Memoized Collatz sequence

Question

Here is one of my programs that utilized memoization and array to improve performance and memory usage. The performance seems satisfactory but the memory usage is ridiculous and I can't figure out what's wrong:

{-# LANGUAGE BangPatterns #-}
import Data.Functor
import Data.Array (Array)
import qualified Data.Array as Arr
import Control.DeepSeq

genColtzArr n = collatzArr
    where collatzArr = Arr.array (1, n) $ take n $ map (\v -> (v, collatz v 0)) [1..] 
          collatz 1 !acc  = 1 + acc
          collatz !m !acc
              | even m    = go (m `div` 2) acc
              | otherwise = go (3 * m + 1) acc
              where go !l !acc
                      | l <= n    = let !v = collatzArr Arr.! l in 1 + acc + v
                      | otherwise = collatz l $ 1 + acc

collatz here means this guy. This function is supposed to receive a number n, and then return an array indexing from 1 to n, and in which each cell contains the length of the link from the index to 1 by applying Collatz formula.

But the memory usage of this method is so high. Here is the profiler result (ghc option -prof -fprof-auto -rtsopts, run time option +RTS -p, n == 500000):

total alloc = 730,636,136 bytes  (excludes profiling overheads)

COST CENTRE              MODULE  %time %alloc

genColtzArr.collatz      Main     40.4   34.7
genColtzArr.collatz.go   Main     25.5   14.4


COST CENTRE                      MODULE                    no.     entries  %time %alloc   %time %alloc     

      genColtzArr                Main                      105           1    0.0    0.0    74.7   72.1
       genColtzArr.collatzArr    Main                      106           1    8.0   20.8    74.7   72.1
        genColtzArr.collatzArr.\ Main                      107      500000    0.9    2.2    66.8   51.3
         genColtzArr.collatz     Main                      109     1182582   40.4   34.7    65.9   49.1
          genColtzArr.collatz.go Main                      110     1182581   25.5   14.4    25.5   14.4

Please note that -O2 is not a desired answer. I want to figure out what's the problem in this program and in general, how should I spot time and memory inefficiencies in Haskell code. Specifically, I have no idea why this code, with tail recursion and bang pattern, can consume so much memory.

UPDATE1:

the same code with -s produces this:

   1,347,869,264 bytes allocated in the heap
     595,901,528 bytes copied during GC
     172,105,056 bytes maximum residency (7 sample(s))
         897,704 bytes maximum slop
             315 MB total memory in use (0 MB lost due to fragmentation)

                                     Tot time (elapsed)  Avg pause  Max pause
  Gen  0      2408 colls,     0 par    0.412s   0.427s     0.0002s    0.0075s
  Gen  1         7 colls,     0 par    0.440s   0.531s     0.0759s    0.1835s

  INIT    time    0.000s  (  0.000s elapsed)
  MUT     time    0.828s  (  0.816s elapsed)
  GC      time    0.852s  (  0.958s elapsed)
  RP      time    0.000s  (  0.000s elapsed)
  PROF    time    0.000s  (  0.000s elapsed)
  EXIT    time    0.004s  (  0.017s elapsed)
  Total   time    1.684s  (  1.791s elapsed)

  %GC     time      50.6%  (53.5% elapsed)

  Alloc rate    1,627,861,429 bytes per MUT second

  Productivity  49.4% of total user, 46.4% of total elapsed

so it takes 300 meg. that is still too large.

Answer 1

It doesn't use 300 megabytes of heap, it peaks at a little over 20 megabytes. Total allocation is not peak allocation and Haskell has cheap short-lived allocations so total alloc isn't always a good heuristic for GC time or steady-state heap usage. The heap profiling stuff is giving data designed for tuning code rather than for analytics and total alloc is often more helpful when comparing code for overall memory usage.

Here's a screenshot from a memory profile:

I don't think it necessarily bit you here, but in future, add explicit types to functions you are benchmarking. See here for an example of why you want to do that.

Here's the command I wrote to profile your code:

$ rm -f collatz
$ stack ghc -- -prof -fprof-auto -rtsopts -O2 collatz.hs -o collatz 
$ ./collatz +RTS -hc -p
$ hp2ps collatz.hp
$ evince collatz.ps

Last bits are just converting the hp heap profiling data to a postscript file and then I'm opening it in my PDF reader.

Answer 2

Here is one of my programs that utilized memoization and array to improve performance and memory usage. The performance seems satisfactory but the memory usage is ridiculous and I can't figure out what's wrong:

I don't actually know the answer, but I'd guess that you used memoization improperly. GHC already evaluates a lambda expression at most once per top-level function.

The bang patterns there are almost certainly the reason that GHC is taking up too much memory. Haskell is a lazy language; take advantage of this. Indeed, using strictness in combination with immutability (and yes, arrays are unfortunately immutable here) is probably just not a good idea.

Here is an (admittedly advanced) example using recursion-schemes:

import Data.Functor.Foldable

coalgebra :: Int -> Either [Int] (ListF Int Int)
coalgebra 1 = Left [1]
coalgebra n
    | n `mod` 2 == 0 = Right $ Cons n (div n 2)
    | otherwise = Right $ Cons n (3 * n + 1)

collatz :: Int -> [Int]
collatz = elgot embed coalgebra

In general, while such an aggressively functional style will not always be the fastest, it will provide the best control flow and run 1-2x slower than a Rust implementation (for small enough inputs).

Indeed, the more involed

{-# LANGUAGE TemplateHaskell #-}

import Data.Foldable.Functor
import Language.Haskell.TH.Syntax
import CollatzTH

coalgebra :: Int -> Either [Int] (ListF Int Int)
coalgebra 1 = Left [1]
coalgebra 2 = Left $( lift (collatzTH 2) )
coalgebra 3 = Left $( lift (collatzTH 3) )
coalgebra 6 = Left $( lift (collatzTH 6) )
coalgebra 7 = Left $( lift (collatzTH 7) )
coalgebra 9 = Left $( lift (collatzTH 9) )
coalgebra 18 = Left $( lift (collatzTH 18) )
coalgebra n
    | n `mod` 2 == 0 = Right $ Cons n (div n 2)
    | otherwise = Right $ Cons n (3 * n + 1)

collatz :: Int -> [Int]
collatz = elgot embed elgotCoalgebra

We have to put collatzTH in another module, because TemplateHaskell is a pain in the butt.

collatzTH :: Int -> [Int]
collatzTH 1 = [1]
collatzTH n
    | n `mod` 2 == 0 = n:(collatzPatternTH (div n 2))

This is not memoized, nor is it exactly a supercompiler, but it is in fact faster than the naive Rust implementation for values I tested < 2223.