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Packed arrays are very useful because they save memory and generally allow speedier and more efficient calculation times. If a list of data currently stored as a packed array is unpacked, it can slow things down considerably. It has been suggested that using certain built-in functions has the advantage that unpacking doesn't occur.

Short of some trial-and-error spelunking using PackedArrayQ from the Developer Utilites package, are there any guidelines or best practices for using functions and coding styles that avoid these performance hits from inadvertant unpacking?

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3 Answers 3

up vote 26 down vote accepted

I will try to list some cases I can recall. The unpacking will happen when:

  • The result, or any intermediate step, is a ragged (irregular) array. For example

     Range /@ Range[4]
    

    To avoid this, you can try to use regular structures, perhaps padding your arrays with zeros appropriately

  • The result (or any intermediate step) contains numbers of different types, such as a mix of integers and reals, or when some of the elements are not of numeric types at all (symbols, expressions)

    This usually happens by mistake for 1D lists. For multi-dimensional lists, there are several ways out. One is to convert all numbers to a single type (e.g. Reals to Integers or vice versa), when that is feasible. One such example is here.

    Another way out is to store an array parts separately. For example, you have two arrays of the same length, but different element types, which logically belong together (such as a result of Tally operation on reals, for example, as illustrated below). While our usual reaction would be to store it in transposed (and thus unpacked) form, one can also store them as {list1,list2}, which will be unpacked, but the parts list1 and list2 inside it will remain packed - just don't transpose it. One example of such treatment is here

    This trick can be generalized to even ragged arrays. In the already cited post, I used it to convert an imported ragged array to a more space-efficient form, with elements being packed arrays, with

    packed =  Join @@Map[Developer`ToPackedArray, list]
    
  • Some of the numbers don't fit into the numerical precision limits (for example, very big integers). This can be insidious, because this may be data-dependent and happen in the middle of a computation, and it may not be clear what is going on.

    Here, you can try to predict in advance whether or not this is likely, but other than that, there is little of what can be done, short of changing the algorithm.

  • The packed array is a part of an expression used with some rule-based code and subject to pattern-matching attempts. This will happen in cases when the match is not established before the pattern-matcher comes to the array. Here is an example:

    Cases[f[g[Range[5]]], g[l_List] :> g[l^2], Infinity]
    
    During evaluation of In[14]:= Developer`FromPackedArray::unpack: 
    Unpacking array in call to f. >>
    

    {g[{1, 4, 9, 16, 25}]}

    while this does not unpack:

    f[g[Range[5]]] /. g[l_List] :> g[l^2]
    

    f[g[{1, 4, 9, 16, 25}]]

    This happened because Cases searches depth-first (and therefore reaches elements before heads, and then must unpack), while ReplaceAll replaces from expressions to sub-expressions. I discussed this extensively here.

    This situation is typical for the pattern-matching - it will generally unpack. Note also that the pattern-matching goes inside held expressions, and will unpack even there:

    FreeQ[Hold[Evaluate@Range[10]], _Integer]
    

    The only way I know to generally prevent it is to make sure that the pattern will either match or be rejected before the pattern-matcher comes to a given packed array. Note that there are certain exceptions, e.g. like this:

    MatchQ[Range[10], {__Integer}]
    

    In which case, there is no unpacking.

  • In certain cases, you will not see the unpacking message, but the result returned by a function may be packed or unpacked, depending on its type. Here is an example:

    tst = RandomInteger[10,20]
    

    {6,9,9,4,6,4,0,9,7,1,3,2,2,0,7,2,1,0,7,5}

    ntst = N@tst;
    
    tally = Tally[tst]
    

    {{6,2},{9,3},{4,2},{0,3},{7,3},{1,2},{3,1},{2,3},{5,1}}

    ntally = Tally[ntst]
    

    {{6.,2},{9.,3},{4.,2},{0.,3},{7.,3},{1.,2},{3.,1},{2.,3},{5.,1}}

    Developer`PackedArrayQ/@{tally,ntally}
    

    {True,False}

    You can see that the ntally was returned as an unpacked array, because it contains elements of different types, and there was no message to tell us about it, since indeed, nothing was unpacked - the result is a new array.

    As I metnioned already, one way here is to separate frequencies and elements themselves, and store them separately packed.

  • As elaborated by @Mr.Wizard, Apply leads to unpacking. This also refers to Apply at level 1 (@@@). The way out here is just not to use Apply - chances are, that you can achieve your goal by other means, with packed lists.

  • Map will unpack on short lists, with lengths smaller than "SystemOptions"->"CompileOptions"->"MapCompileLength". This may come as a surprise, since we are used to the fact that Map does not unpack. For example, this unpacks:

    Map[#^2 &, Range[10]] 
    

    The way out here would be to change the system options ("MapCompileLength") accordingly, to cover your case, or (perhaps even better), to manually pack the list with Developer`ToPackedArray after Map is finished. This often does not matter much for small lists, but sometimes it does.

