Shifting big numbers

问题

X = 712360810625491574981234007851998 is represented using a linked list and each node is an unsigned int

Is there a fast way to do X << 8 X << 591 other than X * 2^8 X * 2^591 ?

回答1:

Bit shifting is very easy in any arbitrary number of bits. Just remember to shift the overflowed bits to the next element. That's all

Below is a left shift by 3 example

uint64_t i1, i2, i3, o1, o2, o3; // {o3, o2, o1} = {i3, i2, i1} << 3;

o3 = i3 << 3 | i2 >> (32 - 3);
o2 = i2 << 3 | i1 >> (32 - 3);
o1 = i1 << 3;

Similar for shifting right, just iterate in the reverse direction.

Edit:

It seems that you're using base 10⁹ for your large number, so binary shifting does not apply here. "Shifting" left/right N digits in a base B is equivalent to multiplying the number by B^N and B^-N respectively. You can't do binary shift in decimal and vice versa

If you don't change your base then you have only one solution, that's multiplying the number by 2⁵⁹¹. If you want to shift like in binary you must change to a base that is a power of 2 like base 2³² or base 2⁶⁴

A general solution would be like this, with the limbs stored in little-endian and each digit is in base 2^{CHAR_BIT*sizeof(T)}

template<typename T>
void rshift(std::vector<T>& x, std::size_t shf_amount) // x >>= shf_amount
{
    constexpr std::size_t width = CHAR_BIT*sizeof(T);
    if (shf_amount > width)
        throw;

    const std::size_t shift     = shf_amount % width;
    const std::size_t limbshift = shf_amount / width;

    std::size_t i = 0;
    for (; i < x.size() - limbshift - 1; ++i)
        x[i] = (x[i + limbshift] >> shift) | (x[i + 1 + limbshift] << (width - shift));
    x[i++] = x[i + limbshift] >> shift;
    for (; i < x.size() ; ++i)
        x[i] = 0;
}

Moreover from the tag you're likely using a linked-list for storing the limbs which is not cache-friendly due to elements scattering all around the memory space, and it also wastes a lot of memory due to the next pointers. In fact you shouldn't use linked list in most real life problems

• Bjarne Stroustrup says we must avoid linked lists
• Why you should never, ever, EVER use linked-list in your code again
• Number crunching: Why you should never, ever, EVER use linked-list in your code again
• Bjarne Stroustrup: Why you should avoid Linked Lists
• Are lists evil?—Bjarne Stroustrup

回答2:

I think, if you use 64 bit integer the code should be

    o3 = i3 << 3 | i2 >> (32 - 3);
    .....

来源：https://stackoverflow.com/questions/10526840/shifting-big-numbers