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    [原]mt19937是什么鬼?

    caimouse发表于 2017-02-18 16:20:14
    love 0

    今天看一个C++的例子,突然看到这个mt19937,起先还以为是什么地方搞错了,怎么会有这个怪的名称呢?这个名称是mt1937? 代表1937年?心里一开始有这个疑问。代码如下:

    std::random_device rd;
    		std::mt19937 gen(rd());
    		std::uniform_int_distribution<> dist(-10, 10);
    		std::vector<int> v;
    		generate_n(back_inserter(v), 20, bind(dist, gen));
    
    		std::cout << "Before sort: ";
    		copy(v.begin(), v.end(), std::ostream_iterator<int>(std::cout, " "));
    
    		selection_sort(v.begin(), v.end());
    
    		std::cout << "\nAfter sort: ";
    		copy(v.begin(), v.end(), std::ostream_iterator<int>(std::cout, " "));
    		std::cout << '\n';

    后来通过查看MSDN以及网络相关的文章,才了解到这个是最新的计算随机数的算法。



    Mersenne Twister算法译为马特赛特旋转演算法,是伪随机数发生器之一,其主要作用是生成伪随机数。此算法是Makoto Matsumoto (松本)和Takuji Nishimura (西村)于1997年开发的,基于有限二进制字段上的矩阵线性再生。可以快速产生高质量的伪随机数,修正了古老随机数产生算法的很多缺陷。Mersenne Twister这个名字来自周期长度通常取Mersenne质数这样一个事实。常见的有两个变种Mersenne Twister MT19937和Mersenne Twister MT19937-64。
    Mersenne Twister算法的原理:Mersenne Twister算法是利用线性反馈移位寄存器(LFSR)产生随机数的,LFSR的反馈函数是寄存器中某些位的简单异或,这些位也称之为抽头序列。一个n位的LFSR能够在重复之前产生2^n-1位长的伪随机序列。只有具有一定抽头序列的LFSR才能通过所有2^n-1个内部状态,产生2^n - 1位长的伪随机序列,这个输出的序列就称之为m序列。为了使LFSR成为最大周期的LFSR,由抽头序列加上常数1形成的多项式必须是本原多项式。一个n阶本原多项式是不可约多项式,它能整除x^(2*n-1)+1而不能整除x^d+1,其中d能整除2^n-1。例如(32,7,5,3,2,1,0)是指本原多项式x^32+x^7+x^5+x^3+x^2+x+1,把它转化为最大周期LFSR就是在LFSR的第32,7,5,2,1位抽头。利用上述两种方法产生周期为m的伪随机序列后,只需要将产生的伪随机序列除以序列的周期,就可以得到(0,1)上均匀分布的伪随机序列了。
    Mersenne Twister有以下优点:随机性好,在计算机上容易实现,占用内存较少(mt19937的C程式码执行仅需624个字的工作区域),与其它已使用的伪随机数发生器相比,产生随机数的速度快、周期长,可达到2^19937-1,且具有623维均匀分布的性质,对于一般的应用来说,足够大了,序列关联比较小,能通过很多随机性测试。
    马特赛特旋转演算法产生一个伪随机数,一般为MtRand()。

    从这段话里可以看到它是2的19937次方,所以它的名称就来源这里。

    在STL标准库定义如下:

    typedef mersenne_twister_engine<uint_fast32_t,
      32,624,397,31,0x9908b0df,11,0xffffffff,7,0x9d2c5680,15,0xefc60000,18,1812433253>
      mt19937;

    这个算法在C++里简单地实现如下:

    #include <stdint.h>
    
    // Define MT19937 constants (32-bit RNG)
    enum
    {
        // Assumes W = 32 (omitting this)
        N = 624,
        M = 397,
        R = 31,
        A = 0x9908B0DF,
    
        F = 1812433253,
    
        U = 11,
        // Assumes D = 0xFFFFFFFF (omitting this)
    
