1. Welcome to CSCA67 Discrete Mathematics for Computer Scientists
Anna Bretscher and Richard Pancer
SY110 Bretscher Fridays 10:10-12pm
SY110 Pancer Fridays 13:10-14pm

2. Evaluation Assignments 4 each worth 10%
Late assignments will be accepted up to 24 hrs late with a penalty of 25%
You are encouraged to discuss the problems with other students however, the actual write up must be an individual effort
You must be able to reproduce any solution that you submit. The penalty for cheating ranges from a zero on the assignment to suspension from the university

3. Evaluation
Term Test
Week 7 or 8 worth 20%
Final Exam
Worth 40%

4. Resources
Course Slides Posted each week. Print them and bring to class. Website Check the announcements daily. Textbook Stein, Drysdale and Bogart, Discrete Mathematics for Computer Scientists Office Hours Tutorials

5. Course Expectations Expectations of the lecturer
Give clear, organized lectures
Assign fair, challenging assignments that ensure that you, the student, understand the material
Be available for help in office hours
Help every student achieve their goals in the course (this requires your help!)

6. Course Expectations Expectations of the student
Attend lectures and participate
Bring course notes to class
Review lecture notes after each class, not just before the exam
Complete homework fully, neatly and independently
Have respect for your classmates and lecturers

7. Course Expectations What does neatly mean? Staple sheets
Write legibly (if you are incapable of this skill, please type)
Your work should be of a quality that you would feel comfortable giving to your boss in a work environment.

8. Discrete Mathematics Who needs it?
Anyone in computer science or a mathematical science
Why?
In CS we need to be able to
speak precisely without ambiguity
analyze problems and
formulate solutions
apply the concepts associated with probability, graph theory and counting theory.

9. CS is Applied Mathematics!
Specifically, we will work on:
Thinking abstractly
Expressing ourselves precisely
Arguing logically – i.e., inferring conclusions that necessarily result from assumptions
Writing rigorous solutions
Learning how mathematics and computer science work together

10. Where Does Mathematics Appear in Computer Science?
Computer Graphics
Multivariable calculus, physics-based modelling
Digital Signal Processing
Multivariable calculus, (eg., speech understanding)
Numerical Analysis
Multivariable calculus, linear algebra
Cryptography
Number theory
Networking Algorithms
Graph theory, statistics, combinatorics, probability, set theory

11. Where Does Mathematics Appear in Computer Science?
Databases
Set theory, logic
Artificial Intelligence
Programming Languages
Formal Methods
Set theory, logic for the specification and verification of hardware and software; (e.g., nuclear, aviation – NASA!)

12. Course Outline Proofs 3 weeks Graph Theory Counting Probability

13. How Do I Become Good At This Stuff?
Same way that you become a good hockey player, calculus student, violinist…
practice, practice, practice...

14. Weighted Job Scheduling
Problem
Given a set of jobs where
each job has a value or weight
jobs have a start time and duration
Schedule the jobs so that
the total value is maximized
scheduled jobs do not overlap
Q. What are some examples of this type of problem?
A. printer queue, airplanes on runway, server use, scheduling surgeries…

15. Some Examples 10 5 16 3
Time

16. Some Examples 10 19 3 25
Time

17. Finding an Algorithm Q. What is an algorithm?
A. A step-by-step procedure for solving a problem or accomplishing some end especially by a computer.*
Let’s list some potential algorithms to try:
Earliest start time.
Time 10 5 16 3
* Merrian-Webster

18. Potential Algorithms Largest weight first. Other ideas? Time 25 10 163
Time

19. Potential Algorithms
Create all possible orders by largest weight first.
Eg. Option 1: Pick 25.
Option 2: Pick 16. Pick 10. Pick 3. Total = 29.
Time 10 25 16 3 11
Q. Does this work?
A. Now Option 2 is Pick 16. Pick 11. Total = 27.

