Python Programming - IX. On Randomness

IX. ON RANDOMNESS
Engr. RANEL O. PADON

PYTHON PROGRAMMING TOPICS
I

•Introduction to Python Programming

II

•Python Basics

III

•Controlling the Program Flow

IV

•Program Components: Functions, Classes, Packages, and Modules

V

•Sequences (List and Tuples), and Dictionaries

VI

•Object-Based Programming: Classes and Objects

VII

•Customizing Classes and Operator Overloading

VIII

•Object-Oriented Programming: Inheritance and Polymorphism

IX

•Randomization Algorithms

X

•Exception Handling and Assertions

XI

•String Manipulation and Regular Expressions

XII

•File Handling and Processing

XIII

•GUI Programming Using Tkinter

ON RANDOMNESS
Randomization Algorithms
* Random numbers are used for testing the performance of programs,
creating scientific simulations, and so on.

* A pseudorandom number generator (PRNG) is an algorithm for
generating a sequence of numbers that approximates the properties
of random numbers.
* The sequence is not truly random in that it is completely
determined by a relatively small set of initial values (random seed)

ON RANDOMNESS Randomization Algorithms
Early Algorithm
Middle-Square Method
* take any number, square it, remove the middle digits of the
resulting number as the "random number", then use that number as
the seed for the next iteration.
* squaring the number "1111" yields "1234321", which can be written
as "01234321", an 8-digit number being the square of a 4-digit
number. This gives "2343" as the "random" number. Repeating this
procedure gives "4896" as the next result, and so on.

1.) Blum Blum Shub (cryptographically sound, but slow)
2.) Mersenne Twister
- fast with good statistical properties,
- used when cyptography is not an issue
- used in Python
3.) Linear Congruential Generator
- commonly-used in compilers

and many, many others

2.) Mersenne Twister Algorithm
is based on a matrix linear recurrence over a finite binary field
provides for fast generation of very high-quality pseudorandom
numbers, having been designed specifically to rectify many of the
flaws found in older algorithms.
used in PHP, Ruby and Python
name derives from the fact that period length is chosen to be a
Mersenne prime

ON RANDOMNESS Mersenne Twister Algorithm
It has a very long period of 219937 − 1. While a long period is
not a guarantee of quality in a random number generator, short
periods (such as the 232 common in many software packages)
can be problematic.
It is k-distributed to 32-bit accuracy for every 1 ≤ k ≤ 623

It passes numerous tests for statistical randomness, including the
Diehard tests. It passes most, but not all, of the even more
stringent TestU01 Crush randomness tests.

(as used in the random module of Python)
# Create a length 624 list to store the state of the generator
MT = [0 for i in xrange(624)]
index = 0

# To get last 32 bits
bitmask_1 = (2 ** 32) - 1
# To get 32. bit
bitmask_2 = 2 ** 31
# To get last 31 bits
bitmask_3 = (2 ** 31) - 1

def initialize_generator(seed):
"Initialize the generator from a seed"
global MT
global bitmask_1
MT[0] = seed
for i in xrange(1,624):
MT[i] = ((1812433253 * MT[i-1]) ^ ((MT[i-1] >> 30) + i)) &
bitmask_1

def extract_number():
""“ Extract a tempered pseudorandom number based on the indexth value, calling generate_numbers() every 624 numbers ""“
global index
global MT
if index == 0:
generate_numbers()
y = MT[index]
y ^= y >> 11
y ^= (y << 7) & 2636928640
y ^= (y << 15) & 4022730752
y ^= y >> 18
index = (index + 1) % 624
return y

def generate_numbers():
"Generate an array of 624 untempered numbers"
global MT
for i in xrange(624):
y = (MT[i] & bitmask_2) + (MT[(i + 1 ) % 624] & bitmask_3)
MT[i] = MT[(i + 397) % 624] ^ (y >> 1)
if y % 2 != 0:
MT[i] ^= 2567483615
if __name__ == "__main__":
from datetime import datetime
now = datetime.now()
initialize_generator(now.microsecond)
for i in xrange(100):
"Print 100 random numbers as an example"
print extract_number()

ON RANDOMNESS Linear Congruential Generator
represents one of the oldest and best-known pseudorandom number
generator algorithms.

the theory behind them is easy to understand, and they are easily
implemented and fast.

The basic idea is to multiply the last number with a factor a, add a
constant c and then modulate it by m.
Xn+1 = (aXn + c) mod m.
where X0 is the seed.

ON RANDOMNESS Random Seed

Random Seed
* a number (or vector) used to initialize a pseudorandom number
generator
* crucial in the field of computer security.
* having the seed will allow one to obtain the secret encryption key
in a pseudorandomly generated encryption values

ON RANDOMNESS Random Seed

Random Seed

* two or more systems using matching pseudorandom number
algorithms and matching seeds can generate matching sequences of
non-repeating numbers which can be used to synchronize remote
systems, such as GPS satellites and receivers.

a=3
c=9
m = 16
xi = 0
def seed(x):
global xi
xi = x

Random
Number
Generator

def rng():
global xi
xi = (a*xi + c) % m
return xi
for i in range(10):
print rng()

LCG (Standard Parameters)