        S = 7,
        B = 0x9D2C5680,
    
        T = 15,
        C = 0xEFC60000,
    
        L = 18,
    
        MASK_LOWER = (1ull << R) - 1,
        MASK_UPPER = (1ull << R)
    };
    
    static uint32_t  mt[N];
    static uint16_t  index;
    
    // Re-init with a given seed
    void Initialize(const uint32_t  seed)
    {
        uint32_t  i;
    
        mt[0] = seed;
    
        for ( i = 1; i < N; i++ )
        {
            mt[i] = (F * (mt[i - 1] ^ (mt[i - 1] >> 30)) + i);
        }
    
        index = N;
    }
    
    static void Twist()
    {
        uint32_t  i, x, xA;
    
        for ( i = 0; i < N; i++ )
        {
            x = (mt[i] & MASK_UPPER) + (mt[(i + 1) % N] & MASK_LOWER);
    
            xA = x >> 1;
    
            if ( x & 0x1 )
                xA ^= A;
    
            mt[i] = mt[(i + M) % N] ^ xA;
        }
    
        index = 0;
    }
    
    // Obtain a 32-bit random number
    uint32_t ExtractU32()
    {
        uint32_t  y;
        int       i = index;
    
        if ( index >= N )
        {
            Twist();
            i = index;
        }
    
        y = mt[i];
        index = i + 1;
    
        y ^= (mt[i] >> U);
        y ^= (y << S) & B;
        y ^= (y << T) & C;
        y ^= (y >> L);
    
        return y;
    }

    相关网站:

    http://www.cppblog.com/Chipset/archive/2009/01/19/72330.html


    boost库的实现:

    /* boost random/mersenne_twister.hpp header file
     *
     * Copyright Jens Maurer 2000-2001
     * Copyright Steven Watanabe 2010
     * Distributed under the Boost Software License, Version 1.0. (See
     * accompanying file LICENSE_1_0.txt or copy at
     * http://www.boost.org/LICENSE_1_0.txt)
     *
     * See http://www.boost.org for most recent version including documentation.
     *
     * $Id: mersenne_twister.hpp 74867 2011-10-09 23:13:31Z steven_watanabe $
     *
     * Revision history
     *  2001-02-18  moved to individual header files
     */
    
    #ifndef BOOST_RANDOM_MERSENNE_TWISTER_HPP
    #define BOOST_RANDOM_MERSENNE_TWISTER_HPP
    
    #include <iosfwd>
    #include <istream>
    #include <stdexcept>
    #include <boost/config.hpp>
    #include <boost/cstdint.hpp>
    #include <boost/integer/integer_mask.hpp>
    #include <boost/random/detail/config.hpp>
    #include <boost/random/detail/ptr_helper.hpp>
    #include <boost/random/detail/seed.hpp>
    #include <boost/random/detail/seed_impl.hpp>
    #include <boost/random/detail/generator_seed_seq.hpp>
    
