Random Numbers Until now, all programs have behaved deterministically - completely predictable and repeatable based on input values Some applications (e.g. games) require unpredictable behavior - nondeterministic One method for nondeterminism die toss where one of several outcomes possible

Random Numbers Number is random if there is no way to determine in advance what value it will have among a set of equally probable possibilites E.g. on a standard unloaded die random number between 1 and 6 equally possible Cannot generate number randomly on computer uses deterministic behavior

Pseudo-random Numbers Use a deterministic procedure to generate “random” numbers Numbers must appear random behave like random numbers statistically be sufficiently difficult to predict in advance Random numbers generated by an algorithmic process by a computer are called pseudo-random numbers

Pseudo-random number in C++ <cstdlib> includes function rand and constant RAND_MAX prototype int rand(); generates pseudo-random number between 0 and RAND_MAX, inclusive Value of RAND_MAX depends on system Ours uses 32 767 but your program should not rely on a given value but on the symbolic constant

Using “rand” to flip a coin Want to use the result of rand that is a number between 0 and RAND_MAX to generate Heads or Tails Use if (rand() <= RAND_MAX / 2) cout << ”Heads”; else cout << ”Tails”;

Wrong Method to flip a coin Why not use: if (rand() % 2 == 0) cout << ”Heads”; etc. No guarantee of distribution of even and odd numbers, only even distribution along the range 0 to RAND_MAX

Using “rand” to Roll a Die int RollDie(void) { if (rand() < RAND_MAX / 6) return 1; if (rand() < RAND_MAX*2/6) return 2; if (rand() < RAND_MAX*3/6) return 3; if (rand() < RAND_MAX*4/6) return 4; if (rand() < RAND_MAX*5/6) return 5; return 6; } Unfortunately uses new value of “rand” in each “if”, thus losing original.

Using “rand” to Roll a Die (2) int RollDie() { int r = rand(); if (r < RAND_MAX / 6) return 1; if (r < RAND_MAX*2/6) return 2; if (r < RAND_MAX*3/6) return 3; if (r < RAND_MAX*4/6) return 4; if (r < RAND_MAX*5/6) return 5; return 6; } Unfortunately RAND_MAX*2/6 gives strange value due to integer arithmetic Better to use RAND_MAX/6.0*2.0

Generalize the Problem Want a function to choose random number between two values, inclusive int RandomInteger(int low, int high); Four-step process 1. Normalize rand to number d where 0 <= d < 1 2. Scale d by multiplying by size of desired range 3. Truncate number back to an integer 4. Translate integer so range begins at lower bound

RandomInteger int RandomInteger(int low, int high) { int k; double d; d = rand() / ((double) RAND_MAX + 1.0); k = (int) (d * (high - low + 1)); return (low + k); }

Random Problem? Running randtest twice gives same values Why? rand function always returns same sequence of values on every execution Why? So program behavior is repeatable for debugging But - want non-deterministic behavior Can modify behavior

Seeding Random Numbers Initial value for random number generator same each time program is run Next value depends on previous Change initial value, change behavior Function srand in stdlib allows seed Using time from time.h allows “random” seed srand( (int) time(NULL)); Provide operation in interface

Header files and implementation files Example Random Two steps: Create an interface - random.h Write implementation - random.cpp

Interface Contents Documentation on use Function prototypes Constant definitions Type definitions (later) Three lines at top: #ifndef _name_h #define _name_h at end: #endif

Preliminary random.h #ifndef _random_h #define _random_h /* Function: RandomInteger * Usage: n = RandomInteger(low,high); * ----------------------------------- * This function returns a random integer * in the range low to high, inclusive. */ int RandomInteger(int low, int high); #endif

Preliminary random.cpp #include <stdlib> #include ”random.h” int RandomInteger(int low, int high) { int k; double d; d = rand() / ((double) RAND_MAX + 1); k = (int) (d * (high - low + 1)); return (low + k); }

Preliminary Client /* Filename: randtest.cpp */ #include <iostream> #include ”random.h” int main(void) { int i; for (i = 0; i < 10; i++) cout << RandomInteger(1,6) << endl; return (0); }