Efficient comparison of small integer vectors

Question

I have small vectors. Each of them is made of 10 integers which are between 0 and 15. This means that every element in a vector can be written using 4 bits. Hence I can concatenate my vector elements and store the whole vector in a single long type (in C, C++, java...)

Vector v1 dominates vector v2 if for each i in 0,...,9, v1[i] >= v2[i]

I want to write a method compare(long v1, long v2) that would return 0 if non of the vectors dominates the other, 1 if the first one dominates and -1 if the second one dominates.

Is there any efficient way to implement compare other than getting every i component and doing 10 times the normal integer comparison?

EDIT

if v1 is exactly the same as v2 returning 1 or -1 are both fine

Answer 1

It's possible to do this using bit-manipulation. Space your values out so that each takes up 5 bits, with 4 bits for the value and an empty 0 in the most significant position as a kind of spacing bit.

Placing a spacing bit between each value stops borrows/carries from propagating between adjacent values and means you can do certain SIMD-like arithmetic operations on the vector just by using regular integer addition or subtraction. We can use subtraction to do a vector comparison.

To do the test you can set all the spacing bits to 1 in one of the vectors and then subtract the second one. If the value in the 4 bits below the spacing bit is greater in the second one then it will carry the bit from the spacing bit and set it to zero in the result, if not then it will remain a one (the first value is greater than or equal to the second). If the first vector dominates the second then all the spacing bits will be one after the subtraction.

Simple demonstration using ints:

#define SPACING_BITS ((1<<4)|(1<<9)|(1<<14)|(1<<19))
int createVector(int v0, int v1, int v2, int v3)
{
    return v0 | (v1 << 5) | (v2 << 10) | (v3 << 15);
}

int vectorDominates(int vectorA, int vectorB)
{
     // returns 1 if vectorA dominates vectorB:
     return (((vectorA | SPACING_BITS) - vectorB) & SPACING_BITS) == SPACING_BITS;
}

int compare(int vectorA, int vectorB)
{
    if(vectorDominates(vectorA, vectorB))
        return 1;
    else if(vectorDominates(vectorB, vectorA))
        return -1;
    return 0;
}

You can extend it to use 64 bit values using 50 bits to store the 10 values. You can also inline the calls to vectorDominates in the compare function.

Demo