    namespace boost {
    namespace random {
    
    /**
     * Instantiations of class template mersenne_twister_engine model a
     * \pseudo_random_number_generator. It uses the algorithm described in
     *
     *  @blockquote
     *  "Mersenne Twister: A 623-dimensionally equidistributed uniform
     *  pseudo-random number generator", Makoto Matsumoto and Takuji Nishimura,
     *  ACM Transactions on Modeling and Computer Simulation: Special Issue on
     *  Uniform Random Number Generation, Vol. 8, No. 1, January 1998, pp. 3-30. 
     *  @endblockquote
     *
     * @xmlnote
     * The boost variant has been implemented from scratch and does not
     * derive from or use mt19937.c provided on the above WWW site. However, it
     * was verified that both produce identical output.
     * @endxmlnote
     *
     * The seeding from an integer was changed in April 2005 to address a
     * <a href="http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/MT2002/emt19937ar.html">weakness</a>.
     * 
     * The quality of the generator crucially depends on the choice of the
     * parameters.  User code should employ one of the sensibly parameterized
     * generators such as \mt19937 instead.
     *
     * The generator requires considerable amounts of memory for the storage of
     * its state array. For example, \mt11213b requires about 1408 bytes and
     * \mt19937 requires about 2496 bytes.
     */
    template<class UIntType,
             std::size_t w, std::size_t n, std::size_t m, std::size_t r,
             UIntType a, std::size_t u, UIntType d, std::size_t s,
             UIntType b, std::size_t t,
             UIntType c, std::size_t l, UIntType f>
    class mersenne_twister_engine
    {
    public:
        typedef UIntType result_type;
        BOOST_STATIC_CONSTANT(std::size_t, word_size = w);
        BOOST_STATIC_CONSTANT(std::size_t, state_size = n);
        BOOST_STATIC_CONSTANT(std::size_t, shift_size = m);
        BOOST_STATIC_CONSTANT(std::size_t, mask_bits = r);
        BOOST_STATIC_CONSTANT(UIntType, xor_mask = a);
        BOOST_STATIC_CONSTANT(std::size_t, tempering_u = u);
        BOOST_STATIC_CONSTANT(UIntType, tempering_d = d);
        BOOST_STATIC_CONSTANT(std::size_t, tempering_s = s);
        BOOST_STATIC_CONSTANT(UIntType, tempering_b = b);
        BOOST_STATIC_CONSTANT(std::size_t, tempering_t = t);
        BOOST_STATIC_CONSTANT(UIntType, tempering_c = c);
        BOOST_STATIC_CONSTANT(std::size_t, tempering_l = l);
        BOOST_STATIC_CONSTANT(UIntType, initialization_multiplier = f);
        BOOST_STATIC_CONSTANT(UIntType, default_seed = 5489u);
      
        // backwards compatibility
        BOOST_STATIC_CONSTANT(UIntType, parameter_a = a);
        BOOST_STATIC_CONSTANT(std::size_t, output_u = u);
        BOOST_STATIC_CONSTANT(std::size_t, output_s = s);
        BOOST_STATIC_CONSTANT(UIntType, output_b = b);
        BOOST_STATIC_CONSTANT(std::size_t, output_t = t);
        BOOST_STATIC_CONSTANT(UIntType, output_c = c);
        BOOST_STATIC_CONSTANT(std::size_t, output_l = l);
        
        // old Boost.Random concept requirements
        BOOST_STATIC_CONSTANT(bool, has_fixed_range = false);
    
    
        /**
         * Constructs a @c mersenne_twister_engine and calls @c seed().
         */
        mersenne_twister_engine() { seed(); }
    
        /**
         * Constructs a @c mersenne_twister_engine and calls @c seed(value).
         */
        BOOST_RANDOM_DETAIL_ARITHMETIC_CONSTRUCTOR(mersenne_twister_engine,
                                                   UIntType, value)
        { seed(value); }
        template<class It> mersenne_twister_engine(It& first, It last)
        { seed(first,last); }
    
        /**
         * Constructs a mersenne_twister_engine and calls @c seed(gen).
         *
         * @xmlnote
         * The copy constructor will always be preferred over
         * the templated constructor.
         * @endxmlnote
         */
        BOOST_RANDOM_DETAIL_SEED_SEQ_CONSTRUCTOR(mersenne_twister_engine,
                                                 SeedSeq, seq)
        { seed(seq); }
    
