Intro to Computer Algorithms Lecture 1

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Intro to Computer Algorithms Lecture 1

• Phillip G. Bradford
• Computer Science
• University of Alabama

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General Lecture Format

• Some power-point slides
• Some white-board writing
• Details are very important
• Lots of Interaction a good thing

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Lecture Outline

• Syllabus
• Learning
• Grading
• Your Expectations
• My Expectations
• Material High Level Overview
• Motivation
• Foundations, History and Perspective
• Applications of Algorithms the Future

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Lecture Outline

• The book by A. Levitin
• Design paradigms approach
• Nice way to think about design analysis
• Classifications of challenges on how they are
  solved
• Commonalities are the solutions
• Lets get started
• Chapter 1.1 to 1.3

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My Expectations

• Exercise gives strength
• This is an exercise course
• Excellent Students give excellent results
• Given a good deal of exercise
• What are your expectations?
• Survey

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Motivation

• How hard is it to solve Computer Challenges?
• Deep Questionheavy ramifications
• Optimization Applications
• Networks
• Financial Models, business issues, etc.
• Real-Time Applications
• Airplanes, trains, cars
• Missiles, Sensors, medical applications, etc.

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Motivation

• Foundational questions
• Sorting searching
• Exhaustive Search
• Apparent hardness of old and new challenges
• Cuts across engineering and many other science
  disciplines

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Algorithms

• What is an algorithm?
• Unambiguous list of instructions that solve a
  challenge
• Gives the right answer on legitimate input
• What about algorithms that use randomness?
• Designed with randomness in mind
• Really, pseudo-random numbers

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Algorithms what to doPage 10, Figure 1.2

• Understand the challenge!
• Design an algorithm
• Easy sub cases first, perhaps
• Prove correctness
• Analyze the algorithm
• Check improve design if analysis warrants
• Iterate, iterate,

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Euclids Algorithm

• The challenge
• Greatest-common divisors
• GCD(12,6) 6
• GCD(17,16) 1
• GCD(20,15) 5
• Euclids method (assume mgtn)
• GCD(m,0) m
• GCD(m,n) GCD(n, m mod n)

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Euclids Algorithm, cont.

• Book example
• GCD(60,24) GCD(24,12) GCD(12,0) 12
• Euclid(m,n)
• While(n ! 0)
• r ? m mod n
• m ? n
• n ? r
• Return m

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Euclids Algorithm

• Correctness?
• Halts?
• Prime-Factor approach
• How?
• Which is better??
• Relative costs?

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Euclids Algorithm Correctness

• Assume m gt n
• What is the mod function?
• Start m nfloor(m/n) m mod n
• So, m - nfloor(m/n) m mod n
• Thus, if pm and p(nfloor(m/n))
• Then p(m mod n)
• What about piqj, for both p and q primes and igt0
  and jgt0 integers?

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Euclids Algorithm Correctness

• How do we know it halts?
• Guaranteed a parameter always decreases at each
  iteration
• Why?
• Answer in HW

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Sieve of Eratosthenes

• Challenge given an integer n, find all primes lt
  n.
• Example n 18, then output 2,3,5,7,11,13,17
• Elimination of prime candidates by potential
  factors
• Claim the largest potential factor
  floor(sqroot(n))

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Types of Problems

• Sorting, Searching
• Stable Sorts
• String Processing
• Graph problems
• Combinatorial problems
• TSP
• Geometric Problems
• Numerical Problems

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Homework set

• Due next Thursday
• Prove Euclids algorithm always halts on positive
  integer inputs mgtn
• Page 18 5, 9
• Page 24 1, 4,
• Some Old HW briefly reviewed on Tuesday
• Page 17, 1
• Page 18, 3, 8

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Lets Have a Great Semester!

• Good work pays off!
• Make use of my office hours!
• Consider me your algorithms coach!