I thought I understood the basics of fold performance in Haskell, as described in foldr, foldl, foldl' on the Haskell Wiki and many other places. In particular, I learned that for accumulating functions, one should use foldl', to avoid space leaks, and that the standard library functions are written to respect this. So I presumed that simple accumulators like length, applied to simple lists like replicate n 1, should require constant space (or at least sub-linear) in the length of the list. My intuition was that on sufficiently simple lists, they would behave roughly like a for loop in an imperative language.
But today I found that this seems not to hold in practice. For instance, length $ replicate n 1 seems to use space linear in n. In ghci:
ghci> :set +s
ghci> length $ replicate (10^6) 1
1000000
(0.02 secs, 56,077,464 bytes)
ghci> length $ replicate (10^7) 1
10000000
(0.08 secs, 560,078,360 bytes)
ghci> length $ replicate (10^8) 1
100000000
(0.61 secs, 5,600,079,312 bytes)
ghci> length $ replicate (10^9) 1
1000000000
(5.88 secs, 56,000,080,192 bytes)
Briefly, my question is: Do length and other strict folds really use linear space? If so, why? And is it inevitable? Below are more details of how I’ve played around trying to understand this, but they’re probably not worth reading — the tl;dr is that the linear-space usage seems to persist whatever variations I try.
(I originally used sum as the example function. As Willem Van Onsem points out, that was a badly-chosen example as default instances aren’t actually strict. However, the main question remains, since as noted below, this occurs with plenty of other functions that really are based on strict folds.)
Replacing length with foldl' (\n _ -> n+1) 0 appears to make performance worse by a constant factor; space usage still seems to be linear.
Versions defined with foldl and foldr had worse memory usage (as expected), but only by a small constant factor, not asymptotically worse (as most discussions seem to suggest).
Replacing length with sum, last, or other simple accumulators, or with the obvious definitions of these using foldl', also doesn’t seem to change the linear space usage.
Using [1..n] as the test list, and other similar variations, also seems to make no significant difference.
Switching between the general versions of sum, foldl', etc from Data.Foldable, the specialised ones in Data.List, and local versions defined directly by pattern-matching, also seems to make no difference.
Compiling instead of working in ghci also only seemed to improve space usage by a constant factor.
Switching between several recent versions of GHC — 8.8.4, 8.10.5, and 9.0.1 — also seemed to make no significant difference.
"Do they use linear space" is a slightly unclear question. Usually when we talk about the space an algorithm uses, we're talking about its working set: the maximum amount of memory it needs all at once. "If my computer only had X bytes of memory, could I run this program?" But that's not what GHCI's :set +s measures. It measures the sum of all memory allocations made, including those that were cleaned up partway through. And what is the biggest use of memory in your experiment? The list itself, of course.
So you've really just measured the number of bytes that a list of size N takes up. You can confirm this by using last instead of length, which I hope you'll agree allocates no intermediate results, and is strict. It takes the same amount of memory using your metric as length does - length does no extra allocation for the sums.
But a bigger problem is that GHCI is not an optimizing compiler. If you care about performance characteristics at all, GHCI is the wrong tool. Instead, use GHC with -O2, and turn on GHC's profiler.
import System.Environment (getArgs)
main = do
n <- read . head <$> getArgs
print $ length (replicate (10^n) 1)
And running it:
$ ghc -O2 -prof -fprof-auto stackoverflow.hs
$ ./stackoverflow 6 +RTS -p
1000000
$ grep "total alloc" stackoverflow.prof
total alloc = 54,856 bytes (excludes profiling overheads)
$ ./stackoverflow 9 +RTS -p
1000000000
$ grep "total alloc" stackoverflow.prof
total alloc = 55,008 bytes (excludes profiling overheads)
we can see that space usage is roughly constant despite a thousand-fold increase in input size.
Will Ness correctly points out in a comment that -s would be a better measuring tool than -p.
Replacing sum with foldl' (+) 0 here, then performance improves noticeably in both time and space (which is itself a surprise; shouldn’t the standard sum be at least as efficient?) — but only by a constant factor; space usage still seems to be linear.
