Discrete Math for Computing II

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Discrete Math for Computing II

• Introduction

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Course information

• CS 3305-501
• Discrete Math for Computing II
• Meets Monday and Wednesday 700-815pm
  at ECSS 2.305

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What is this course about?

• We will cover more advanced concepts in discrete
  mathematics
• Discrete mathematics is the area of math that
  deals with counting and discrete structures
• Enumerate the objects in a specified collection
  or set
• Model collections of objects with inherent
  relations between them

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Course components

• Since discrete math is math, part of the course
  will further introduce you to mathematical
  reasoning, the art of understanding logical
  arguments and constructing proofs
• Discrete math is also fundamental for computer
  science, so we will cover numerous algorithms
  that specify step-by-step solutions

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Course applications

• Discrete mathematics offers ways to construct
  models for real applications
• In computer science, it allows us to understand
• data structures
• interaction between processes and agents
• computer networks, including the internet
• Other fields biology, linguistics, sociology,
  chemistry, medicine, business

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Instructor

• Vasileios Hatzivassiloglou
• Associate Professor, Computer Science
• Founding Professor, Bioengineering
• Research focus Discover knowledge from massive
  amounts of raw data
• a fairly applied field

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Research Interests

• Computational linguistics
• Computer modeling of human language
• Bioinformatics and medical informatics
• Computer modeling of molecular biology
  interactions, and knowledge modeling for clinical
  applications
• Supporting technologies text analysis, machine
  learning, data mining
• Applications web search, summarization, question
  answering

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Course topics

• Advanced counting Recurrences and the
  Inclusion-Exclusion Principle
• Relations properties of binary relations,
  representing relations, equivalence relations,
  partial orders
• Graphs graph representation, isomorphism, Euler
  paths, shortest path algorithms, planar graphs,
  graph coloring
• Trees tree applications, tree traversal, trees
  and sorting, spanning trees

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Course learning objectives

• Ability to construct and solve recurrence
  relations
• Ability to use the principle of inclusion and
  exclusion to solve problems
• Ability to understand binary relations and their
  applications
• Ability to recognize and use equivalence
  relations and partial orderings
• Ability to use and construct graphs and graph
  terminology
• Ability to apply the graph theory concepts of
  Euler and Hamilton circuits
• Ability to identify and use planar graphs and
  shortest path problems
• Ability to use and construct trees and tree
  terminology
• Ability to use and construct binary search trees

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Intended audience

• Undergraduate students in computer science
  seeking the theoretical foundations for later
  work in the field
• Undergraduate students in other fields seeking an
  understanding of basic modeling principles for
  discrete systems

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Prerequisites

• CS 2305, Discrete Math for Computing I
• You should know
• Propositional logic and proofs
• Basic mathematical structures sets, functions,
  sequences
• The concept of an algorithm
• Integers and matrices
• Mathematical induction and recursion
• Basic counting principles permutations and
  combinations
• Elementary concepts of discrete probability

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Contact information

• Office hours Monday and Wednesday 600-700pm
• Additional office hours by appointment
• Office location ECSS 3.406
• vh_at_hlt.utdallas.edu
• [phone]
• Teaching Assistant TBA

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Expected work load

• Easy problem sets given after each class
• Two in-class quizzes with one-week notification
• Four to five homework sets
• Each set requires 10-12 hours to solve
• Two weeks to turn in each homework set
• Mid-term exam
• Final exam

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Class participation

• I encourage active class participation
• Feel free to ask questions!
• Lectures are structured to allow you to provide
  input
• I will frequently ask questions
• Many of the questions will be challenging do
  not be afraid of being wrong!

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Academic honesty

• I expect you to respect UTDs policies and each
  other by not collaborating with each other in
  examinations and homework assignments
• Solve problems independently, do not copy from
  solutions obtained from elsewhere
• Minimum penalty is an F for the assignment and
  referral to the dean of students

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Grading

• Class participation 20
• Quizzes 10 (total)
• Homework assignments 30 (total)
• Midterm 15
• Final exam 25

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Textbook

• Discrete Mathematics and its Applications, 6th
  edition, by Kenneth H. Rosen
• McGraw-Hill, 2007
• ISBN 978-0-07-288008-3
• Available at the UTD bookstore, and also at
  Barnesandnoble.com (147) or Amazon.com (142)
• Recommended Students Solution Guide to the
  above (48)

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Additional book buying options

• Off-campus bookstore
• 561 W. Campbell Road, 201
• 972-907-8398
• http//www.offcampusbooks.bkstr.com/
• Used 127 main book, 44 solutions
• Stanza Textbooks
• 581 W. Campbell Road, 111
• 972-231-2665

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Textbook coverage

• We will cover selected topics from chapters
• 7 Advanced Counting Techniques
• 8 Relations
• 9 Graphs
• 10 Trees
• Additional application examples will be discussed
  by the instructor

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Counting

• Counting is
• the process of determining the number of objects
  in a specified set
• This set is typically specified by some
  properties, often as a model of a real-life
  collection of objects
• set of UTD IP addresses
• set of protein variants produced by a given part
  of DNA

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Why do we count?

• Determining the size of a set has implications
  for the complexity of tasks involving that set
• how much storage is needed (physical or in
  computer memory)
• how much additional work will be required
• In the particular case of algorithms, counting
  the number of steps taken allows us to estimate
  the computational burden of that algorithm

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Why do we count?

• Often, counting the size of a set enables us to
  calculate probabilities involving that set, which
  enables probabilistic reasoning and decision
  theory
• Examples
• Should one fold in Texas Holdem?
• Can Obama pass meaningful insurance reform?
• What is the genetic diversity of a species?

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Simple counting techniques

• Combinations of elementary objects
• can be extended in size without interactions
  between the objects
• example how many 5-out-of-7 card hands have four
  aces?
• Permutations of elementary objects, where only
  the order varies

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Advanced counting

• Sometimes it is impossible or very difficult to
  express the size of a set as a combination or
  permutation
• Typically this happens when there are
  restrictions between successive objects, or when
  only some of the objects combine to produce new
  ones

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Example one

• How many bit strings of length n do not have two
  consecutive zeros?
• interdependence of successive objects (bits in
  this case) makes straightforward combinatorial
  analysis difficult

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Example two

• Rabbits on an island reproduce at a rate of one
  new pair for every existing pair at least two
  months old. How many rabbits are there after n
  months?
• Only some of the rabbits (those two months old or
  older) reproduce every given month this number
  increases every month

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Using sequences

• We model our problem as a set that depends on a
  parameter n
• length of the bit string, number of months
• The size of a set that depends on a parameter n
  is a function of n
• Since n is an integer, the above function is a
  sequence an