        // compiler-generated copy ctor and assignment operator are fine
    
        /** Calls @c seed(default_seed). */
        void seed() { seed(default_seed); }
    
        /**
         * Sets the state x(0) to v mod 2w. Then, iteratively,
         * sets x(i) to
         * (i + f * (x(i-1) xor (x(i-1) rshift w-2))) mod 2<sup>w</sup>
         * for i = 1 .. n-1. x(n) is the first value to be returned by operator().
         */
        BOOST_RANDOM_DETAIL_ARITHMETIC_SEED(mersenne_twister_engine, UIntType, value)
        {
            // New seeding algorithm from 
            // http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/MT2002/emt19937ar.html
            // In the previous versions, MSBs of the seed affected only MSBs of the
            // state x[].
            const UIntType mask = (max)();
            x[0] = value & mask;
            for (i = 1; i < n; i++) {
                // See Knuth "The Art of Computer Programming"
                // Vol. 2, 3rd ed., page 106
                x[i] = (f * (x[i-1] ^ (x[i-1] >> (w-2))) + i) & mask;
            }
        }
        
        /**
         * Seeds a mersenne_twister_engine using values produced by seq.generate().
         */
        BOOST_RANDOM_DETAIL_SEED_SEQ_SEED(mersenne_twister_engine, SeeqSeq, seq)
        {
            detail::seed_array_int<w>(seq, x);
            i = n;
    
            // fix up the state if it's all zeroes.
            if((x[0] & (~static_cast<UIntType>(0) << r)) == 0) {
                for(std::size_t j = 1; j < n; ++j) {
                    if(x[j] != 0) return;
                }
                x[0] = static_cast<UIntType>(1) << (w-1);
            }
        }
    
        /** Sets the state of the generator using values from an iterator range. */
        template<class It>
        void seed(It& first, It last)
        {
            detail::fill_array_int<w>(first, last, x);
            i = n;
    
            // fix up the state if it's all zeroes.
            if((x[0] & (~static_cast<UIntType>(0) << r)) == 0) {
                for(std::size_t j = 1; j < n; ++j) {
                    if(x[j] != 0) return;
                }
                x[0] = static_cast<UIntType>(1) << (w-1);
            }
        }
      
        /** Returns the smallest value that the generator can produce. */
        static result_type min BOOST_PREVENT_MACRO_SUBSTITUTION ()
        { return 0; }
        /** Returns the largest value that the generator can produce. */
        static result_type max BOOST_PREVENT_MACRO_SUBSTITUTION ()
        { return boost::low_bits_mask_t<w>::sig_bits; }
        
        /** Produces the next value of the generator. */
        result_type operator()();
    
        /** Fills a range with random values */
        template<class Iter>
        void generate(Iter first, Iter last)
        { detail::generate_from_int(*this, first, last); }
    
        /**
         * Advances the state of the generator by @c z steps.  Equivalent to
         *
         * @code
         * for(unsigned long long i = 0; i < z; ++i) {
         *     gen();
         * }
         * @endcode
         */
        void discard(boost::uintmax_t z)
        {
            for(boost::uintmax_t j = 0; j < z; ++j) {
                (*this)();
            }
        }
    
    #ifndef BOOST_RANDOM_NO_STREAM_OPERATORS
        /** Writes a mersenne_twister_engine to a @c std::ostream */
        template<class CharT, class Traits>
        friend std::basic_ostream<CharT,Traits>&
        operator<<(std::basic_ostream<CharT,Traits>& os,
                   const mersenne_twister_engine& mt)
        {
            mt.print(os);
            return os;
        }
        
        /** Reads a mersenne_twister_engine from a @c std::istream */
        template<class CharT, class Traits>
        friend std::basic_istream<CharT,Traits>&
        operator>>(std::basic_istream<CharT,Traits>& is,
                   mersenne_twister_engine& mt)
        {
            for(std::size_t j = 0; j < mt.state_size; ++j)
                is >> mt.x[j] >> std::ws;
            // MSVC (up to 7.1) and Borland (up to 5.64) don't handle the template
            // value parameter "n" available from the class template scope, so use
            // the static constant with the same value
            mt.i = mt.state_size;
            return is;
        }
    #endif
    
        /**
         * Returns true if the two generators are in the same state,
         * and will thus produce identical sequences.
         */
        friend bool operator==(const mersenne_twister_engine& x,
                               const mersenne_twister_engine& y)
        {
            if(x.i < y.i) return x.equal_imp(y);
            else return y.equal_imp(x);
        }
        