The sum is implemented as [src]:
sum :: Num a => t a -> a
sum = getSum #. foldMap Sum
It thus makes use of the Sum data type and its Monoid instance such that mappend = (+) and mempty = 0. foldMap works right associative, indeed:
Map each element of the structure into a monoid, and combine the results with (<>). This fold is right-associative and lazy in the accumulator. For strict left-associative folds consider foldMap' instead.
foldMap is thus implemented with foldr [src]:
foldMap :: Monoid m => (a -> m) -> t a -> m
{-# INLINE foldMap #-}
-- This INLINE allows more list functions to fuse. See #9848.
foldMap f = foldr (mappend . f) mempty
While foldl' will indeed have a (much) smaller memory footprint, and likely be more efficient, a reason to work with foldr is that for Peano numbers for example, one can make use of lazyness, and thus the head normal form will look like S(…) where … might not be evaluated (yet).
foldr can also terminate earlier. If for example you make a sum for a certain algebraic structure, it is possible that we can terminate the looping earlier.
Related
I have a program which traverses an expression tree that does algebra on probability distributions, either sampling or computing the resulting distribution.
I have two implementations computing the distribution: one (computeDistribution) nicely reusable with monad transformers and one (simpleDistribution) where I concretize everything by hand. I would like to not concretize everything by hand, since that would be code duplication between the sampling and computing code.
I also have two data representations:
type Measure a = [(a, Rational)]
-- data Distribution a = Distribution (Measure a) deriving Show
newtype Distribution a = Distribution (Measure a) deriving Show
When I use the data version with the reusable code, computing the distribution of 20d2 (ghc -O3 program.hs; time ./program 20 > /dev/null) takes about one second, which seems way too long. Pick higher values of n at your own peril.
When I use the hand-concretized code, or I use the newtype representation with either implementation, computing 20d2 (time ./program 20 s > /dev/null) takes the blink of an eye.
Why?
How can I find out why?
My knowledge of how Haskell is executed is almost nil. I gather there's a graph of thunks in basically the same shape as the program, but that's about all I know.
I figure with newtype the representation of Distribution is the same as that of Measure, i.e. it's just a list, whereas with the data version each Distribution is kinda' like a single-field record, except with a pointer to the contained list, and so the data version has to perform more allocations. Is this true? If true, is this enough to explain the performance difference?
I'm new to working with monad transformer stacks. Consider the Let and Uniform cases in simpleDistribution — do they do the same as the walkTree-based implementation? How do I tell?
Here's my program. Note that Uniform n corresponds to rolling an n-sided die (in case the unary-ness was surprising).
Update: based on comments I simplified my program by removing everything not contributing to the performance gap. I made two semantic changes: probabilities are now denormalized and all wonky and wrong, and the simplification step is gone. But the essential shape of my program is still there. (See question edit history for the non-simplified program.)
Update 2: I made further simplifications, reducing Distribution down to the list monad with a small twist, removing everything to do with probabilities, and shortening the names. I still observe large performance differences when using data but not newtype.
import Control.Monad (liftM2)
import Control.Monad.Trans (lift)
import Control.Monad.Reader (ReaderT, runReaderT)
import System.Environment (getArgs)
import Text.Read (readMaybe)
main = do
args <- getArgs
let dieCount = case map readMaybe args of Just n : _ -> n; _ -> 10
let f = if ["s"] == (take 1 $ drop 1 $ args) then fast else slow
print $ f dieCount
fast, slow :: Int -> P Integer
fast n = walkTree n
slow n = walkTree n `runReaderT` ()
walkTree 0 = uniform
walkTree n = liftM2 (+) (walkTree 0) (walkTree $ n - 1)
data P a = P [a] deriving Show
-- newtype P a = P [a] deriving Show
class Monad m => MonadP m where uniform :: m Integer
instance MonadP P where uniform = P [1, 1]
instance MonadP p => MonadP (ReaderT env p) where uniform = lift uniform
instance Functor P where fmap f (P pxs) = P $ fmap f pxs
instance Applicative P where
pure x = P [x]
(P pfs) <*> (P pxs) = P $ pfs <*> pxs
instance Monad P where
(P pxs) >>= f = P $ do
x <- pxs
case f x of P fxs -> fxs
How can I find out why?
This is, in general, hard.
The extreme way to do it is to look at the core code (which you can produce by running GHC with -ddump-simpl). This can get complicated really quickly, and it's basically a whole new language to learn. Your program is already big enough that I had trouble learning much from the core dump.
The other way to find out why is to just keep using GHC and asking questions and learning about GHC optimizations until you recognize certain patterns.
Why?
In short, I believe it's due to list fusion.
NOTE: I don't know for sure that this answer is correct, and it would take more time/work to verify than I'm willing to put in right now. That said, it fits the evidence.
