11

We are implementing a matrix compression library based on a modified two dimensional grammar syntax. Now we have two approaches for our data types -which one will be better in case of memory usage? (we want to compress something ;)).

The grammars contain NonTerminals with exactly 4 Productions or a Terminal on the righthand side. We will need the names of Productions for equality checks and grammar minimization.

The First:

-- | Type synonym for non-terminal symbols
type NonTerminal = String

-- | Data type for the right hand side of a production
data RightHandSide = DownStep NonTerminal NonTerminal NonTerminal NonTerminal | Terminal Int

-- | Data type for a set of productions
type ProductionMap = Map NonTerminal RightHandSide

data MatrixGrammar = MatrixGrammar {
    -- the start symbol
    startSymbol :: NonTerminal,
    -- productions
    productions :: ProductionMap    
    } 

Here our RightHandSide data saves only String names to determine the next productions, and what we do not know here is how Haskell saves these strings. For example the [[0, 0], [0, 0]] matrix has 2 productions:

a = Terminal 0
aString = "A"
b = DownStep aString aString aString aString
bString = "B"
productions = Map.FromList [(aString, a), (bString, b)]

So the question here is how often is the String "A" really saved? Once in aString, 4 times in b and once in productions or just once in aString and the others just hold "cheaper" references?

The Second:

data Production = NonTerminal String Production Production Production Production
                | Terminal String Int 

type ProductionMap = Map String Production

here the term "Terminal" is a bit misleading because its actually the production that has a terminal as right hand side. The same Matrix:

a = Terminal "A" 0
b = NonTerminal "B" a a a a
productions = Map.fromList [("A", a), ("B", b)]

and the similar question: how often is the production a saved internally by Haskell? Possibly we will drop the names inside the productions if we don't need them, but we are not sure right now about this.

So lets say we have a grammar with about 1000 productions. Which approach will consume less memory?

Finally a question about integers in Haskell: Currently we are planning on having name as Strings. But we could easily switch to integer names because with 1000 productions we will have names with more then 4 chars (which i assume is 32 bit?). How does Haskell handle this. Is an Int always 32 Bit and Integer allocates memory that it really needs?

I also read through this: Devising test of Haskell's value/reference semantics - but I can't figure out what that exactly means for us - I'm more of a imperative java child then good functional programmer :P

7

You can expand your matrix grammar into an ADT with perfect sharing with a little bit of trickery:

{-# LANGUAGE DeriveFunctor, DeriveFoldable, DeriveTraversable #-}

import Data.Map
import Data.Foldable
import Data.Functor
import Data.Traversable

-- | Type synonym for non-terminal symbols
type NonTerminal = String

-- | Data type for the right hand side of a production
data RHS a = DownStep NonTerminal NonTerminal NonTerminal NonTerminal | Terminal a
  deriving (Eq,Ord,Show,Read,Functor, Foldable, Traversable)

data G a = G NonTerminal (Map NonTerminal (RHS a))
  deriving (Eq,Ord,Show,Read,Functor)

data M a = Q (M a) (M a) (M a) (M a) | T a
  deriving (Functor, Foldable, Traversable)

tabulate :: G a -> M a
tabulate (G s pm) = loeb (expand <$> pm) ! s where
  expand (DownStep a11 a12 a21 a22) m = Q (m!a11) (m!a12) (m!a21) (m!a22)
  expand (Terminal a)               _ = T a

loeb :: Functor f => f (f b -> b) -> f b
loeb x = xs where xs = fmap ($xs) x

Here I generalized your grammars to allow for any data type, not just Int, and tabulate will take the grammar and expand it by folding it in upon itself using loeb.

loeb is described in an article by Dan Piponi

The resulting expansion as an ADT physically takes no more memory than the original grammar -- in fact it takes a fair bit less, because it doesn't need the extra log-factor for the Map spine, and it doesn't need to store the strings at all.

Unlike the naive expansion, using loeb lets me 'tie the knot' and share the thunks for all occurrences of the same non-terminal.

If you want to dip more into the theory of all of this, we can see that RHS could be turned into a base functor:

data RHS t nt = Q nt nt nt nt | L t

and then my M type is just the fixed point of that Functor.

M a ~ Mu (RHS a)

while G a would consist of a chosen string and a map from strings to (RHS String a).

We can then expand G into M by lookup up the entry in a map of expanded strings lazily.