        /**
         * Returns true if the two generators are in different states.
         */
        friend bool operator!=(const mersenne_twister_engine& x,
                               const mersenne_twister_engine& y)
        { return !(x == y); }
    
    private:
        /// \cond show_private
    
        void twist();
    
        /**
         * Does the work of operator==.  This is in a member function
         * for portability.  Some compilers, such as msvc 7.1 and
         * Sun CC 5.10 can't access template parameters or static
         * members of the class from inline friend functions.
         *
         * requires i <= other.i
         */
        bool equal_imp(const mersenne_twister_engine& other) const
        {
            UIntType back[n];
            std::size_t offset = other.i - i;
            for(std::size_t j = 0; j + offset < n; ++j)
                if(x[j] != other.x[j+offset])
                    return false;
            rewind(&back[n-1], offset);
            for(std::size_t j = 0; j < offset; ++j)
                if(back[j + n - offset] != other.x[j])
                    return false;
            return true;
        }
    
        /**
         * Does the work of operator<<.  This is in a member function
         * for portability.
         */
        template<class CharT, class Traits>
        void print(std::basic_ostream<CharT, Traits>& os) const
        {
            UIntType data[n];
            for(std::size_t j = 0; j < i; ++j) {
                data[j + n - i] = x[j];
            }
            if(i != n) {
                rewind(&data[n - i - 1], n - i);
            }
            os << data[0];
            for(std::size_t j = 1; j < n; ++j) {
                os << ' ' << data[j];
            }
        }
    
        /**
         * Copies z elements of the state preceding x[0] into
         * the array whose last element is last.
         */
        void rewind(UIntType* last, std::size_t z) const
        {
            const UIntType upper_mask = (~static_cast<UIntType>(0)) << r;
            const UIntType lower_mask = ~upper_mask;
            UIntType y0 = x[m-1] ^ x[n-1];
            if(y0 & (static_cast<UIntType>(1) << (w-1))) {
                y0 = ((y0 ^ a) << 1) | 1;
            } else {
                y0 = y0 << 1;
            }
            for(std::size_t sz = 0; sz < z; ++sz) {
                UIntType y1 =
                    rewind_find(last, sz, m-1) ^ rewind_find(last, sz, n-1);
                if(y1 & (static_cast<UIntType>(1) << (w-1))) {
                    y1 = ((y1 ^ a) << 1) | 1;
                } else {
                    y1 = y1 << 1;
                }
                *(last - sz) = (y0 & upper_mask) | (y1 & lower_mask);
                y0 = y1;
            }
        }
    
        /**
         * Given a pointer to the last element of the rewind array,
         * and the current size of the rewind array, finds an element
         * relative to the next available slot in the rewind array.
         */
        UIntType
        rewind_find(UIntType* last, std::size_t size, std::size_t j) const
        {
            std::size_t index = (j + n - size + n - 1) % n;
            if(index < n - size) {
                return x[index];
            } else {
                return *(last - (n - 1 - index));
            }
        }
    
        /// \endcond
    
        // state representation: next output is o(x(i))
        //   x[0]  ... x[k] x[k+1] ... x[n-1]   represents
        //  x(i-k) ... x(i) x(i+1) ... x(i-k+n-1)
    
        UIntType x[n]; 
        std::size_t i;
    };
    