First off, we can check whether this slowdown you're seeing is a result of something truly fundamental vs a GHC optimization triggering or not by running in O0, that is, without optimizations. In this mode, both Distribution representations result in about the same (excruciatingly long) runtime. This leads me to believe that it's not the data representation that is inherently the problem but rather there's an optimization that's triggered with the newtype version that isn't with the data version.
When GHC is run in -O1 or higher, it engages certain rewrite rules to fuse different folds and maps of lists together so that it doesn't need to allocate intermediate values. (See https://markkarpov.com/tutorial/ghc-optimization-and-fusion.html#fusion for a decent tutorial on this concept as well as https://stackoverflow.com/a/38910170/14802384 which additionally has a link to a gist with all of the rewrite rules in base.) Since computeDistribution is basically just a bunch of list manipulations (which are all essentially folds), there is the potential for these to fire.
The key is that with the newtype representation of Distribution, the newtype wrapper is erased during compilation, and the list operations are allowed to fuse. However, with the data representation, the wrappers are not erased, and the rewrite rules do not fire.
Therefore, I will make an unsubstantiated claim: If you want your data representation to be as fast as the newtype one, you will need to set up rewrite rules similar to the ones for list folding but that work over the Distribution type. This may involve writing your own special fold functions and then rewriting your Functor/Applicative/Monad instances to use them.
I got a Stack_overflow error in my OCaml program lately. If I turn on backtracing, I see the exception is raised by a "primitive operation" "pervasives.ml", line 270. I went into the OCaml source code and saw that line 270 defines the function # (i.e. list append). I don't get any other information from the backtrace, not even where the exception gets thrown in my program. I switched to bytecode and tried ocamldebug, and it doesn't help (no backtrace generated).
I thought this is an extremely weird situation. The only places in my program where I used a list is (a) building a list containing integers 1 to 1000000, (b) in-order traversing a RBT and putting the result into a list, and (c) printing a list of integers containing ostensibly 1000000 numbers. I've tested all functions and none of them contain could an infinite loop, and I thought 1000000 isn't even a huge number. Moreover, I've tried the equivalent of my program in Haskell (GHC), Scala and SML (MLton), and all of those versions worked perfectly and in a reasonably short amount of time. So, the question is, what could be going on? Can I debug it?
The # operator is not tail-recursive in the OCaml standard library,
let rec ( # ) l1 l2 =
match l1 with
[] -> l2
| hd :: tl -> hd :: (tl # l2)
Thus calling it with large lists (as the left argument) will overflow your stack.
It could be possible, that you're building your list by appending a new element to the end of the already generated list, e.g.,
let rec init n x = if n > 0 then init (n-1) x # [x] else []
This has time complexity n^2 and will consume n slots in the stack space.
Concerning the general question - how to debug such stack overflows, my usual recipe is to reduce the stack size, so that the problem is triggered as soon as possible before the trace is bloated, e.g.,
OCAMLRUNPARAM=b,l=1024 ocaml ./test.ml
If you're compiling your OCaml code to the native code, then you need to pass the -g option to the compiler, so that it can produce backtraces. Also, in the native execution, the size of the stack is controlled by the operating system and should be set using the corresponding mechanism of your OS, for example with ulimit in GNU/Linux, e.g., ulimit -s 1024.
As a bonus track, the following init function is tail recursive and will have O(N) time complexity and will take O(1) stack space:
let init n x =
let rec loop n xs =
if n = 0 then xs else loop (n-1) (x :: xs) in
loop n []
The idea is to use an accumulator list and build the list in the heap space.
If you don't like thinking about tail-recursiveness then you can use Janestreet Base library (or Core), or Batteries library. They both provide tail-recursive versions of the init function, as well as guarantees that all other functions are tail-recursive.
List functions in the standard library are optimised for small lists and are not necessarily tail-recursive; with the partial justification that lists are not an efficient data structure for storing large amount of data (note that Haskell lists are lazy and thus are quite different than OCaml eager lists).
In particular, if you get a stackoverflow error using #, you are quite probably implementing an algorithm with a quadratic time-complexity due to the fact that #'s complexity is linear in the size of its left argument.
They are probably far better data structure than list for your problem, if you want iteration the sequence library or any other forms of iterator would be far more efficient for instance.