Good One

Using
java.util.Random
parameters

a = 25214903917
c = 11
m = 2**48
xi = 1000
def seed(x):
global xi
xi = x
def rng():
global xi
xi = (a*xi + c) % m
return xi
def rng_bounded(low, high):
return low + rng()%(high - low+1)
for i in range(10):
rng_bounded(1, 10)

ON RANDOMNESS Minimal Standard
MINSTD by Park & Miller (1988)
known as the Minimal Standard Random number generator
a very good set of parameters for LCG:

m = 231 − 1 = 2,147,483,647 (a Mersenne prime M31)
a = 75 = 16,807 (a primitive root modulo M31)
c=0
often the generator that used for the built in random number function in
compilers and other software packages.
used in Apple CarbonLib, a procedural API for developing Mac OS X
applications

Using
MINSTD
parameters

a = 7**5
c=0
m = 2**31 - 1
xi = 1000
def seed(x):
global xi
xi = x
def rng():
global xi
xi = (a*xi + c) % m
return xi
return low + rng() % (high - low+1)
for i in range(10):
rng_bounded(1, 2)

Tossing 1 Coin
using LCG MINSTD
(1 million times)

from __ future__ import division
if __name__ == '__main__':
main()
a = 7**5
c=0
m = 2**31 - 1
xi = 1000
def seed(x):
global xi
xi = x

def rng():
global xi
xi = (a*xi + c) % m
return xi

Tossing 1 Coin
using LCG MINSTD
(1 million times)

(heads, tails, count) = (0, 0, 0)
for i in range (1, 1000001):
count += 1
coin rng_bounded(1,2)

if coin == 1:
heads += 1
else:
tails += 1

Tossing 1 Coin
using LCG MINSTD
(1 million times)

print "heads is ", heads/count
print "tails is ", tails/count

Tossing 1 Coin
using Python randrange()
(1 million times)

import random
if __name__ == '__main__':
main()
(heads, tails, count) = (0, 0, 0)
random.seed(1000)
for i in range (1,1000001):
count +=1
coin = random.randrange(1,3)

Tossing 1 Coin
(1 million times)

if coin == 1:
heads += 1
else:
tails += 1

print "heads is ", heads/count
print "tails is ", tails/count

Tossing 2 Coins
using LCG MINSTD
(1 million times)

if __name__ == '__main__':
main()
a = 7**5
c=0
m = 2**31 - 1
xi = 1000
def seed(x):
global xi
xi = x

def rng():
global xi
xi = (a*xi + c) % m
return xi

Tossing 2 Coins
using LCG MINSTD
(1 million times)


(heads, tails, combo, count) = (0, 0, 0, 0)
count += 1
coin1 = rng_bounded(1,2)
coin2 = rng_bounded(1,2)
sum = coin1 + coin2
if sum == 2:
heads += 1
elif sum == 4:
tails += 1
else:
combo += 1

Tossing 2 Coins
using LCG MINSTD
(1 million times)

print "head is ", heads/count
print "tail is ", tails/count
print "combo is ", combo/count

Tossing 2 Coins
(1 million times)

import random
if __name__ == '__main__':
main()
(heads, tails, combo, count) = (0, 0, 0, 0)
random.seed(1000)
count +=1
coin1 = random.randrange(1,3)
coin2 = random.randrange(1,3)
sum = coin1 + coin2

Tossing 2 Coins
(1 million times)

if sum == 2:
heads += 1

elif sum == 4:
tails += 1
else:
combo += 1
print "head is ", heads/count
print "tail is ", tails/count
print "combo is ", combo/count

ON RANDOMNESS Results Summary
Almost Similar Results
Tossing 1 Coin using LCG MINSTD vs Python randrange() (1 million times)

Tossing 2 Coins using LCG MINSTD vs Python randrange() (1 million times)

Tossing 1 Coin using LCG MINSTD (1000 times)
More Elegant Way (Using List)

Tossing 2 Coins
using LCG MINSTD
(1 million times)
More Elegant Way
(Using List)

from __future__ import division

if __name__ == '__main__':
main()
xi = 1000
def seed(x):
global xi
xi = x
def rng(a=7**5, c=0, m = 2**31 - 1):
global xi
xi = (a*xi + c) % m
return xi

Tossing 2 Coins
using LCG MINSTD
(1 million times)
More Elegant Way

tosses = []
for i in range (1, 1000001):
tosses += [rngNiRanie(1, 2) + rngNiRanie(1, 2)]
print "heads count is ", tosses.count(2) / len(tosses)
print "tails count is ", tosses.count(4) / len(tosses)
print "combo count is ", tosses.count(3) / len(tosses)

ON RANDOMNESS
a = 25214903917
c = 11
m = 2**48
xi = 1000
def seed(x):
global xi
xi = x
def rng():
global xi
xi = (a*xi + c) % m
return xi
def ranie(lowerBound, upperBound):
#ano laman nito? #
for i in range(10):
ranie(1,10)

REFERENCES

 Deitel, Deitel, Liperi, and Wiedermann - Python: How to Program (2001).

 Disclaimer: Most of the images/information used here have no proper source citation, and I do
not claim ownership of these either. I don’t want to reinvent the wheel, and I just want to reuse
and reintegrate materials that I think are useful or cool, then present them in another light,
form, or perspective. Moreover, the images/information here are mainly used for
illustration/educational purposes only, in the spirit of openness of data, spreading light, and
empowering people with knowledge. 

Python Programming - IX. On Randomness

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