C++ - Why is boost::hash_combine the best way to combine hash-values?

It being the "best" is argumentative.

It being "good", or even "very good", at least superficially, is easy.

seed ^= hasher(v) + 0x9e3779b9 + (seed<<6) + (seed>>2);

We'll presume seed is a previous result of hasher or this algorithm.

^= means that the bits on the left and bits on the right all change the bits of the result.

hasher(v) is presumed to be a decent hash on v. But the rest is defence in case it isn't a decent hash.

0x9e3779b9 is a 32 bit value (it could be extended to 64 bit if size_t was 64 bit arguably) that contains half 0s and half 1s. It is basically a random series of 0s and 1s done by approximating particular irrational constant as a base-2 fixed point value. This helps ensure that if the hasher returns bad values, we still get a smear of 1s and 0s in our output.

(seed<<6) + (seed>>2) is a bit shuffle of the incoming seed.

Imagine the 0x constant was missing. Imagine the hasher returns the constant 0x01000 for almost every v passed in. Now, each bit of the seed is spread out over the next iteration of the hash, during which it is again spread out.

The seed ^= (seed<<6) + (seed>>2) 0x00001000 becomes 0x00041400 after one iteration. Then 0x00859500. As you repeat the operation, any set bits are "smeared out" over the output bits. Eventually the right and left bits collide, and carry moves the set bit from "even locations" to "odd locations".

The bits dependent on the value of an input seed grows relatively fast and in complex ways as the combine operation recurses on the seed operation. Adding causes carries, which smear things even more. The 0x constant adds a bunch of pseudo-random bits that make boring hash values occupy more than a few bits of the hash space after being combined.

It is asymmetric thanks to addition (combining the hashes of "dog" and "god" gives different results), it handles boring hash values (mapping characters to their ascii value, which only involves twiddling a handful of bits). And, it is reasonably fast.

Slower hash combines that are cryptographically strong can be better in other situations. I, naively, would presume that making the shifts be a combination of even and odd shifts might be a good idea (but maybe addition, which moves even bits from odd bits, makes that less of a problem: after 3 iterations, incoming lone seed bits will collide and add and cause a carry).

The downside to this kind of analysis is that it only takes one mistake to make a hash function really bad. Pointing out all the good things doesn't help that much. So another thing that makes it good now is that it is reasonably famous and in an open-source repository, and I haven't heard anyone point out why it is bad.

It's not the best, surprisingly to me it's not even particularily good. The main problem is the bad distribution, which is not really the fault of boost::hash_combine in itself, but in conjunction with a badly distributing hash like std::hash which is most commonly implemented with the identity function.

Figure 2: The effect of a single bit change in one of two random 32 bit numbers on the result of boost::hash_combine

To demonstrate how bad things can become these are the collisions for points on a 32x32 grid when using hash_combine as intended, and with std::hash:

# hash      x₀   y₀  x₁  y₁ ...
3449074105  6   30   8  15
3449074104  6   31   8  16
3449074107  6   28   8  17
3449074106  6   29   8  18
3449074109  6   26   8  19
3449074108  6   27   8  20
3449074111  6   24   8  21
3449074110  6   25   8  22

For a well distributed hash there should be none, statistically. Using bit-rotations instead of bit-shifts and xor instead of addition one could easily create a similar hash_combine that preserves entropy better. But really what you should do is use a good hash function in the first place, then after that a simple xor is sufficient to combine the seed and the hash, if the hash encodes the position in the sequence. For ease of implementation the following hash is not and to make hash_combine non commutative any non-commutative and bijective operation is sufficient, only then use xor to combine the values. I chose a asymmetric binary rotation because it is cheap.

#include <limits>
#include <cstdint>

template<typename T>
T xorshift(const T& n,int i){
  return n^(n>>i);
}

uint32_t distribute(const uint32_t& n){
  uint32_t p = 0x55555555ul; // pattern of alternating 0 and 1
  uint32_t c = 3423571495ul; // random uneven integer constant; 
  return c*xorshift(p*xorshift(n,16),16);
}

uint64_t hash(const uint64_t& n){
  uint64_t p = 0x5555555555555555;     // pattern of alternating 0 and 1
  uint64_t c = 17316035218449499591ull;// random uneven integer constant; 
  return c*xorshift(p*xorshift(n,32),32);
}

// if c++20 rotl is not available:
template <typename T,typename S>
typename std::enable_if<std::is_unsigned<T>::value,T>::type
constexpr rotl(const T n, const S i){
  const T m = (std::numeric_limits<T>::digits-1);
  const T c = i&m;
  return (n<<c)|(n>>((T(0)-c)&m)); // this is usually recognized by the compiler to mean rotation, also c++20 now gives us rotl directly
}

template <class T>
inline size_t hash_combine(std::size_t& seed, const T& v)
{
    return rotl(seed,std::numeric_limits<size_t>::digits/3) ^ distribute(std::hash<T>(v));
}

The seed is rotated once before combining it to make the order in which the hash was computed relevant.

The hash_combine from boost needs two operations less, and more importantly no multiplications, in fact it's about 5x faster, but at about 2 cyles per hash on my machine the proposed solution is still very fast and pays off quickly when used for a hash table. There are 118 collisions on a 1024x1024 grid (vs. 982017 for boosts hash_combine + std::hash), about as many as expected for a well distributed hash function and that is all we can ask for.

Now even when used in conjunction with a good hash function boost::hash_combine is not ideal. If all entropy is in the seed at some point some of it will get lost. There are 2948667289 distinct results of boost::hash_combine(x,0), but there should be 4294967296 .

In conclusion, they tried to create a hash function that does both, combining and cascading, and fast, but ended up with something that does both just good enough to not be recognised as bad immediately.