With all the caveat stated before, it is relatively straightforward to redefine tail-recursive but inefficient version of the standard library function, e.g. :
let (#!) x y = List.rev_append (List.rev x) y
Another option is to use the containers library or any of the extended standard libraries (batteries or base essentially): all of those libraries reimplement tail-recursive version of list functions.
Closed. This question needs to be more focused. It is not currently accepting answers.
Want to improve this question? Update the question so it focuses on one problem only by editing this post.
Closed 8 years ago.
Improve this question
Lazy evaluation is said to be a way of delaying a process until the first time it is needed. This tends to avoid repeated evaluations and thats why I would imagine that is performing a lot faster.
Functional language like Haskell (and JavaScript..?) have this functionality built-in.
However, I don't understand how and why other 'normal' approaches (that is; same functionality but not using lazy evaluation) are slower.. how and why do these other approaches do repeated evaluations? Can someone elaborate on this by giving simple examples and explaining the mechanics of each approach?
Also, according to Wikipedia page about lazy evaluation these are said to be the advantages of this approach:
Performance increases by avoiding needless calculations, and error
conditions in evaluating compound expressions
The ability to construct potentially infinite data structures
The ability to define control flow (structures) as abstractions
instead of primitives
However, can we just control the calculations needed and avoid repeating the same ones? (1)
We can use i.e. a Linked List to create an infinite data structure (2)
Can we do (3) already..??? We can define classes/templates/objects and use those instead of primitives (i.e JavaScript).
Additionally, it seems to me that (at least from the cases i have seen), lazy evaluation goes hand-to-hand with recursion and using the 'head' and 'tail' (along with others) notions. Surely, there are cases where recursion is useful but is lazy evaluation something more than that...? more than a recursive approach to solving a problem..? Streamjs is JavaScript library that uses recursion along with some other simple operations (head,tail,etc) to perform lazy evaluation.
It seems i can't get my head around it...
Thanks in advance for any contribution.
I'll show examples in both Python 2.7 and Haskell.
Say, for example, you wanted to do a really inefficient sum of all the numbers from 0 to 10,000,000. You could do this with a for loop in Python as
total = 0
for i in range(10000000):
total += i
print total
On my computer, this takes about 1.3s to execute. If instead, I changed range to xrange (the generator form of range, lazily produces a sequence of numbers), it takes 1.2s, only slightly faster. However, if I check the memory used (using the memory_profiler package), the version with range uses about 155MB of RAM, while the xrange version uses only 1MB of RAM (both numbers not including the ~11MB Python uses). This is an incredibly dramatic difference, and we can see where it comes from with this tool as well:
Mem usage Increment Line Contents
===========================================
10.875 MiB 0.004 MiB total = 0
165.926 MiB 155.051 MiB for i in range(10000000):
165.926 MiB 0.000 MiB total += i
return total
This says that before we started we were using 10.875MB, total = 0 added 0.004MB, and then for i in range(10000000): added 155.051MB when it generated the entire list of numbers [0..9999999]. If we compare to the xrange version:
Mem usage Increment Line Contents
===========================================
11.000 MiB 0.004 MiB total = 0
11.109 MiB 0.109 MiB for i in xrange(10000000):
11.109 MiB 0.000 MiB total += i
return total
So we started with 11MB and for i in xrange(10000000): added only 0.109MB. This is a huge memory savings by only adding a single letter to the code. While this example is fairly contrived, it shows how not computing a whole list until the element is needed can make things a lot more memory efficient.
Python has iterators and generators which act as a sort of "lazy" programming for when you need to yield sequences of data (although there's nothing stopping you from using them for single values), but Haskell has laziness built into every value in the language, even user-defined ones. This lets you take advantage of things like data structures that won't fit in memory without having to program complicated ways around that fact. The canonical example would be the fibonacci sequence:
fibs = 1 : 1 : zipWith (+) fibs (tail fibs)
which very elegantly expresses this famous sequence to define a recursive infinite list generating all fibonacci numbers. It's CPU efficient because all values are cached, so each element only has to be computed once (compared to a naive recursive implementation)1, but if you calculate too many elements your computer will eventually run out of RAM because you're now storing this huge list of numbers. This is an example where lazy programming lets you have CPU efficiency, but not RAM efficiency. There is a way around this, though. If you were to write
fib :: Int -> Integer
fib n = let fibs = 1 : 1 : zipWith (+) fibs (tail fibs) in fibs !! n
then this runs in near-constant memory, and does so very quickly, but memoization is lost as subsequent calls to fib have to recompute fibs.
