Use distribution function with rejection sampling.

common/rng_sfmt.cpp

@@ -1,7 +1,10 @@
 #include "rng_sfmt.h"
 #include <QDateTime>
-#include <stdlib.h>
+#include <algorithm>
-#include <iostream>
+
+// This is from gcc sources, namely from fixincludes/inclhack.def
+// On C++11 systems, <cstdint> could be included instead.
+#define UINT64_MAX (~(uint64_t)0)
 
 RNG_SFMT::RNG_SFMT(QObject *parent)
 	: RNG_Abstract(parent)
@@ -10,13 +13,72 @@ RNG_SFMT::RNG_SFMT(QObject *parent)
 	sfmt_init_gen_rand(&sfmt, QDateTime::currentDateTime().toTime_t());
 }
 
+/**
+ * Much thought went into this, please read this comment before you modify the code.
+ * Let SFMT() be an alias for sfmt_genrand_uint64() aka SFMT's rand() function.
+ *
+ * SMFT() returns a uniformly distributed pseudorandom number from 0 to UINT64_MAX.
+ * As SFMT() operates on a limited integer range, it is a _discrete_ function.
+ *
+ * We want a random number from a given interval [min, max] though, so we need to
+ * implement the (discrete) cumulative distribution function SFMT(min, max), which
+ * returns a random number X from [min, max].
+ *
+ * This CDF is by formal definition:
+ * SFMT(X; min, max) = (floor(X) - min + 1) / (max - min + 1)
+ *
+ * To get out the random variable, solve for X:
+ * floor(X) = SFMT(X; min, max) * (max - min + 1) + min - 1
+ * So this is, what getNumber(min, max) should look like.
+ * Problem: SFMT(X; min, max) * (max - min + 1) could produce an integer overflow,
+ * so it is not safe.
+ *
+ * One solution is to divide the universe into buckets of equal size depending on the
+ * range [min, max] and assign X to the bucket that contains the number generated
+ * by SFMT(). This equals to modulo computation and is not satisfying:
+ * If the buckets don't divide the universe equally, because the bucket size is not
+ * a divisor of 2, there will be a range in the universe that is biased because one
+ * bucket is too small thus will be chosen less equally!
+ *
+ * This is solved by rejection sampling:
+ * As SFMT() is assumed to be unbiased, we are allowed to ignore those random numbers
+ * from SFMT() that would force us to have an unequal bucket and generate new random
+ * numbers until one number fits into one of the other buckets.
+ * This can be compared to an ideal six sided die that is rolled until only sides
+ * 1-5 show up, while 6 represents something that you don't want. So you basically roll
+ * a five sided die.
+ */
 unsigned int RNG_SFMT::getNumber(unsigned int min, unsigned int max)
 {
-	// To make the random number generation thread safe, a mutex is created around the generation.
+	// This all makes no sense if min > max.
+	// So in debug mode Q_ASSERT will print a warning...
+	Q_ASSERT(min <= max);
+	// ... and at runtime min and max will be swapped; this should never happen.
+	if(min > max)
+	  std::swap(min, max);
+
+	// First compute the diameter (aka size, length) of the [min, max] interval
+	const unsigned int diameter = max - min + 1;
+
+	// Compute how many buckets (each in size of the diameter) will fit into the
+	// universe.
+	// If the division has a remainder, the result is floored automatically.
+	const uint64_t buckets = UINT64_MAX / diameter;
+
+	// Compute the last valid random number. All numbers beyond have to be ignored.
+	// If there was no remainder in the previous step, limit is equal to UINT64_MAX.
+	const uint64_t limit = diameter * buckets;
+
+	uint64_t rand;
+	// To make the random number generation thread-safe, a mutex is created around
+	// the generation. Outside of the loop of course, to avoid lock/unlock overhead.
 	mutex.lock();
-	uint64_t r = sfmt_genrand_uint64(&sfmt);
+	do {
+		rand = sfmt_genrand_uint64(&sfmt);
+	} while (rand >= limit);
 	mutex.unlock();
 
-	// return a random number from the interval [min, max]
+	// Now determine the bucket containing the SFMT() random number and after adding
-	return (unsigned int) (r % (max - min + 1) + min);
+	// the lower bound, a random number from [min, max] can be returned.
+	return (unsigned int) (rand / buckets + min);
 }