    /// \cond show_private
    
    #ifndef BOOST_NO_INCLASS_MEMBER_INITIALIZATION
    //  A definition is required even for integral static constants
    #define BOOST_RANDOM_MT_DEFINE_CONSTANT(type, name)                         \
    template<class UIntType, std::size_t w, std::size_t n, std::size_t m,       \
        std::size_t r, UIntType a, std::size_t u, UIntType d, std::size_t s,    \
        UIntType b, std::size_t t, UIntType c, std::size_t l, UIntType f>       \
    const type mersenne_twister_engine<UIntType,w,n,m,r,a,u,d,s,b,t,c,l,f>::name
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, word_size);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, state_size);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, shift_size);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, mask_bits);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(UIntType, xor_mask);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, tempering_u);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(UIntType, tempering_d);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, tempering_s);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(UIntType, tempering_b);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, tempering_t);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(UIntType, tempering_c);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, tempering_l);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(UIntType, initialization_multiplier);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(UIntType, default_seed);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(UIntType, parameter_a);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, output_u );
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, output_s);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(UIntType, output_b);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, output_t);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(UIntType, output_c);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(std::size_t, output_l);
    BOOST_RANDOM_MT_DEFINE_CONSTANT(bool, has_fixed_range);
    #undef BOOST_RANDOM_MT_DEFINE_CONSTANT
    #endif
    
    template<class UIntType,
             std::size_t w, std::size_t n, std::size_t m, std::size_t r,
             UIntType a, std::size_t u, UIntType d, std::size_t s,
             UIntType b, std::size_t t,
             UIntType c, std::size_t l, UIntType f>
    void
    mersenne_twister_engine<UIntType,w,n,m,r,a,u,d,s,b,t,c,l,f>::twist()
    {
        const UIntType upper_mask = (~static_cast<UIntType>(0)) << r;
        const UIntType lower_mask = ~upper_mask;
    
        const std::size_t unroll_factor = 6;
        const std::size_t unroll_extra1 = (n-m) % unroll_factor;
        const std::size_t unroll_extra2 = (m-1) % unroll_factor;
    
        // split loop to avoid costly modulo operations
        {  // extra scope for MSVC brokenness w.r.t. for scope
            for(std::size_t j = 0; j < n-m-unroll_extra1; j++) {
                UIntType y = (x[j] & upper_mask) | (x[j+1] & lower_mask);
                x[j] = x[j+m] ^ (y >> 1) ^ ((x[j+1]&1) * a);
            }
        }
        {
            for(std::size_t j = n-m-unroll_extra1; j < n-m; j++) {
                UIntType y = (x[j] & upper_mask) | (x[j+1] & lower_mask);
                x[j] = x[j+m] ^ (y >> 1) ^ ((x[j+1]&1) * a);
            }
        }
        {
            for(std::size_t j = n-m; j < n-1-unroll_extra2; j++) {
                UIntType y = (x[j] & upper_mask) | (x[j+1] & lower_mask);
                x[j] = x[j-(n-m)] ^ (y >> 1) ^ ((x[j+1]&1) * a);
            }
        }
        {
            for(std::size_t j = n-1-unroll_extra2; j < n-1; j++) {
                UIntType y = (x[j] & upper_mask) | (x[j+1] & lower_mask);
                x[j] = x[j-(n-m)] ^ (y >> 1) ^ ((x[j+1]&1) * a);
            }
        }
        // last iteration
        UIntType y = (x[n-1] & upper_mask) | (x[0] & lower_mask);
        x[n-1] = x[m-1] ^ (y >> 1) ^ ((x[0]&1) * a);
        i = 0;
    }
    /// \endcond
    
    template<class UIntType,
             std::size_t w, std::size_t n, std::size_t m, std::size_t r,
             UIntType a, std::size_t u, UIntType d, std::size_t s,
             UIntType b, std::size_t t,
             UIntType c, std::size_t l, UIntType f>
    inline typename
    mersenne_twister_engine<UIntType,w,n,m,r,a,u,d,s,b,t,c,l,f>::result_type
    mersenne_twister_engine<UIntType,w,n,m,r,a,u,d,s,b,t,c,l,f>::operator()()
    {
        if(i == n)
            twist();
        // Step 4
        UIntType z = x[i];
        ++i;
        z ^= ((z >> u) & d);
        z ^= ((z << s) & b);
        z ^= ((z << t) & c);
        z ^= (z >> l);
        return z;
    }
    
