Probabilistic Algorithms

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Probabilistic Algorithms

• Lecture 32
• CS 312

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Schedule

• Probabilistic algorithms for 4 lectures
• today (introduction and Buffons needle)
• 2 on Monte Carlo alg. to test primality
• 1 on Las Vegas for 8 queens
• 1 lecture about emergant computation and formal
  verification
• 1 review lecture

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Final Exam

• In the testing center
• Will include significant portion of multiple
  choice.
• Some traditional multiple choice
• samples will be given out.
• Some work out the problem multiple choice
• show work on separate paper for part. credit.

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Objectives

• Compare Las Vegas and Monte Carlo algorithms.
• Justify the use of probabilistic algorithms.

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Numerical Prob. Algs.

• Have a confidence interval
• 90 sure the answer lies between x and y.
• Gets better if run longer

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Monte Carlo Prob. Algs.

• High probability of a correct answer
• Low probability of incorrect answer
• No warning when an incorrect answer is given.

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Monte Carlo Gradebook

• Suppose your grade is a 95.
• If you repeatedly ask a 3/4-correct Monte Carlo
  algorithm
• Whats my grade?
• Youll (probably) get
• 95, 95, 90, 95, 70, 95, 95, 95, 95, 100, 95 ...

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Las Vegas Prob. Algs.

• Never gives incorrect answers
• Sometimes gives no answer

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Las Vegas Gradebook

• Suppose your grade is a 95.
• If you repeatedly ask a 3/4-correct Las Vegas
  algorithm
• Whats my grade?
• Youll (probably) get
• 95, dont know, 95, 95, dont know, 95, 95, 95,
  95, 95, dont know, 95

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Why?

• Suppose you really need to solve a problem.
• fate of the free world rests on your shoulders.
• Suppose the problem is really hard.
• random neutron diffusion in fissile material
• Suppose you dont have enough computing resources
  to solve the problem.
• ENIAC is all youve got.

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Buffons Needle

• 1707-1788, French
• Invented binomial theorem
• Noted natural historian
• Most famous for throwing needles
• over his shoulder to estimate Pi.

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Buffons Needle

• A needle of length m thrown onto a floor with
  lines every 2m has a 1/pi chance of landing on a
  line.
• So if n needles are thrown, and k hit a line,
  then n/k is an estimate for pi.
• Numerical probabilistic algorithm
• more needles give better estimate.
• 100x more needles for another digit of precision.