete Mathematics
  • K. Joshi, Foundations of Discrete Mathematics
  • R. Mc Eliece, R. Ash, C. Ash, Introduction to Discrete Mathematics
  • N.L. Biggs, Discrete Mathematics
  • I. Anderson, A First Course in Combinatorial Mathematics
  • S. Barrett, Discrete Mathematics, Numbers and Beyond
  • R.J. Wilson, Introduction to Graph Theory
  • S. Foldes, Fundamental Structures of Algebra and Discrete Mathematics


  Useful Links