A more complex example can be found here, where the author shows how to use lazy programming and recursion in Haskell to perform dynamic programming with arrays, a feat that most initially think is very difficult and requires mutation, but Haskell manages to do very easily with "tying the knot" style recursion. It results in both CPU and RAM efficiency, and does so in fewer lines than I'd expect in C/C++.
All this being said, there are plenty of cases where lazy programming is annoying. Often you can build up huge numbers of thunks instead of computing things as you go (I'm looking at you, foldl), and some strictness has to be introduced to attain efficiency. It also bites a lot of people with IO, when you read a file to a string as a thunk, close the file, and then try to operate on that string. It's only after the file is closed that the thunk gets evaluated, causing an IO error to occur and crashes your program. As with anything, lazy programming is not without its flaws, gotchas, and pitfalls. It takes time to learn how to work with it well, and to know what its limitations are.
1) By "naive recursive implementation", I mean implementing the fibonacci sequence as
fib :: Integer -> Integer
fib 0 = 1
fib 1 = 1
fib n = fib (n-1) + fib (n-2)
With this implementation, you can see the mathematical definition very clearly, it's very much in the style of inductive proofs, you show your base cases and then the general case. However, if I call fib 5, this will "expand" into something like
fib 5 = fib 4 + fib 3
= fib 3 + fib 2 + fib 2 + fib 1
= fib 2 + fib 1 + fib 1 + fib 0 + fib 1 + fib 0 + fib 1
= fib 1 + fib 0 + fib 1 + fib 1 + fib 0 + fib 1 + fib 0 + fib 1
= 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
= 8
When instead we'd like to share some of those computations, that way fib 3 only gets computed once, fib 2 only gets computed once, etc.
By using a recursively defined list in Haskell, we can avoid this. Internally, this list is represented something like this:
fibs = 1 : 1 : zipWith (+) fibs (tail fibs)
= 1 : 1 : zipWith (+) (f1:f2:fs) (f2:fs)
^--------------------^ ^ ^
^-------------------|-------|
= 1 : 1 : 2 : zipWith (+) (f2:f3:fs) (f3:fs)
^--------------------^ ^ ^
^-------------------|-------|
= 1 : 1 : 2 : 3 : zipWith (+) (f3:f4:fs) (f4:fs)
^--------------------^ ^ ^
^-------------------|-------|
So hopefully you can see the pattern forming here, as the list is build, it keeps pointers back to the last two elements generated in order to compute the next element. This means that for the nth element computed, there are n-2 additions performed. Even for the naive fib 5, you can see that there are more additions performed than that, and the number of additions will continue to grow exponentially. This definition is made possible through laziness and recursions, letting us turn an O(2^n) algorithm into an O(n) algorithm, but we have to give up RAM to do so. If this is defined at the top level, then values are cached for the lifetime of the program. It does mean that if you need to refer to the 1000th element repeatedly, you don't have to recompute it, just index it.
On the other hand, the definition
fib :: Int -> Integer
fib n =
let fibs = 1 : 1 : zipWith (+) fibs (tail fibs)
in fibs !! n
uses a local copy of fibs every time fib is called. We don't get caching between calls to fib, but we do get local caching, leaving our complexity O(n). Additionally, GHC is smart enough to know that we don't have to keep the beginning of the list around after we've used it to calculate the next element, so as we traverse fibs looking for the nth element, it only needs to hold on to 2-3 elements and a thunk pointing at the next element. This saves us RAM while computing it, and since it isn't defined at a global level it doesn't eat up RAM over the lifetime of the program. It's a tradeoff between when we want to spend RAM and CPU cycles, and different approaches are better for different situations. These techniques are applicable to much of Haskell programming in general, not just for this sequence!
Lazy evaluation is not, in general, faster. When it's said that lazy evaluation is more efficient, it is because when you consider Lambda Calculus (which is essentially what your Haskell programs are once the compiler finishes de-sugaring them) as a system of terms and reduction rules, then applying those rules in the order specified by the rules of a call-by-name with sharing evaluation policy always applies the same or fewer reduction rules than when you follow the rules in the order specified by call-by-value evaluation.
