Transcript Probabilistic Analysis and Randomized Algorithms • Example Problem • Probabilistic Analysis

Probabilistic Analysis and
Randomized Algorithms
• Example Problem
– Hiring problem
• Probabilistic Analysis
– Indicator variables
– Linearity of expectation
• Randomized Algorithms
• Online Hiring Problem
Hiring Problem
• Input
– A sequence of n candidates for a position
– Each has a distinct quality rating that we can determine in an
interview
• Algorithm
– Current = 0;
– For k = 1 to n
• If candidate(k) is better than Current, hire(k) and Current = k;;
• Cost:
– Number of hires
• Worst-case cost is n
Probabilistic Analysis
• Assume a probability distribution
• Analyze item of interest over probability
distribution
• Analysis implies result based on probability
distribution
• Caveats
– Specific inputs may have much worse performance
– If distribution is wrong, analysis may give misleading
picture
Analyze Hiring Problem
• Assume a probability distribution
– Each of the n! permutations is equally likely
• Analyze item of interest over probability
distribution
– Let Xi = probability that ith interviewed candidate is
hired
– E[Xi] = ? And why?
– Si = 1 to n Xi = ?
– Linearity of expectations: Can sum expectations of
variables to get their expected sum even if variables are
dependent on each other
Alternative analysis
• Analyze item of interest over probability
distribution
– Let Xi = probability that ith best candidate is
hired
– E[Xi] = ? And why?
– Si = 1 to n Xi = ?
Questions
• What is the probability you will hire n times?
• What is the probability you will hire exactly
twice?
• Biased Coin
– Suppose you want to output 0 with probability ½ and 1
with probability ½
– You have a coin that outputs 1 with probability p and 0
with probability 1-p for some unknown 0 < p < 1
– Can you use this coin to output 0 and 1 fairly?
– What is the expected running time to produce the fair
output as a function of p?
Hat Check Problem
• N people give their hats to a hat-check
person at a restaurant
• The hat-check person returns the hats to the
people randomly
• What is the expected number of people who
get their own hat back?
Randomized Algorithm
• Randomize the algorithm so that it works
well with high probability on all inputs
• In this case, randomize the order that
candidates arrive
• Universal hash functions: randomize
selection of hash function to use
Online Hiring Problem
• Input
– A sequence of n candidates for a position
– Each has a distinct quality rating that we can determine in an interview
• We know total ranking of interviewed candidates, but not with respect to
candidates left to interview
– We can hire only once
• Online Algorithm Hire(k,n)
– Current = 0;
– For j = 1 to k
• If candidate(j) is better than Current, Current = j;
– For j = k+1 to n
• If candidate(j) is better than current, hire(j) and return
• Questions:
– What is probability we hire the best qualified candidate given k?
– What is best value of k to maximize above probability?
Online Hiring Analysis
• Let Si be the probability that we successfully hire
the best qualified candidate AND this candidate
was the ith one interviewed
• Let M(j) = the candidate in 1 through j with
highest score
• What needs to happen for Si to be true?
– Best candidate is in position i: Bi
– No candidate in positions k+1 through i-1 are hired: Oi
– These two quantities are independent, so we can
multiply their probabilities to get Si
Computing S
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Bi = 1/n
Oi = k/(i-1)
Si = k/(n(i-1))
S = Si>k Si = k/n Si>k 1/(i-1) is probability of
success
• k/n (Hn – Hk): roughly k/n (ln n – ln k)
• Maximized when k = n/e
• Leads to probability of success of 1/e