chimera

Lazy infinite compact streams with cache-friendly 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 re-evaluation.

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 cache-friendly 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 re-evaluation 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 non-Word 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., cantor-pairing. 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.

#!/usr/bin/env cabal
{- cabal:
build-depends: base, chimera, memoize, time
-}
{-# 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 64-bit 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 on-demand 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 fixed-size data).

Additional resources

• Lazy streams with O(1) access, London Haskell, 25.02.2020.