CSE 502N Fundamentals of Computer Science

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CSE 502NFundamentals of Computer Science

• Summer 2006
• (Some slides modified from David Taylor and
  Richard Souvenir)

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Course Information

• http//www.cse.wustl.edu/cse502
• Instructor Robert Glaubius
• TA TBA
• Highlights of course website
• Office Hours
• Syllabus
• Grading
• Coding Standards

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Pertinent Quote

• Computer Science is no more about computers than
  astronomy is about telescopes.- Edsger W.
  Dijkstra

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Course Overview

• We will focus on three core classes in the
  undergraduate C.S.curriculum
• Algorithms and data structures (CSE 241) (50)
• Sorting, searching, graph problems, and analysis
• Logic and discrete mathematics (CSE 240) (40)
• Recurrences, Boolean algebra, proof techniques,
  induction
• Introduction to digital logic and computer design
  (CSE 260M) (10)
• Logic design, computer architecture, finite state
  machines
• Fast-paced introduction to an extremely broad
  field
• Demands that you independently explore the
  concepts at greater depth
• Plan on significantly more out-of-class study
  time
• Ask questions early and often
• Get your moneys worth out of me!
• We will quickly build on concepts, so dont get
  behind!

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Closest-Pair Problem

• Your company has been asked to design an
  air-traffic control safetysystem by the FAA.
• The system must identify the closest two
  aircraftout of all the aircraft within radar
  range.
• For a set P containing n points find the closest
  pair of points (p1, p2), wherethe distance
  between p1 (x1, y1) and p2 (x2, y2) is the
  Euclidean distance

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What is an algorithm?

• CLR, pg. 1 An algorithm is any well-defined
  computational procedure that takes some value, or
  set of values, as input and produces some value,
  or set of values, as output. An algorithm is
  thus a sequence of computational steps that
  transform the input into the output.
• Easy answer Recipe

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What makes a recipe good?

• Resulting food is what you said it would be
• Hopefully, it should taste good
• Recipe is convenient (fast)
• These are the same criteria for algorithms
• Correctness
• Speed

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By the end of the summer

• When given a problem (such as FAA scenario), you
  should be able to
• Propose fast algorithms
• Using formal notation
• Prove the algorithms are correct
• Determine how fast the algorithms are
• Determine if youve designed the fastest
  algorithm possible

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Algorithms

• Well focus on two criteria for algorithms
• Correctness
• Speed
• How do we know if algorithm is correct?
• How do we know how fast our algorithm is?
• What does algorithm speed depend on?
• Some time function T(?, ?, , ?)
• What are the units of T?
• T(n) will be our notation for time of algorithm
  (in number of operations) as a function of the
  input size, n.

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Suggestions

• Your company has been asked to design an
  air-traffic control safetysystem by the FAA.
• The system must identify the closest two
  aircraftout of all the aircraft within radar
  range.
• For a set P containing n points find the closest
  pair of points (p1, p2), wherethe distance
  between p1 (x1, y1) and p2 (x2, y2) is the
  Euclidean distance

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Divide-and-Conquer Algorithm

• Let X be an array of points in P sorted by
  x-coordinate
• Let Y be an array of points in P sorted by
  y-coordinate
• Divide Find a vertical line l that bisects P
  into two sets PL (points on left) and PR (points
  on right)
• Divide arrays X and Y into XL and XR , YL and YR

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Divide-and-Conquer Alg. (contd)

• Find closest pair in PL and PR
• dL closest pair on left side
• dR closest pair on the right side
• Let d min(dL, dR)
• Are we done?

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Divide-and-Conquer Alg. (contd)

• Check points around the centerline
• Only a distance d away from the line
• Let Y contain these points sorted by y-coordinate

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Divide-and-Conquer Alg. (contd)

• Let d be the minimum distance between any two
  points in Y
• Closest pair is minimum of d and d
• Is this algorithm correct?
• Is this algorithm fast?
• How fast?
• Can you implement this algorithm?

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Points in Y

• How many other points do we need to measure the
  distance to?
• Y only spans length d on each side of the
  centerline in the x-direction
• No need to measure distance to any point higher
  than d in the y-direction
• So, how many points can fit in this d x 2d
  rectangle?
• Points on one side of line must be at least d
  distance from each other. Why?

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So how fast is this?
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Take-Home Message

• Algorithms place a fundamental limit on the
  scalability of the solution
• Possible to increase performance by orders of
  magnitude
• At best, CPU performance doubles every 18 months
  (Moores Law)
• Elegant algorithms are much easier to analyze and
  implement than ad-hoc solutions

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Coming Attractions

• Thursday
• Running time analysis
• Calculate worst-case (or expected) performance
• Saves a lot of implementation effort if the
  algorithm is not scalable
• Read CLR 3, 4.2 and Rosen 1.8
• Upcoming
• Proofs
• More formally assures that a solution is correct
• Important for critical applications (air traffic
  control, etc.)