Thursday Week 9

Random Numbers 1/15

Random numbers are useful

in computer games:
- may want aliens to move in a random pattern
- the layout of a dungeon may be randomly generated
- may want to introduce unpredictability
in physics/applied maths:
- carry out simulations to determine behaviour
- e.g. models of molecules are often assume to move randomly
in testing:
- stress test components by bombarding them with random data
- random data is often seen as unbiased data
  - gives average performance (e.g. in sorting algorithms)
in cryptography

... Random Numbers 2/15

But how can a computer pick a number at random?

it cannot

Software can only produce pseudo random numbers.

a pseudo random number is one that is predictible
(although it may appear unpredictable)

A Random Number Generator 3/15

The most widely-used technique is called the Linear Congruential Generator (LCG)

it uses a recurrence relation:
- X_n+1 = (a·X_n + c) mod m, where:
  - m is the "modulus"
  - a, 0 < a < m is the "multiplier"
  - c, 0 ≤ c ≤ m is the "increment"
  - X₀ is the "seed"
- if c=0 it is called a multiplicative congruential generator

... A Random Number Generator 4/15

LCG is not good for applications that need extremely high-quality random numbers

the period length is too short
- the length of the sequence at which point the sequence repeats itself
- the longer the period length, the lower the predictability
- a short period means the numbers are correlated
the maximum period length of LCG is m (so it should be very large)

... A Random Number Generator 5/15

Trivial example:

for simplicity assume c=0
so the formula is X_n+1 = a·X_n mod m

try a=11=X₀, m=31, which generates the sequence:


11, 28, 29, 9, 6, 4, 13, 19, 23, 5, 24, 16, 21, 14, 30, 20, 3, 2, 22, 25,
27, 18, 12, 8, 26, 7, 15, 10, 17, 1, 11, 28, 29, 9, 6, 4, 13, 19, 23, 5,
24, 16, 21, 14, 30, 20, 3, 2, 22, 25, 27, 18, 12, 8, 26, 7, 15, 10, 17, 1,
11, 28, 29, 9, 6, 4, 13, 19, 23, 5, 24, 16, 21, 14, 30, 20, 3, 2, 22, 25, 27,
18, 12, 8, 26, 7, 15, 10, 17, 1, 11, 28, 29, 9, 6, 4, 13, 19, 23, 5, 24, 16,
21, 14, 30, 20, 3, 2, 22, 25, 27, 18, 12, 8, 26, 7, 15, 10, 17, 1, 11, 28,
29, 9, 6, 4, 13, 19, 23, 5, 24, 16, 21, 14, 30, 20, 3, 2, 22, 25, 27, 18,
12, 8, 26, 7, 15, 10, 17, 1, ...

all the integers from 1 to 30 are here ... but notice the period length

... A Random Number Generator 6/15

Another trivial example:

again let c=0

try a=12=X₀ and m=30

that is, X_n+1 = 12·X_n mod 30
which generates the sequence:


12, 24, 18, 6, 12, 24, 18, 6, 12, 24, 18, 6, 12, 24, 18, 6, 12, 24, 18, 6,
12, 24, 18, 6, 12, 24, 18, 6, 12, 24, 18, 6, ...

notice the period length ... clearly a terrible sequence

... A Random Number Generator 7/15

It is a complex task to pick good numbers. A bit of history:

Lewis, Goodman and Miller (1969) suggested

X_n+1 = 7⁵·X_n mod (2³¹-1)
note:
- 7⁵ is 16807
- 2³¹-1 is 2147483674
- X₀ = 0 is not a good seed value!!

gcc uses a LCG-based algorithm that's slightly more involved; for details, including a short C program that produces the exact same pseudo-random numbers for any given seed value, see here

... A Random Number Generator 8/15

man random or man srandom gives lots of information

Two functions are required:

srandom(int seed) // sets its argument as the seed

random() // uses a LCG technique to generate random 
         // numbers in the range 0 .. RAND_MAX

where the constant RAND_MAX is defined in stdlib.h
(depends on the computer: on the CSE network, RAND_MAX = 2147483647)

The period of this random number generator is very large
approximately 16 · ((2³¹) - 1)

Exercise: Random Numbers 9/15

Write a program randolf.c that

calls srandom() and random()
- to generate random numbers in the range 0 .. RANGE, for some value of RANGE

... Exercise: Random Numbers 10/15

To convert to a number between 0 .. RANGE

compute the remainder after division by RANGE+1

Using the remainder to compute a random number is not the best way:

can generate a 'better' random number by using a more complex division
but good enough for the purposes of this course

Seeding 11/15

There is one significant problem with randolf.c:

every time we run it, we get exactly the same sequence of 'random' numbers
why?

To vary the output, can give the random seeder a starting point that varies with time

an example of such a starting point is the current time, time(NULL)
(this is different from the UNIX command time, used to measure program running time)

time(NULL) // returns the time as the number of seconds
           // since the Epoch, 1970-01-01 00:00:00 +0000

// time(NULL) on October 1st, 2015, 12:59pm was 1443668340
// time(NULL) about a minute later was 1443668402

Exercise: Seeding 12/15

Modify randolf.c by

calling time() to vary the seed

Note: this function is provided by the time.h library

Exercise: Coin Tossing 13/15

A common application of random numbers is simulation

Another common application is generating 'average' data to perform as testcases

Write a program that

simulates coin tossing
has 50% chance of outputting a 'H', and 50% chance of outputting a 'T'
accepts the number of tosses as a command-line argument
keeps a count of the number of heads and tails

Measure its performance using the UNIX time command (for a million toin cosses, 10 million, 100 million, ...)

... Exercise: Coin Tossing 14/15

What you can observe:

the runtime increases by a factor of almost exactly 10 when the number increases by 10
- the algorithm has linear performance:
  - its execution time is proportional to the value of the input N
    - double N, double the time
    - triple N, triple the time
    - increase N by a factor 10, it takes 10 times as long to execute
    - etc

Tips for Next Week's Lab 15/15

BSTs, Files, Random Numbers

A (recursive) function to delete all leaves in a binary search tree
- think about the base case first:
  What is it, what should happen, what should the function return?
- then think about recursive cases
Merge 3 files, line by line
- keep going after reaching EOF in one file if there are still lines left in the other(s)
- stop once all 3 files are EOF
Generate random words
- choose each letter randomly between 'a'..'z'
- for the second part, design algorithm to reverse a linked list in-place (that is, without using a second list)

Produced: 5 Oct 2016