20. So...What’s the Answer?
Use a mathematical tool called Dynamic Programming.
You can read about it in a text such as Algorithm Design by Kleinberg and Tardos.
Or visit this link:
programming-2x2.pdf
Or take CSCC73.

21. A Simpler Problem Interval Scheduling Problem
Given a set of jobs where
jobs have a start time and finish time
Schedule the jobs so that
the number of jobs scheduled is maximized
jobs do not overlap
Notation
J: The set of jobs
ji: The ith job
si: The start time of the ith job
fi: The finish time of the ith job

22. Scheduling The Jobs Challenge What is our algorithm? Example
Let students work for a few minutes. Ask them to get their neighbour to try and find a counterexample. Remind them what a counterexample is. Explain what is meant by “general” - ie, works for more than just the given example, works for all cases.
Challenge
First person to come up with a correct algorithm wins chocolate.

23. Scheduling The Jobs What is our algorithm? Example Ordering Schedule
Sort the jobs by increasing start time. Schedule each job if there are no conflicts.
Example j1 j2 j3 j4 j5 j6 j7
Some students may suggest trying the job with earliest start time.
Ordering
j1, j3, j6, j4, j7, j2, j5
Schedule
j1, j7, j5

24. Scheduling The Jobs What is our algorithm?
Sort the jobs by increasing start time. Schedule each job if there are no conflicts.
Does this algorithm always find the best solution? j2 j1 j3 j4 j5 j6 j7
Now show that this doesn’t always give the optimal answer.
Ordering
j1, j3, j6, j4, j7, j2, j5
Schedule j1

25. Scheduling The Jobs Other ideas?
Does this algorithm always find the best solution?
Some students may say by shortest duration - show that this doesn’t always work as well. simple example is two long jobs consecutively and one short one that overlaps a little with each of the other two. If they say something else, get the class to decide if there is a counter example.
Ordering
Schedule

26. Scheduling The Jobs What is our algorithm? Example Ordering Schedule
Sort the jobs by increasing finish time. Schedule each job if there are no conflicts.
Example j1 j2 j3 j4 j5 j6 j7
Now show an algorithm that does always find the optimal.
Ordering
j3, j6, j1, j7, j4, j2, j5
Schedule
j3, j7, j5

27. Is It Correct? Q. How do we know our algorithm is correct?
A. Prove it!
One common proof technique is “Proof by Contradiction”
Idea
Play Devil’s Advocate
Assume our solution is not the best
This means there is a better solution B
Show that our algorithm’s solution is as good as B by making B equivalent to our solution.
Point out that in previous cases we were able to disprove algorithms by giving a counter example. How do we know that a counter example doesn’t exist? perhaps we just haven’t thought of a good one yet...this is why we need to prove that our solution is the best possible.

28. The Proof Let S be the schedule our algorithm creates.
S = (s1, s2, s3, ... , sn)
Let B be a better solution.
B = (b1, b2, b3, ... , bm)
Q. What do we know about m and n?
A. m >= n
Q. What do we know about b1 and s1?
A. The finish time for s1 is less than or equal to the finish time for b1.
in order of non-decreasing finish time

29. The Proof S = (s1, s2, s3, ... , sn) B = (b1, b2, b3, ... , bm)
Q. If the finish time for s1 is less then or equal to the finish time for b1, what can we say about B* = (s1, b2, b3, ... , bm)?
A. |B*| = |B|. So B* is as good a solution as B but is more similar to S.
Q. How should the proof finish?
A. Repeat the argument with S’ = (s2, s3,…, sn) and B’ = (b2, b3,…, bm).
Q. Why does this work?
A. Induction - more on this later.

30. Challenge Another Scheduling Problem
Given a set of employees, want to set up a meeting that everyone can attend. Each person has a calendar which says whether they are available for any given time slot during the day.
Give an efficient algorithm to schedule the meeting so that everyone can attend (if possible).