Generating random integer from a range

Question

I need a function which would generate a random integer in given range  including border values   I don t unreasonable quality randomness requirements  I have four requirements    I need it to be fast  My project needs to generate millions  or sometimes even tens of millions  of random numbers and my current generator function has proven to be a bottleneck  I need it to be reasonably uniform  use of rand   is perfectly fine   the min-max ranges can be anything from  lt 0  1  to  lt -32727  32727   it has to be seedable    I currently have following C   code   output   min    rand      int  max - min    RAND MAX    The problem is  that it is not really uniform - max is returned only when rand     RAND MAX  for Visual C   it is 1 32727   This is major issue for small ranges like  lt -1  1   where the last value is almost never returned   So I grabbed pen and paper and came up with following formula  which builds on the  int  n   0 5  integer rounding trick      But it still doesn t give me uniform distribution  Repeated runs with 10000 samples give me ratio of 37 50 13 for values values -1  0  1   Could you please suggest better formula   or even whole pseudo-random number generator function

User · Answer

The formula for this is very simple  so try this expression    int num    int  rand      max - min    min       Where rand   returns a random number between 0 0 and 1 0

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assume min and max are int values    and   means include this value    and   means not include this value   using above to get the right value using c   rand    reference  for      define  visit   https   en wikipedia org wiki Interval  mathematics   for rand and srand function or RAND MAX define  visit   http   en cppreference com w cpp numeric random rand    min  max   int randNum   rand      max - min   1    min    min  max   int randNum   rand      max - min    min   1    min  max   int randNum   rand      max - min    min    min  max   int randNum   rand      max - min - 1    min   1

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The simplest  and hence best  C    using the 2011 standard  answer is   include  lt random gt   std  random device rd         only used once to initialise  seed  engine std  mt19937 rng rd           random-number engine used  Mersenne-Twister in this case  std  uniform int distribution lt int gt  uni min max      guaranteed unbiased  auto random integer   uni rng     No need to re-invent the wheel  No need to worry about bias  No need to worry about using time as random seed

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The following expression should be unbiased if I am not mistaken    std  floor    max - min   1 0     rand       min    I am assuming here that rand   gives you a random value in the range between 0 0 and 1 0 NOT including 1 0 and that max and min are integers with the condition that min  lt  max

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Notice that in most suggestions the initial random value that you have got from rand   function  which is typically from 0 to RAND MAX  is simply wasted  You are creating only one random number out of it  while there is a sound procedure that can give you more   Assume that you want  min max  region of integer random numbers  We start from  0  max-min   Take base b max-min 1  Start from representing a number you got from rand   in base b   That way you have got floor log b RAND MAX   because each digit in base b  except possibly the last one  represents a random number in the range  0  max-min    Of course the final shift to  min max  is simple for each random number r min   int n   NUM DIGIT-1  while n  gt   0        r n    res   b      res -  r n       res    b      n--      If NUM DIGIT is the number of digit in base b that you can extract and that is   NUM DIGIT   floor log b RAND MAX     then the above is as a simple implementation of extracting NUM DIGIT random numbers from 0 to b-1 out of one RAND MAX random number providing b  lt  RAND MAX

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In this thread rejection sampling was already discussed  but I wanted to suggest one optimization based on the fact that rand     2 something does not introduce any bias as already mentioned above     The algorithm is really simple    calculate the smallest power of 2 greater than the interval length randomize one number in that  new  interval  return that number if it is less than the length of the original interval   reject otherwise    Here s my sample code   int randInInterval int min  int max        int intervalLen   max - min   1        now calculate the smallest power of 2 that is  gt   than  intervalLen      int ceilingPowerOf2   pow 2  ceil log2 intervalLen          int randomNumber   rand     ceilingPowerOf2    this is  as uniform as rand         if  randomNumber  lt  intervalLen          return min   randomNumber         ok      return randInInterval min  max        reject sample and try again      This works well especially for small intervals  because the power of 2 will be  nearer  to the real interval length  and so the number of misses will be smaller     PS Obviously avoiding the recursion would be more efficient  no need to calculate over and over the log ceiling    but I thought it was more readable for this example

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Here is an unbiased version that generates numbers in  low  high    int r  do     r   rand      while  r  lt    unsigned int  RAND MAX    1     high   1 - low    return r    high   1 - low    low    If your range is reasonably small  there is no reason to cache the right-hand side of the comparison in the do loop

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I recommend the Boost Random library  it s super detailed and well-documented  lets you explicitly specify what distribution you want  and in non-cryptographic scenarios can actually outperform a typical C library rand implementation

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int RandU int nMin  int nMax        return nMin    int   double rand      RAND MAX 1     nMax-nMin 1        This is a mapping of 32768 integers to  nMax-nMin 1  integers  The mapping will be quite good if  nMax-nMin 1  is small  as in your requirement   Note however that if  nMax-nMin 1  is large  the mapping won t work  For example - you can t map 32768 values to 30000 values with equal probability   If such ranges are needed - you should use a 32-bit or 64-bit random source  instead of the 15-bit rand    or ignore rand   results which are out-of-range