This is sort of the dual of what is done in the data-reify package, which can take such a base functor, and something like M and recover the moral equivalent of your G from it. They use a different type for the non-terminal names, which is basically just an Int.

data Graph e = Graph [(Unique, e Unique)] Unique

and provide a combinator

reifyGraph :: MuRef s => s -> IO (Graph (DeRef s))

which can be used with an appropriate instance on the above data types to get a graph (MatrixGrammar) out of an arbitrary matrix. It won't do deduplication of identical but separately stored quadrants, but it'll recover all of the sharing that is present in the original graph.

8

In Haskell, the String type is an alias for [Char], which is a regular Haskell list of Char, not a vector or array. Char is a type that holds a single Unicode character. String literals are, unless you use a language extension, values of String type.

I think you can guess from the above that String is not a very compact or otherwise efficient representation. Common alternative representations for strings include the types supplied by Data.Text and Data.ByteString.

For extra convenience, you can use -XOverloadedStrings so that you can use string literals as representations of an alternative string type, such as provided by Data.ByteString.Char8. That's probably the most space-efficient way to conveniently use strings as identifiers.

As far as Int goes, it's a fixed-width type, but there is no guarantee about how wide it is except that it must be wide enough to hold the values [-2^29 .. 2^29-1]. This suggests it's at least 32 bits, but does not rule out being 64 bits. Data.Int has some more specific types, Int8-Int64, which you can use if you need a specific width.

Edit to add information

I don't believe the semantics of Haskell specify anything about data sharing either way. You shouldn't expect two String literals, or two of any constructed data, to refer to the same 'canonical' object in memory. If you were to bind a constructed value to a new name (with let, a pattern match, etc.) both names would most likely refer to the same data, but whether they do or not is not really visible due to the immutable nature of Haskell data.

For the sake of storage efficiency, you could intern the strings, which essentially stores a canonical representation of each in a lookup table of some sort, typically a hash table. When you intern an object, you get a descriptor for it back, and you can compare those descriptors with others to see if they're the same much more cheaply than you could strings, and they're also often much smaller.

For a library that does interning, you could use https://github.com/ekmett/intern/

As for deciding which integer size to use at run-time, it's fairly easy to write code that depends on Integral or Num type classes instead of concrete numeric types. Type inference will give you the most general types it can automatically. You could then have a few different functions with types explicitly narrowed to specific numeric types that you could choose one of at runtime to do the initial setup, and thereafter all of the other polymorphic functions would work the same on any of them. E.g.:

polyConstructor :: Integral a => a -> MyType a
int16Constructor :: Int16 -> MyType Int16
int32Constructor :: Int32 -> MyType Int32

int16Constructor = polyConstructor
int32Constructor = polyConstructor

Edit: More information about interning

If you only want to intern strings, you could create a new type that wraps a string (preferably a Text or ByteString) and a small integer together.

data InternedString = { id :: Int32, str :: Text }
instance Eq InternedString where
    {x, _ } == {y, _ }  =  x == y

intern :: MonadIO m => Text -> m InternedString

What 'intern' does is look up the string in a weak-reference HashMap where Texts are keys and InternedStrings are values. If a match is found, 'intern' returns the value. If not, it creates a new InternedString value with the original Text and a unique integer id (which is why I included the MonadIO constraint; it could use a State monad or some unsafe operation instead to get the unique id; there are many possibilities) and stores it in the map before returning it.

Now you get a fast compare based on the integer id and only have one copy of each unique string.

Edward Kmett's intern library applies the same principle, more or less, in a much more general way so that whole structured data terms are hashed, stored uniquely, and given a fast comparison operation. It is a bit daunting and not particularly documented, but he might be willing to help if you ask; or you could just try your own string interning implementation first to see if it helps enough.

  • Thanks for your answer so far. Is it possible to determine which int size we should use at runtime? I hope someone else can give some input on the problem with the copies :) – Dennis Ich Aug 8 '13 at 19:39
  • Thanks for the added information. I will take a look there. Just to get it right this descriptors u are speaking of are something like a reference that gets hashed and can be compared? Did you work with this your self? Can you maybe say how "more complicated" it gets with this because on the first look it seems i have to be very careful then with defining the grammars ;) – Dennis Ich Aug 8 '13 at 22:26
  • 1
    The author of that library is a very advanced Haskell user known for quality work, but I have not used that particular library. It is a very general-purpose "hash cons" implementation, which will store and allow for representation sharing in any constructed data type, not just strings. Look at his example directory for a problem kind of like yours, and you can see how the equality functions are implemented. – Levi Pearson Aug 9 '13 at 8:14

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