Randomized algorithms

1. Randomized algorithms Hiring problem We always want the best hire for a job! • Using employment agency to send one candidate at a time • Each day, we interview one candidate • We must decide immediately if to hire candidate and if so, fire previous In other words … • Always want the best hire… • Cost to interview (low) • Cost to fire/hire… (expensive) CSC317

2. Hire-Assistant(n) 1. best = 0 //least qualified candidate 2. for i = 1 to n 3. interviewcandidate i 4. ifcandidate i betterthan best 5. best= i 6. hire candidate i • Always want the best hire… fire if better candidate comes along • Cost to interview (low) Ci • Cost to fire/hire… (expensive) Ch Ch> Ci n total number of candidates m total number of hires O (?) CSC317

3. Hire-Assistant(n) 1. best = 0 //least qualified candidate 2. for i = 1 to n 3. interviewcandidate i 4. ifcandidate i betterthan best 5. best= i 6. hire candidate i • Always want the best hire… fire if better candidate comes along • Cost to interview (low) Ci • Cost to fire/hire… (expensive) Ch Ch> Ci n total number of candidates m total number of hires O (?) different cost of hiring Not run time but cost of hiring, but flavor of max problem CSC317

4. Always want the best hire… fire if better candidate comes along • Cost to interview (low) Ci • Cost to fire/hire… (expensive) Ch Ch> Ci n total number of candidates m total number of hires O (cin + chm) Different type of cost – not run time, but cost of hiring. But when is it most expensive? When candidates interview in reverse order, worst first… CSC317

5. Hiring problem 1. best = 0 //least qualified candidate 2. fori = worst_candidatetobest_candidate… O (cin + chn) = O(chn) When is this most expensive? When candidates interview in reverse order, worst first… CSC317

6. Always want the best hire… fire if better candidate comes along • Cost to interview (low) Ci • Cost to fire/hire… (expensive) Ch Ch> Ci n total number of candidates m total number of hires When is it least expensive? CSC317

7. Hiring problem 1. best = 0 //least qualified candidate 2. fori = best_candidatetoworst_candidate… O (cin + ch) When is this leatexpensive? When candidates interview in order, best first… But we don’t know who is best a priori CSC317

8. Hiring problem • Cost to interview (low Ci) • Cost to fire/hire … (expensive Ch) • n number of candidates • m hired O (cin + chm) depends on order of candidates Independent of order of candidates What about if we randomly invite candidates? CSC317

9. Hiring problem and randomized algorithms O (cin + chm) depends on order of candidates • Can’t assume in advance that candidates are in random order – bias might exist. • We can add randomization to our algorithm – by asking the agency to send names in advance and randomize CSC317

10. Hiring problem - randomized We always want the best hire for a job! • Employment agency sends list of n candidates in advance • Each day, we choose randomly a candidate from the list to interview • We do not rely on the agency to send us randomly, but rather take control in algorithm CSC317

11. Randomized hiring problem(n) 1. randomly permute the list of candidates 2. Hire-Assistant(n) What do we mean by randomly permute (we need a random number generator)? Random(a,b) Function that returns an integer between a and b, inclusive, where each integer has equal probability Most programs: pseudorandom-number generator CSC317

12. Including random number generator Random(a,b) Function that returns an integer between a and b, inclusive, where each integer has equal probability So, Random(0,1)? 0 with P = 0.5 1 with P = 0.5 CSC317

13. Including random number generator Random(a,b) Function that returns an integer between a and b, inclusive, where each integer has equal probability So, Random(3,7)? 3with P = 0.2 4with P = 0.2 5with P = 0.2 6with P = 0.2 7with P = 0.2 Like rolling a (7-3+1) dice CSC317

14. How do we random permutations? Randomize-in-place(A,n) 1. For i = 1 to n 2. swap A[i] with A[Random(i,n)] A = 1 2 3 4 5 6 A = 3 2 1 4 5 6 A = 3 6 14 5 2 A = 3 6 1 4 5 2 … i Random choice from A[i,…,n] CSC317

15. Probability review • Sample space S: space of all possible outcomes (e.g. that can happen in a coin toss) • Probability of each outcome: p(i) ≥ 0 • Example: coin toss S = {H, T} CSC317

16. Probability review • Event: a subset of all possible outcomes; subset of sample space S • Probability of event: • Example: coin toss that give heads E = {H} • Example: rolling two dice CSC317

17. Probability review • Example: rolling two dice • List all possible outcomes • S={(1,1},(1,2),(1,3), …(2,1),(2,2),(2,3), …(6,6)} • List the event or subset of outcomes for which the sum of the dice is 7 • E = {(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)} • Probability of event that sum of dice 7: 6/36 = 1/6 CSC317

18. Probability review • Random variable: attaches value to an outcome/s Example: value of one dice Example: sum of two dice P(X = x): Probability that random variable X takes on value x random variable value of random variable P(X = x): Probability that sum of two dice has value x, e.g. P(X = 7) = 1/6 CSC317

19. Probability review • Expectation or Expected Value of Random Variables: Average sum over each possible values, weighted by probability of that value 5 3 1 2 3 4 5 6 7 8 9 10 11 12 CSC317

20. Easier way? • Linearity of Expectation • random variables, then: Does not require independence of random variables. Example: X1and X2 represent the values of a dice CSC317

21. Probability review • Indicator Random Variable (indicating if occurs…) if A occurs vice versa Lemma: For event A Example: Expected number Heads when flipping fair coin CSC317

22. Probability review • Indicator Random Variable & Linearity of Expectation if A occurs vice versa Example: Expected number Heads when flipping 3 fair coins CSC317