The reason that this theoretical result does not make lazy evaluation faster in general is that the translation to a linear sequential machine model with a memory access bottleneck tends to make all the reductions performed much more expensive! Initial attempts at implementing this model on computers led to programs that executed orders of magnitude more slowly than typical eagerly-evaluating language implementations. It has taken a lot of research and engineering into techniques for implementing lazy evaluation efficiently to get Haskell performance to where it is today. And the fastest Haskell programs take advantage of a form of static analysis called "strictness analysis" which attempts to determine at compile time which expressions will always be needed so that they can be evaluated eagerly rather than lazily.
There are still some cases where straightforward implementations of algorithms will execute faster in Haskell due to only evaluating terms that are needed for the result, but even eager languages always have some facility for evaluating some expressions by need. Conditionals and short-circuiting boolean expressions are ubiquitous examples, and in many eager languages, one can also delay evaluation by wrapping an expression in an anonymous function or some other sort of delaying form. So you can typically use these mechanisms (or even more awkward rewrites) to avoid evaluating expensive things that won't be necessary in an eager language.
The real advantage of Haskell's lazy evaluation is not a performance-related one. Haskell makes it easier to pull expressions apart, re-combine them in different ways, and generally reason about code as if it were a system of mathematical equations instead of being a sequentially-evaluated set of machine instructions. By not specifying any evaluation order, it forced the developers of the language to avoid side-effects that rely on a simple evaluation ordering, such as mutation or IO. This in turn led to a host of elegant abstractions that are generally useful and might not have been developed into usability otherwise.
The state of Haskell is now such that you can write high-level, elegant algorithms that make better re-use of existing higher-order functions and data structures than in nearly any other high-level typed language. And once you become familiar with the costs and benefits of lazy evaluation and how to control when it occurs, you can ensure that the elegant code also performs very well. But getting the elegant code to a state of high performance is not necessarily automatic and may require a bit more thought than in a similar but eagerly-evaluated language.
The concept of "lazy evaluation" is only about 1 thing, and only about that 1 thing:
The ability to postpone evaluation of something until needed
That's it.
Everything else in that wikipedia article follows from it.
Infinite data structures? Not a problem. We'll just make sure we don't actually figure out what the next element is until you actually ask for it. For instance, asking some code what the next value after X is, if the operation to perform is just to increase X by 1, will be infite. If you create a list containing all those values, it's going to fill your available memory in the computer. If you only figure out what the next value is when asked, not so much.
Needless calculations? Sure. You can return an object containing a lot of properties that when asked will provide you with some value. If you don't ask (ie. never inspect the value of a given property), the calculation necessary to figure out the value of that property will never be done.
Control flow ... ? Not at all sure what that is about.
The purpose of lazy evaluation of something is exactly as I stated to begin with, to avoid evaluating something until you actually need it. Be it the next value of something, the value of a property, whatever, adding support for lazy evaluation might conserve CPU cycles.
What would the alternative be?
I want to return an object to the calling code, containing any number of properties, some of which might be expensive to calculate. Without lazy evaluation, I would have to calculate the values of all those properties either:
Before constructing the object
After constructing the object, on the first time you inspected a property
After constructing the object, every time you inspected that property
With lazy evaluation you usually end up with number 2. You postpone evaluating the value of that property until some code inspects it. Note that you might cache the value once evaluated, which would save CPU cycles when inspecting the same property more than once, but that is caching, not quite the same, but in the same line of work: optimizations.
I can't figure out why m1 is apparently memoized while m2 is not in the following:
m1 = ((filter odd [1..]) !!)
m2 n = ((filter odd [1..]) !! n)
m1 10000000 takes about 1.5 seconds on the first call, and a fraction of that on subsequent calls (presumably it caches the list), whereas m2 10000000 always takes the same amount of time (rebuilding the list with each call). Any idea what's going on? Are there any rules of thumb as to if and when GHC will memoize a function? Thanks.
GHC does not memoize functions.
It does, however, compute any given expression in the code at most once per time that its surrounding lambda-expression is entered, or at most once ever if it is at top level. Determining where the lambda-expressions are can be a little tricky when you use syntactic sugar like in your example, so let's convert these to equivalent desugared syntax:
m1' = (!!) (filter odd [1..]) -- NB: See below!
m2' = \n -> (!!) (filter odd [1..]) n
(Note: The Haskell 98 report actually describes a left operator section like (a %) as equivalent to \b -> (%) a b, but GHC desugars it to (%) a. These are technically different because they can be distinguished by seq. I think I might have submitted a GHC Trac ticket about this.)
