1. Some Uses of Probability Randomized algorithms –for CS in general –for games and robotics in particular Testing Simulation Solving probabilistic problems by simulation

2. Random Number Generators –Actually pseudo-random –Seed Same sequence from same seed Often time is used. Many examples on web. Custom random number generators exist. Can be used in algorithms.

3. A Simple Random Number Generator random_seed = random_seed * 1103515245 +12345; r = (int)(random_seed / 65536) % 32768;

4. Java Random Numbers double x= Math.random() –Generates a random double value in the range from 0.0 up to, but not including 1. –That calculation uses the current time of day to seed the random number generator such that each execution of a program yields a different sequence of random values.

5. Scaling double x = (B – A) * Math.random + A –Gives double value A  num < B int n = (int) ((B – A) * Math.random()) + A –Gives integer value A  num < B

6. Java Random Class import java.util.Random; Random g = new Random(); Random g2 = new Random(19580427); int r = g.nextInt(); –value any int int r = g.nextInt(n); –value 0  r < n double r = g.nextDouble() –value 0  r < 1

7. Quality of Generators Java implementation so so. –Careful about successive calls using time to set seed if calls will be within 1 millisecond of each other Best probably Mersenne Twister –available on net to replace Random

8. Actions Based on Probabilities Assign probabilities to each action. Probabilities must add to 1. Roulette wheel method –Range of random number generator is size of roulette wheel. –Give each action a section of the roulette wheel proportional to its probability. –Generate a random number. (Run the wheel.)

9. Example P(go straight) = 50% P(turn left) = 30% P(turn right) = 20% Use random numbers from 0 – 1.000 Roulette wheel –0 ≤ Go straight <.50 –.50 ≤ turn left <.80 –.80 ≤ turn right < 1.00

10. Random Sequence

11. Java Shuffle The Collections class which can be found within the java.util namespace provides two methods which suffle the elements of a Collection. –static void shuffle(List list) static void shuffle(List list, Random rnd) Collections.shuffle(sl);

12. Example import java.util.*; public class ShuffleTest{ public static void main(String[] args){ List sl = new ArrayList (); sl.add("One"); sl.add("Two"); sl.add("Three"); Collections.shuffle(sl); } }

13. Sampling (Polling) Use random sequence but stop after needed number. Generate random numbers but check for repetition

14. Basic Approaches Monte Carlo –the resources used are bounded but the final result may not be 100% accurate. Las Vegas –always produces the correct result or it informs about the failure. In other words, it does not gamble with the verity of the result -- - it only gambles with the resources used for the computation.

15. Simulated Annealing A meta-algorithm for global optimization. Meta- –details differ for different problems Tries to avoid getting stuck in local optimum. Probablistic Jumping frog analogy –frog tires and gets caught by deepest pit

16. Simulated Annealing (2) Use a parameter called the temperature –Start with a high temperature (frog is fresh) Jump –Replace current solution with a “nearby” solution that is chosen based on a temperature dependent transition probability. Keep jumping as the temperature is decreased –The frog gets tired

17. Minimizing f Transition probability depends on temperature and the difference of the f values of the original and the new state. For high temperatures, the jump to higher f is about as likely as the jump to lower f. (frog is fresh) As temperatures lower, the jump to lower f is much more likely (frog tires)

18. Simulated Annealing s := s0; e := E(s) // Initial state, energy. k := 0 // Energy evaluation count. while k emax // While time remains & not // good enough: sn := neighbour(s) // Pick some neighbour. en := E(sn) // Compute its energy. if random() < P(e, en, temp(k/kmax)) then // Should we // move to it? s := sn; e := en // Yes, change state. k := k + 1 // One more evaluation done return s // Return current solution