  • Map will also unpack for any function which it can not auto-compile:

    ClearAll[f];
    f[x_] := x^2;
    Map[f, Range[1000]]
    

    while this does not unpack:

    Map[#^2 &, Range[1000]]
    

    The solution here is to avoid using rule-based functions in such cases. Sometimes one can also, perhaps, go with some more exotic options, such as using something similar to a withGlobalFunctions macro from this answer (which expands certain rule-based functions at run-time).

  • Functions like Array and Table will produce unpacked arrays for functions or expressions which they can not auto-compile. They will not produce any warnings. For example:

    Array[f, {1000}] // Developer`PackedArrayQ
    

    False

  • Similar situation for other functions which have special compile options.

    In all these cases, the same advice: make your functions/expressions (auto)compilable, and / or change the system settings. Sometimes you can also manually pack the resulting list afterwords, as an alternative.

  • While this reiterates on one of the previous points, innocent-looking functions which combine packed arrays of different types will often unpack both:

    Transpose[{Range[10], N@Range[10]}]
    

    In cases like this, often (also as mentioned already) you can live with such lists as they are, without transposing them. Then, the sub-lists will remain packed.

  • When you use Save to save some symbol's definitions and Get to get them back, packed arrays will be generally unpacked during Save. This is not the case with DumpSave, which is highly recommended for that. Also, Compress does not unpack.

  • Import and Export will often not preserve packed arrays. The situation is particularly grave with Import, since often it takes huge memory (and time) to import some data, which could be stored as a packed array, but is not recognized as such.

There are probably many more cases. I intend to add to this list once I recall some more, and invite others to contribute. One characteristic feature of unpacking is, however, general: whenever a final result or some intermediate expressions can not be represented as regular arrays (tensors) of the same basic type (Integer, Real or Complex), most of the time unpacking will happen.

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1  
. . . Or that. :-) –  Mr.Wizard May 7 '12 at 18:39
    
@Mr.Wizard Thanks for the upvote:). I voted for you answer too. –  Leonid Shifrin May 7 '12 at 18:40
1  
Good catch on the pattern matchers... Exactly the reason that Image is not using packed array as internal data representation. Also, there are internal functions that use FreeQ/Cases quite intensively--for instance, Manipulate... –  Yu-Sung Chang May 8 '12 at 4:09
    
@Yu-SungChang Thanks, I was not aware of these details. Makes perfect sense. –  Leonid Shifrin May 8 '12 at 8:55

I'll be the first to simply mention that you can use On["Packing"] and then observe any unpacking that occurs in the course of evaluation. I'm not sure there is a more systematic way to approach this, other than to compile a list of functions that do or do not preserve packing.

There are always a few surprises for me, such as PadRight on a ragged array composed of packed vectors not returning a packed array, but I don't think there is any way to guess that beforehand without prior knowledge of the implementation.


I guess one guideline is that functions that operate on a list or array directly often handle packed arrays, while Mapping or Applying usually do not.

For example:

a = Range@30;
Tr[a]
Total[a]
Plus @@ a

Tr and Total do not unpack, but Plus @@ does. Likewise:

a = RandomInteger[99, {5, 5}]
Total[a, {2}]
Plus @@@ a
Tr /@ a
Plus @@ # & /@ a

Total does not unpack; Plus @@@ and Tr /@ unpack one level; the last line fully unpacks.

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4  
On["Packing"] is extremely useful, although it may give you quite a false positive. Some short answers can be packed and tend to be immediately unpacked in the course of evaluation. For instance, Developer`PackedArrayQ[Dimensions[{a, b, c, d}]]. And most internal functions that calls Dimensions unpack it immediately, which usually causes the unpacking message--and mostly nothing to worry about. –  Yu-Sung Chang May 8 '12 at 4:13
    
@Yu-Sung Chang very good tip; thank you! –  Mr.Wizard May 8 '12 at 5:38
    
@Yu-SungChang Where can I find further information about your statement? E.g., does FoldList internally call Dimensions or does it truly unpack packed arrays? - Does this warrant its own question? –  Andrea Colonna Jun 21 '12 at 16:20
1  
Comment for google, humans may ignore: SetSystemOptions::sysname: "UnpackMessage" is not a known SystemOption." SetSystemOptions::obs: "Developer`SetSystemOptions" has been superseded by SetSystemOptions, and is now obsolete. It will not be included in future versions of Mathematica. " Developer`SetSystemOptions["UnpackMessage" -> True] –  ssch Aug 11 '13 at 15:07

This is not really a guideline, but it is so basic/common I felt it deserved to be mentioned.

Be careful when using Set with Part

<< Developer`
packed4 = packed3 = packed2 = packed = ConstantArray[0, 100];
packed[[{1, 2, 3, 4, 5}]] = {6, 7, 8, 9, 3};
packed2[[1 ;; 5]] = {6, 7, 8, 9, 3};
packed3[[1]] = 1;
packed4[[{1, 2, 3, 4, 5}]] = {6, 7, 8, 9, 3} // ToPackedArray;
PackedArrayQ /@ {packed, packed2, packed3, packed4}
{False, False, True, True}
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1  
Definitely worth a mention. +1 Also see: Why does list assignment with a packed array result in unpacked values? –  Mr.Wizard May 31 at 8:49

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