Emil Sekerinski, McMaster University, Winter Term 16/17 COMP SCI 1MD3 Introduction to Programming

Transcription

1 Emil Sekerinski, McMaster University, Winter Term 16/17 COMP SCI 1MD3 Introduction to Programming

2 Games: HiLo, Entertainment: "random quote of the day", recommendations, Simulation: traffic, queues (supermarket, banks), finance, environment, biology, Numerical computation: integration, Efficiency: sorting, Networks: protocols, Cryptography Randomness appears e.g. as radio noise and can be traced back to the laws of quantum physics. While dedicated hardware to generate true random numbers exists, it is more common and for most purposes sufficient to generate pseudorandom numbers.

3 Suppose all 4 cars stop at the same moment. What algorithm have drivers to follow to allow them to eventually pass the intersection while avoiding collision? One driver has randomly to allow the one to their left to pass. Suppose two computers transmit to a WiFi base station at almost the same time. For physical reasons, a collision of the signals can be detected only after the transmission. How can such a collision be resolved? Retransmission occurs in a random time within an interval. If there is again a collision, the interval is doubled.

4 Pseudorandom numbers are generated by a deterministic algorithm. For many purposes, they are sufficiently indistinguishable from truly random numbers. Algorithm for the linear congruential generator: Given x o, compute x n+1 = (a x n + c) % m seed multiplier increment modulus

5 The randomness properties and the length of the period depend on the choice of a, c, m. For m, 2 32 or 2 64 are convenient as they correspond to word sizes. A full period is obtained for all seed values if: c and m are relatively prime (only 1 divides both evenly), a - 1 is divisible by all prime factors of m, a - 1 is a multiple of 4 if m is a multiple of 4. Knuth uses: a = , c = def lcg(old): # linear congruential generator a = c = return (a * old + c) % 2**64

6 The last random value is kept as a global variable, preserved between calls to randint As the seed value, the current time in microseconds is taken On importing this module, the seed is initialized from datetime import datetime x = (datetime.today()).microsecond def randint(bound): global x a = c = x = (a * x + c) % 2**64 return x % bound

7 Python provides a function random() that generates a random float uniformly in [0.0, 1.0) by using Mersenne Twister rather than LCG, producing 53-bit precision floats and has a period of seed([a]) Initializes the generator with the given integer or the time randint(a, b) Generate integer N such that a N b choice(seq) Return random element from the sequence random() Generate float N such that 0 N < 1.0 uniform(a, b) Generate float N such that a N b Python supports other distributions are well, e.g. exponential

8 The problem is named after the host of the TV show "Let's Make a Deal". Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? Joe does not switch the door. Sue switches the door.

9 Joe_1 and Joe_2 return the two choices of Joe. The result of the first choice is stored in the global variable J and is available for the second choice. from random import * def Joe_1(): global J # Joe's first choice J = choice([1, 2, 3]) return J def Joe_2(goat): global J return J def Sue_1(): global S # Sue's first choice S = choice([1, 2, 3]) return S def Sue_2(goat): global S # S!= goat # return that door that is # neither S nor goat return choice(list({1, 2, 3} \ - {goat, S})) def monty(contestant_1, contestant_2): # hide the prize car = choice([1, 2, 3]) # ask contestant first choice C1 = contestant_1() # open door with goat goat = choice(list({1, 2, 3} - \ {car, C1})) # offer contestant to switch C2 = contestant_2(goat) return C2 == car def JoeWins(): c = 0 for _ in range(1000): if monty(joe_1, Joe_2): c += 1 return c def SueWins(): c = 0 for _ in range(1000): if monty(sue_1, Sue_2): c += 1 return c

10 Integrating a function in a given interval graphically means determining the area between the x-axis and the function. With randomization, this can be approximated for any function, even if an analytical solution is not known. def area(f, a, b, l, u, n): u c = 0 for i in range(n): x = random.uniform(a, b) y = random.uniform(l, u) if 0 < y <= f(x): c = c + 1 elif f(x) <= y < 0: l c = c - 1 return c * (u - l) * (b - a) / n a + b