Re: Resizing hash tables in Guile

[Joris van der Hoeven <TeXmacs@math.u-psud.fr>]

>  > Regarding reshuffling time: Yes, rehuffling means that every operation
>  > isn't O(1), but it *does* mean that they are O(1) on average.  You can
>  > understand this by a trick sometimes used in algorithm run-time
>  > analysis called "amortization":
>  >
>  > The idea is that for every operation on the hash table you "pay" some
>  > extra virtual time, and the sum of this time can then be used to
>  > account for the reshuffling time.  In my implementation, the hash
>  > table is roughly doubled for each rehash.  Before rehashing occurs,
>  > you have inserted N elements.  This has cost you less than cN seconds.
>  > Rehashing is O(2N) = O(N), so we can say it will cost us less than dN
>  > seconds.  If we now pay off d seconds per operation in advance, and
>  > note that the argument above holds equally well for each rehashing
>  > point, we realize that each operation costs less than c + d seconds on
>  > average.  This means that, on average, operations are O(1).
> 
> Inserts are, but lookups aren't necessarily.

Both aren't necessarily, because inserting requires looking up too.

> Lookups being O(1) requires uniformity of bucket sizes.
> Worst case hash table lookup time is still O(N).

You can also store a binary search tree in each of the buckets,
if you think that your hash function is bad.

> And good hashing functions are still hard to write.

I do not really agree. A good hash algorithm for lists (or strings),
which I use in TeXmacs, is to rotate the 32 bit integer hash values of
each of the members by a prime number like 3, 5, 7 or 11 and progressively
take the exclusive or. This seems to lead to bucket sizes as
predicted by probability theory, even for hash tables of size 2^p.

> People overestimate log(N) and overuse O().  When comparing an O(1)
> algorithm to an O(log(N)) algorithm, it really comes down to the
> actual functions involved, and actual problem size, not just the
> asymptotic behavior.  2^32 is over 4,000,000,000.

A factor 10 is still a factor 10 though.
(2^10 ~~ 1000).

> With this many
> items, log(N) is still just 32, so an O(log(N)) algorithm will still
> beat an O(1) algorithm if it's really log_2(N) vs 32.

Yes, but the O(1) is really *table lookup* multiplied by a small
constant here, so this is *fast*. You may adjust the small constant
by choosing an appropriate threshold for "size/nr buckets".

> Also, if a person's relying on O(1) for hash table performance, it might be
> not because they need that on average, but because they need an upper
> bound on the operation time, in which case automatic resizing would
> also violate this, even though it maintains O(1) on average.

This is a more serious drawback of standard hash tables, but,
as I said before, we already have garbage collection in Guile anyway...

> Trees also sort the data for you, which hash tables don't give you.

But you need a compairison operation for that,
which may be even less natural than a hash function.

> So, ideally, one would have a hash table object with & without
> resizing, and various sorts of tree (AVL, red/black, B*, etc) objects.
> insert and delete and map would be methods that work on all of the
> above, with map on trees returning the data in sorted order.  For that
> matter, insert & delete might as well also work on lists...

Agreed: ideally, we have everything :^)



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