Discrete Mathematics for Computer Scientists

Anna Bretscher and Richard Pancer

SY110 Bretscher Fridays 10:10-12pm

SY110 Pancer Fridays 13:10-14pm

Welcome to CSCA67

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

EvaluationTerm Test

– Week 7 or 8 worth 20%

Final Exam– Worth 40%

ResourcesCourse Slides

Posted each week.

Print them and bring to class.

Website

http://www.utsc.utoronto.ca/bretscher/a67/f13

Check the announcements daily.

Textbook

Stein, Drysdale and Bogart, Discrete Mathematics for Computer Scientists

Office Hours

Tutorials

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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!)

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

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.

Discrete MathematicsWho 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.

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

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

Where Does Mathematics Appear in Computer Science?

Databases Set theory, logic

Artificial Intelligence Set theory, logic

Programming Languages

Set theory, logic

Formal Methods Set theory, logic for the specification and verification of hardware and software; (e.g., nuclear, aviation – NASA!)

Course Outline

Proofs

3 weeks

Graph Theory

3 weeks

Counting

3 weeks

Probability

2 weeks

How Do I Become Good At This Stuff?

Same way that you become a good hockey player, calculus student, violinist…

practice, practice, practice...

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Weighted Job SchedulingProblem

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…

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Some Examples

Time

10

5

16 3

16

Some Examples

Time

25

10

19 3

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Finding an AlgorithmQ. 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.

* Merrian-Webster

Time

10

5

16 3

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Potential Algorithms

Largest weight first.

Other ideas?

Time

10

25

16 3

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

Q. Does this work?

A. Now Option 2 is Pick 16. Pick 11. Total = 27.

11

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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: http://www.cs.princeton.edu/~wayne/kleinberg-tardos/06dynamic-programming-2x2.pdf

Or take CSCC73.

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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

NotationJ: The set of jobs

ji: The ith job

si: The start time of the ith job

fi: The finish time of the ith job

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Scheduling The JobsWhat is our algorithm?

Examplej1 j2

j3 j4 j5

j6 j7

ChallengeFirst person to come up with a correct algorithm wins chocolate.

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Scheduling The JobsWhat is our algorithm?

Sort the jobs by increasing start time. Schedule each job if there are no conflicts.

Examplej1 j2

j3 j4 j5

j6 j7

Ordering j1, j3, j6, j4, j7, j2, j5

Schedulej1, j7, j5

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Scheduling The JobsWhat is our algorithm?

Sort the jobs by increasing start time. Schedule each job if there are no conflicts.

j1

j2

j3 j4 j5

j6 j7Ordering

j1, j3, j6, j4, j7, j2, j5

Schedulej1

Does this algorithm always find the best solution?

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Scheduling The JobsOther ideas?

Ordering

Schedule

Does this algorithm always find the best solution?

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Scheduling The JobsWhat is our algorithm?

Sort the jobs by increasing finish time. Schedule each job if there are no conflicts.

Examplej1 j2

j3 j4 j5

j6 j7

Ordering j3, j6, j1, j7, j4, j2, j5

Schedulej3, j7, j5

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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.

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The ProofLet 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

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The ProofS = (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.

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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).