chimera
Lazy infinite compact streams with cachefriendly O(1) indexing
and applications for memoization.
Introduction
Imagine having a function f :: Word > a
,
which is expensive to evaluate. We would like to memoize it,
returning g :: Word > a
, which does effectively the same,
but transparently caches results to speed up repetitive
reevaluation.
There are plenty of memoizing libraries on Hackage, but they
usually fall into two categories:

Store cache as a flat array, enabling us
to obtain cached values in O(1) time, which is nice.
The drawback is that one must specify the size
of the array beforehand,
limiting an interval of inputs,
and actually allocate it at once.

Store cache as a lazy binary tree.
Thanks to laziness, one can freely use the full range of inputs.
The drawback is that obtaining values from a tree
takes logarithmic time and is unfriendly to CPU cache,
which kinda defeats the purpose.
This package intends to tackle both issues,
providing a data type Chimera
for
lazy infinite compact streams with cachefriendly O(1) indexing.
Additional features include:
 memoization of recursive functions and recurrent sequences,
 memoization of functions of several, possibly signed arguments,
 efficient memoization of boolean predicates.
Example 1
Consider the following predicate:
isOdd :: Word > Bool
isOdd n = if n == 0 then False else not (isOdd (n  1))
Its computation is expensive, so we'd like to memoize it:
isOdd' :: Word > Bool
isOdd' = memoize isOdd
This is fine to avoid reevaluation for the same arguments.
But isOdd
does not use this cache internally, going all the way
of recursive calls to n = 0
. We can do better,
if we rewrite isOdd
as a fix
point of isOddF
:
isOddF :: (Word > Bool) > Word > Bool
isOddF f n = if n == 0 then False else not (f (n  1))
and invoke memoizeFix
to pass cache into recursive calls as well:
isOdd' :: Word > Bool
isOdd' = memoizeFix isOddF
Example 2
Define a predicate, which checks whether its argument is
a prime number, using trial division.
isPrime :: Word > Bool
isPrime n = n > 1 && and [ n `rem` d /= 0  d < [2 .. floor (sqrt (fromIntegral n))], isPrime d]
This is certainly an expensive recursive computation and we would like
to speed up its evaluation by wrappping into a caching layer.
Convert the predicate to an unfixed form such that isPrime = fix isPrimeF
:
isPrimeF :: (Word > Bool) > Word > Bool
isPrimeF f n = n > 1 && and [ n `rem` d /= 0  d < [2 .. floor (sqrt (fromIntegral n))], f d]
Now create its memoized version for rapid evaluation:
isPrime' :: Word > Bool
isPrime' = memoizeFix isPrimeF
Example 3
No manual on memoization is complete
without Fibonacci numbers:
fibo :: Word > Integer
fibo = memoizeFix $ \f n > if n < 2 then toInteger n else f (n  1) + f (n  2)
No cleverness involved: just write a recursive function
and let memoizeFix
take care about everything else:
> fibo 100
354224848179261915075
What about nonWord
arguments?
Chimera
itself can memoize only Word > a
functions, which sounds restrictive.
That is because we decided to outsource
enumerating of user's datatypes to other packages, e. g.,
cantorpairing
.
Use fromInteger . fromCantor
to convert data to Word
and toCantor . toInteger
to go back.
Also, Data.Chimera.ContinuousMapping
covers several simple cases,
such as Int
, pairs and triples.
Benchmarks
How important is to store cached data as a flat array instead of a lazy binary tree?
Let us measure the maximal length of Collatz sequence,
using chimera
and memoize
packages.
{# LANGUAGE TypeApplications #}
import Data.Chimera
import Data.Function.Memoize
import Data.Ord
import Data.List
import Data.Time.Clock
collatzF :: Integral a => (a > a) > (a > a)
collatzF f n = if n <= 1 then 0 else 1 + f (if even n then n `quot` 2 else 3 * n + 1)
measure :: (Integral a, Show a) => String > (((a > a) > (a > a)) > (a > a)) > IO ()
measure name memo = do
t0 < getCurrentTime
print $ maximumBy (comparing (memo collatzF)) [0..1000000]
t1 < getCurrentTime
putStrLn $ name ++ " " ++ show (diffUTCTime t1 t0)
main :: IO ()
main = do
measure "chimera" Data.Chimera.memoizeFix
measure "memoize" (Data.Function.Memoize.memoFix @Int)
Here chimera
appears to be 20x faster than memoize
:
837799
chimera 0.428015s
837799
memoize 8.955953s
Magic and its exposure
Internally Chimera
is represented as a boxed vector
of growing (possibly, unboxed) vectors v a
:
newtype Chimera v a = Chimera (Data.Vector.Vector (v a))
Assuming 64bit architecture, the outer vector consists of 65 inner vectors
of sizes 1, 1, 2, 2², ..., 2⁶³. Since the outer vector
is boxed, inner vectors are allocated ondemand only: quite fortunately,
there is no need to allocate all 2⁶⁴ elements at once.
To access an element by its index it is enough to find out to which inner
vector it belongs, which, thanks to the doubling pattern of sizes,
can be done instantly by ffs
instruction. The caveat here is
that accessing an inner vector first time will cause its allocation,
taking O(n) time. So to restore amortized O(1) time we must assume
a dense access. Chimera
is no good for sparse access
over a thin set of indices.
One can argue that this structure is not infinite,
because it cannot handle more than 2⁶⁴ elements.
I believe that it is infinite enough and no one would be able to exhaust
its finiteness any time soon. Strictly speaking, to cope with indices out of
Word
range and memoize
Ackermann function,
one could use more layers of indirection, raising access time
to O(log ⃰ n).
I still think that it is morally correct to claim O(1) access,
because all asymptotic estimates of data structures
are usually made under an assumption that they contain
less than maxBound :: Word
elements
(otherwise you can not even treat pointers as a fixedsize data).
Additional resources