Given this, you can see that in m1', the expression filter odd [1..] is not contained in any lambda-expression, so it will only be computed once per run of your program, while in m2', filter odd [1..] will be computed each time the lambda-expression is entered, i.e., on each call of m2'. That explains the difference in timing you are seeing.
Actually, some versions of GHC, with certain optimization options, will share more values than the above description indicates. This can be problematic in some situations. For example, consider the function
f = \x -> let y = [1..30000000] in foldl' (+) 0 (y ++ [x])
GHC might notice that y does not depend on x and rewrite the function to
f = let y = [1..30000000] in \x -> foldl' (+) 0 (y ++ [x])
In this case, the new version is much less efficient because it will have to read about 1 GB from memory where y is stored, while the original version would run in constant space and fit in the processor's cache. In fact, under GHC 6.12.1, the function f is almost twice as fast when compiled without optimizations than it is compiled with -O2.
m1 is computed only once because it is a Constant Applicative Form, while m2 is not a CAF, and so is computed for each evaluation.
See the GHC wiki on CAFs: http://www.haskell.org/haskellwiki/Constant_applicative_form
There is a crucial difference between the two forms: the monomorphism restriction applies to m1 but not m2, because m2 has explicitly given arguments. So m2's type is general but m1's is specific. The types they are assigned are:
m1 :: Int -> Integer
m2 :: (Integral a) => Int -> a
Most Haskell compilers and interpreters (all of them that I know of actually) do not memoize polymorphic structures, so m2's internal list is recreated every time it's called, where m1's is not.
I'm not sure, because I'm quite new to Haskell myself, but it appears that it's beacuse the second function is parametrized and the first one is not. The nature of the function is that, it's result depends on input value and in functional paradigm especailly it depends ONLY on the input. Obvious implication is that a function with no parameters returns always the same value over and over, no matter what.
Aparently there's an optimizing mechanizm in GHC compiler that exploits this fact to compute the value of such a function only once for whole program runtime. It does it lazily, to be sure, but does it nonetheless. I noticed it myself, when I wrote the following function:
primes = filter isPrime [2..]
where isPrime n = null [factor | factor <- [2..n-1], factor `divides` n]
where f `divides` n = (n `mod` f) == 0
Then to test it, I entered GHCI and wrote: primes !! 1000. It took a few seconds, but finally I got the answer: 7927. Then I called primes !! 1001 and got the answer instantly. Similarly in an instant I got the result for take 1000 primes, because Haskell had to compute the whole thousand-element list to return 1001st element before.
Thus if you can write your function such that it takes no parameters, you probably want it. ;)
I was trying to solve ITA Software's "Word Nubmers" puzzle using a brute force approach. It looks like my Haskell version is more than 10 times slower than a C#/C++ version.
The answer
Thanks to Bryan O'Sullivan's answer, I was able to "correct" my program to acceptable performance. You can read his code which is much cleaner than mine. I am going to outline the key points here.
Int is Int64 on Linux GHC x64. Unless you unsafeCoerce, you should just use Int. This saves you from having to fromIntegral. Doing Int64 on Windows 32-bit GHC is just darn slow, avoid it. (This is in fact not GHC's fault. As mentioned in my blog post below, 64 bit integers in 32-bit programs is slow in general (at least in Windows))
-fllvm or -fvia-C for performance.
Prefer quotRem to divMod, quotRem already suffices. That gave me 20% speed up.
In general, prefer Data.Vector to Data.Array as an "array"
Use the wrapper-worker pattern liberally.
The above points were enough to give me about 100% boost over my original version.
In my blog post, I have detailed a step-by-step illustrated example of how I turned the original program to match Bryan's program. There are other points mentioned there as well.
The original question
(This may sound like a "could you do the work for me" post, but I argue that such a concrete example would be very instructive since profiling Haskell performance is often seen as a myth)
(As noted in the comments, I think I have misinterpreted the problem. But who cares, we can focus on performance in a different problem)
Here's a my version of a quick recap of the problem:
A wordNumber is defined as
wordNumber 1 = "one"
wordNumber 2 = "onetwo"
wordNumber 3 = "onethree"
wordNumber 15 = "onetwothreefourfivesixseveneightnineteneleventwelvethirteenfourteenfifteen"
...
Problem: Find the 51-billion-th letter of (wordNumber Infinity); assume that letter is found at 'wordNumber x', also find 'sum [1..x]'
From an imperative perspective, a naive algorithm would be to have 2 counters, one for sum of numbers and one for sum of lengths. Keep counting the length of each wordNumber and "break" to return the result.