    /**
     * The specializations \mt11213b and \mt19937 are from
     *
     *  @blockquote
     *  "Mersenne Twister: A 623-dimensionally equidistributed
     *  uniform pseudo-random number generator", Makoto Matsumoto
     *  and Takuji Nishimura, ACM Transactions on Modeling and
     *  Computer Simulation: Special Issue on Uniform Random Number
     *  Generation, Vol. 8, No. 1, January 1998, pp. 3-30. 
     *  @endblockquote
     */
    typedef mersenne_twister_engine<uint32_t,32,351,175,19,0xccab8ee7,
        11,0xffffffff,7,0x31b6ab00,15,0xffe50000,17,1812433253> mt11213b;
    
    /**
     * The specializations \mt11213b and \mt19937 are from
     *
     *  @blockquote
     *  "Mersenne Twister: A 623-dimensionally equidistributed
     *  uniform pseudo-random number generator", Makoto Matsumoto
     *  and Takuji Nishimura, ACM Transactions on Modeling and
     *  Computer Simulation: Special Issue on Uniform Random Number
     *  Generation, Vol. 8, No. 1, January 1998, pp. 3-30. 
     *  @endblockquote
     */
    typedef mersenne_twister_engine<uint32_t,32,624,397,31,0x9908b0df,
        11,0xffffffff,7,0x9d2c5680,15,0xefc60000,18,1812433253> mt19937;
    
    #if !defined(BOOST_NO_INT64_T) && !defined(BOOST_NO_INTEGRAL_INT64_T)
    typedef mersenne_twister_engine<uint64_t,64,312,156,31,
        UINT64_C(0xb5026f5aa96619e9),29,UINT64_C(0x5555555555555555),17,
        UINT64_C(0x71d67fffeda60000),37,UINT64_C(0xfff7eee000000000),43,
        UINT64_C(6364136223846793005)> mt19937_64;
    #endif
    
    /// \cond show_deprecated
    
    template<class UIntType,
             int w, int n, int m, int r,
             UIntType a, int u, std::size_t s,
             UIntType b, int t,
             UIntType c, int l, UIntType v>
    class mersenne_twister :
        public mersenne_twister_engine<UIntType,
            w, n, m, r, a, u, ~(UIntType)0, s, b, t, c, l, 1812433253>
    {
        typedef mersenne_twister_engine<UIntType,
            w, n, m, r, a, u, ~(UIntType)0, s, b, t, c, l, 1812433253> base_type;
    public:
        mersenne_twister() {}
        BOOST_RANDOM_DETAIL_GENERATOR_CONSTRUCTOR(mersenne_twister, Gen, gen)
        { seed(gen); }
        BOOST_RANDOM_DETAIL_ARITHMETIC_CONSTRUCTOR(mersenne_twister, UIntType, val)
        { seed(val); }
        template<class It>
        mersenne_twister(It& first, It last) : base_type(first, last) {}
        void seed() { base_type::seed(); }
        BOOST_RANDOM_DETAIL_GENERATOR_SEED(mersenne_twister, Gen, gen)
        {
            detail::generator_seed_seq<Gen> seq(gen);
            base_type::seed(seq);
        }
        BOOST_RANDOM_DETAIL_ARITHMETIC_SEED(mersenne_twister, UIntType, val)
        { base_type::seed(val); }
        template<class It>
        void seed(It& first, It last) { base_type::seed(first, last); }
    };
    
    /// \endcond
    
    } // namespace random
    
    using random::mt11213b;
    using random::mt19937;
    using random::mt19937_64;
    
    } // namespace boost
    
    BOOST_RANDOM_PTR_HELPER_SPEC(boost::mt11213b)
    BOOST_RANDOM_PTR_HELPER_SPEC(boost::mt19937)
    BOOST_RANDOM_PTR_HELPER_SPEC(boost::mt19937_64)
    
    #endif // BOOST_RANDOM_MERSENNE_TWISTER_HPP


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