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If your compiler supports C  0x and using it is an option for you  then the new standard  lt random gt  header is likely to meet your needs   It has a high quality uniform int distribution which will accept minimum and maximum bounds  inclusive as you need   and you can choose among various random number generators to plug into that distribution   Here is code that generates a million random ints uniformly distributed in  -57  365    I ve used the new std  lt chrono gt  facilities to time it as you mentioned performance is a major concern for you    include  lt iostream gt   include  lt random gt   include  lt chrono gt   int main         typedef std  chrono  high resolution clock Clock      typedef std  chrono  duration lt double gt  sec      Clock  time point t0   Clock  now        const int N   10000000      typedef std  minstd rand G      G g      typedef std  uniform int distribution lt  gt  D      D d -57  365       int c   0      for  int i   0  i  lt  N    i           c    d g       Clock  time point t1   Clock  now        std  cout  lt  lt  N sec t1-t0  count    lt  lt    random numbers per second  n       return c      For me  2 8 GHz Intel Core i5  this prints out   2 10268e 07 random numbers per second   You can seed the generator by passing in an int to its constructor       G g seed     If you later find that int doesn t cover the range you need for your distribution  this can be remedied by changing the uniform int distribution like so  e g  to long long        typedef std  uniform int distribution lt long long gt  D    If you later find that the minstd rand isn t a high enough quality generator  that can also easily be swapped out   E g        typedef std  mt19937 G      Now using mersenne twister engine   Having separate control over the random number generator  and the random distribution can be quite liberating   I ve also computed  not shown  the first 4  moments  of this distribution  using minstd rand  and compared them to the theoretical values in an attempt to quantify the quality of the distribution   min   -57 max   365 mean   154 131 x mean   154 var   14931 9 x var   14910 7 skew   -0 00197375 x skew   0 kurtosis   -1 20129 x kurtosis   -1 20001    The x  prefix refers to  expected

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A fast  somewhat better than yours  but still not properly uniform distributed solution is  output   min    rand     static cast lt int gt  max - min   1     Except when the size of the range is a power of 2  this method produces biased non-uniform distributed numbers regardless the quality of rand    For a comprehensive test of the quality of this method  please read this

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Let s split the problem into two parts    Generate a random number n in the range 0 through  max-min   Add min to that number   The first part is obviously the hardest  Let s assume that the return value of rand   is perfectly uniform  Using modulo will add bias to the first  RAND MAX   1     max-min 1  numbers  So if we could magically change RAND MAX to RAND MAX -  RAND MAX   1     max-min 1   there would no longer be any bias   It turns out that we can use this intuition if we are willing to allow pseudo-nondeterminism into the running time of our algorithm  Whenever rand   returns a number which is too large  we simply ask for another random number until we get one which is small enough   The running time is now geometrically distributed  with expected value 1 p where p is the probability of getting a small enough number on the first try  Since RAND MAX -  RAND MAX   1     max-min 1  is always less than  RAND MAX   1    2  we know that p  gt  1 2  so the expected number of iterations will always be less than two for any range  It should be possible to generate tens of millions of random numbers in less than a second on a standard CPU with this technique   EDIT   Although the above is technically correct  DSimon s answer is probably more useful in practice  You shouldn t implement this stuff yourself  I have seen a lot of implementations of rejection sampling and it is often very difficult to see if it s correct or not

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How about the Mersenne Twister  The boost implementation is rather easy to use and is well tested in many real-world applications  I ve used it myself in several academic projects such as artificial intelligence and evolutionary algorithms   Here s their example where they make a simple function to roll a six-sided die    include  lt boost random mersenne twister hpp gt   include  lt boost random uniform int hpp gt   include  lt boost random variate generator hpp gt   boost  mt19937 gen   int roll die         boost  uniform int lt  gt  dist 1  6       boost  variate generator lt boost  mt19937 amp   boost  uniform int lt  gt   gt  die gen  dist       return die        Oh  and here s some more pimping of this generator just in case you aren t convinced you should use it over the vastly inferior rand        The Mersenne Twister is a  random   number  generator invented by Makoto   Matsumoto and Takuji Nishimura  their   website includes numerous   implementations of the algorithm       Essentially  the Mersenne Twister is a   very large linear-feedback shift   register  The algorithm operates on a   19 937 bit seed  stored in an   624-element array of 32-bit unsigned   integers  The value 2 19937-1 is a   Mersenne prime  the technique for   manipulating the seed is based on an   older  twisting  algorithm -- hence   the name  Mersenne Twister        An appealing aspect of the Mersenne   Twister is its use of binary   operations -- as opposed to   time-consuming multiplication -- for   generating numbers  The algorithm also   has a very long period  and good   granularity  It is both fast and   effective for non-cryptographic applications

[c++] Generating random integer from a range

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