The imperative brute-force approach is implemented in C# here: http://ideone.com/JjCb3. It takes about 1.5 minutes to find the answer on my computer. There is also an C++ implementation that runs in 45 seconds on my computer.
Then I implemented a brute-force Haskell version: http://ideone.com/ngfFq. It cannot finish the calculation in 5 minutes on my machine. (Irony: it's has more lines than the C# version)
Here is the -p profile of the Haskell program: http://hpaste.org/49934
Question: How to make it perform comparatively to the C# version? Are there obvious mistakes I am making?
(Note: I am fully aware that brute-forcing it is not the correct solution to this problem. I am mainly interested in making the Haskell version perform comparatively to the C# version. Right now it is at least 5x slower so obviously I am missing something obvious)
(Note 2: It does not seem to be space leaking. The program runs with constant memory (about 2MB) on my computer)
(Note 3: I am compiling with `ghc -O2 WordNumber.hs)
To make the question more reader friendly, I include the "gist" of the two versions.
// C#
long sumNum = 0;
long sumLen = 0;
long target = 51000000000;
long i = 1;
for (; i < 999999999; i++)
{
// WordiLength(1) = 3 "one"
// WordiLength(101) = 13 "onehundredone"
long newLength = sumLen + WordiLength(i);
if (newLength >= target)
break;
sumNum += i;
sumLen = newLength;
}
Console.WriteLine(Wordify(i)[Convert.ToInt32(target - sumLen - 1)]);
-
-- Haskell
-- This has become totally ugly during my squeeze for
-- performance
-- Tail recursive
-- n-th number (51000000000 in our problem) -> accumulated result -> list of 'zipped' left to try
-- accumulated has the format (sum of numbers, current lengths of the whole chain, the current number)
solve :: Int64 -> (Int64, Int64, Int64) -> [(Int64, Int64)] -> (Int64, Int64, Int64)
solve !n !acc#(!sumNum, !sumLen, !curr) ((!num, !len):xs)
| sumLen' >= n = (sumNum', sumLen, num)
| otherwise = solve n (sumNum', sumLen', num) xs
where
sumNum' = sumNum + num
sumLen' = sumLen + len
-- wordLength 1 = 3 "one"
-- wordLength 101 = 13 "onehundredone"
wordLength :: Int64 -> Int64
-- wordLength = ...
solution :: Int64 -> (Int64, Char)
solution !x =
let (sumNum, sumLen, n) = solve x (0,0,1) (map (\n -> (n, wordLength n)) [1..])
in (sumNum, (wordify n) !! (fromIntegral $ x - sumLen - 1))
I've written a gist that contains both a C++ version (a copy of yours from a Haskell-cafe message, with a bug fixed) and a Haskell translation.
Notice that the two are structurally almost identical. When compiled with -fllvm, the Haskell code runs at about half the speed of the C++ code, which is pretty good.
Now let's compare my Haskell wordLength code to yours. You're passing around an extra unnecessary parameter, which is unnecessary (you apparently figured that out when writing the C++ code that I translated). Also, the large number of bang patterns suggests panic; they're almost all useless.
Your solve function is also very confused.
You're passing parameters in three different ways: a regular Int, a 3-tuple, and a list! Whoa.
This function is necessarily not very regular in its behaviour, so while you gain nothing stylistically by using a list to supply your counter, you probably force GHC to allocate memory. In other words, this both obfuscates the code and makes it slower.
By using a tuple for three parameters (for no obvious reason), you're again working hard to force GHC to allocate memory for every step through the loop, when it could avoid doing so if you passed the parameters directly.
Only your n parameter is dealt with in a sensible way, but you don't need a bang pattern on it.
The only parameter that needs a bang pattern is sumNum, because you never inspect its value until after the loop has finished. GHC's strictness analyser will deal with the others. All of your other bang patterns are unnecessary at best, misdirections at worst.
Here are two pointers I could come up with in a quick investigation:
Note that using Int64 is really slow when you are using a 32 bit build of GHC, as is the default for Haskell Platform, currently. This also turned out to be the main villain in a previous performance problem (there I give a few more details).
For reasons I don't quite understand the divMod function does not seem to get inlined. As a result, the numbers are returned on the heap. When using div and mod separately, wordLength' executes purely on the stack as it should be.
Sadly I currently have no 64-bit GHC around to test whether this is enough to solve the problem.