Why are hash table expansions usually done by doubling the size?

I've done a little research on hash tables, and I keep running across the rule of thumb that when there are a certain number of entries (either max or via a load factor like 75%) the hash table should be expanded.

Almost always, the recommendation is to double (or double plus 1, i.e., 2n+1) the size of the hash table. However, I haven't been able to find a good reason for this.

Why double the size, rather than, say, increasing it 25%, or increasing it to the size of the next prime number, or next k prime numbers (e.g., three)?

I already know that it's often a good idea to choose an initial hash table size which is a prime number, at least if your hash function uses modulus such as universal hashing. And I know that's why it's usually recommended to do 2n+1 instead of 2n (e.g., http://www.concentric.net/~Ttwang/tech/hashsize.htm)

However as I said, I haven't seen any real explanation for why doubling or doubling-plus-one is actually a good choice rather than some other method of choosing a size for the new hash table.

(And yes I've read the Wikipedia article on hash tables :) http://en.wikipedia.org/wiki/Hash_table

Solution 1:

Hash-tables could not claim "amortized constant time insertion" if, for instance, the resizing was by a constant increment. In that case the cost of resizing (which grows with the size of the hash-table) would make the cost of one insertion linear in the total number of elements to insert. Because resizing becomes more and more expensive with the size of the table, it has to happen "less and less often" to keep the amortized cost of insertion constant.

Most implementations allow the average bucket occupation to grow to until a bound fixed in advance before resizing (anywhere between 0.5 and 3, which are all acceptable values). With this convention, just after resizing the average bucket occupation becomes half that bound. Resizing by doubling keeps the average bucket occupation in a band of width *2.

Sub-note: because of statistical clustering, you have to take an average bucket occupation as low as 0.5 if you want many buckets to have at most one elements (maximum speed for finding ignoring the complex effects of cache size), or as high as 3 if you want a minimum number of empty buckets (that correspond to wasted space).

Solution 2:

I had read a very interesting discussion on growth strategy on this very site... just cannot find it again.

While 2 is commonly used, it's been demonstrated that it was not the best value. One often cited problem is that it does not cope well with allocators schemes (which often allocate power of twos blocks) since it would always require a reallocation while a smaller number might in fact be reallocated in the same block (simulating in-place growth) and thus being faster.

Thus, for example, the VC++ Standard Library uses a growth factor of 1.5 (ideally should be the golden number if a first-fit memory allocation strategy is being used) after an extensive discussion on the mailing list. The reasoning is explained here:

I'd be interested if any other vector implementations uses a growth factor other than 2, and I'd also like to know whether VC7 uses 1.5 or 2 (since I don't have that compiler here).

There is a technical reason to prefer 1.5 to 2 -- more specifically, to prefer values less than 1+sqrt(5)/2.

Suppose you are using a first-fit memory allocator, and you're progressively appending to a vector. Then each time you reallocate, you allocate new memory, copy the elements, then free the old memory. That leaves a gap, and it would be nice to be able to use that memory eventually. If the vector grows too rapidly, it will always be too big for the available memory.

It turns out that if the growth factor is >= 1+sqrt(5)/2, the new memory will always be too big for the hole that has been left sofar; if it is < 1+sqrt(5)/2, the new memory will eventually fit. So 1.5 is small enough to allow the memory to be recycled.

Surely, if the growth factor is >= 2 the new memory will always be too big for the hole that has been left so far; if it is < 2, the new memory will eventually fit. Presumably the reason for (1+sqrt(5))/2 is...

• Initial allocation is s.
• The first resize is k*s.
• The second resize is k*k*s, which will fit the hole iff k*k*s <= k*s+s, i.e. iff k <= (1+sqrt(5))/2

...the hole can be recycled asap.

It could, by storing its previous size, grow fibonaccily.

Of course, it should be tailored to the memory allocation strategy.