1. Probabilistic Analysis and Randomized Algorithms
Example Problem
Hiring problem
Probabilistic Analysis
Indicator variables
Linearity of expectation
Randomized Algorithms
Online Hiring Problem

2. Hiring Problem Input Algorithm Cost: Worst-case cost is n
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

3. 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

4. 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

5. 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 = ?

6. 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?

7. 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?

8. 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

9. Online Hiring Problem Input Online Algorithm Hire(k,n) Questions:
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?

10. 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

11. Computing S 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