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Suppose I have a list of positive integers:

data={1, 1, 2, 3, 3, 3, 5, 5, 5, 7, 7, 8, 8, 9, 10, 10, 12, 16, 23}

I want to count the number of subsets up to length t (including an empty set) whose total doesn't exceed the value t.

The naive approach would be:

sumZaehl[t_, data_] :=
  Length@Select[Total /@ Subsets[data, t], # <= t &]

But this would not work for larger list, because of the rapidly increasing number of subsets. I have an iterative method which works for larger list too:

sumZaehlIter[t_, data_] :=
Module[{n = Length[data], v, d, i, j},
  For[i = 0, i <= t, i++, v[i] = 1];
  For[i = 1, i <= n, i++,
    d = data[[i]];
    For[j = t, j >= d, j--, 
       v[j] = v[j] + v[j - d]; 

Is there a functional way to realize this?

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I figure that the sublists with a total not greater than 3 are following six: {{},{1},{2},{3},{1,1},{1,2}}. Yet both your sumZaehlIter[3,data] and Leonid's v[3,data] return 10. What are the other four sublists with a total not greater than 3? –  David Carraher Mar 6 '12 at 3:29
@David, I think (with multiplicities of some elements), there are two additional occurences of {3} and one additional occurence for each of {1} and {2,1}. –  kguler Mar 6 '12 at 7:57
@kguler Interesting. So it has nothing to do with the order of the elements within a sublist? –  David Carraher Mar 6 '12 at 10:14
@David, yes; so the complete collection is {{},{1},{1},{2},{3},{3},{3},{1,1},{1,2},{1,2}}. –  kguler Mar 6 '12 at 10:44
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1 Answer

up vote 7 down vote accepted

This is not really the same algorithm, but

v[t_, data_] :=
     v[_?Negative, _] := 0;
     v[_, 0] := 1;
     v[tl_, n_] := v[tl, n] =
        v[tl - data[[n]], n - 1] + v[tl, n - 1];
     v[t, Length[data]]

You may need to increase the $RecursionLimit for larger lists.

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the iterative Module works quite well but I'm interested in functional. With the "naive" way I get errors "Subset::toomany and "must be a machine integer". Why "a machine integer" –  Peter Breitfeld Mar 5 '12 at 20:38
@Peter Your iterative algorithm uses a lot of mutable state. While you probably can achieve the same with the functional constructs, I suspect that the direct translation of it into functional would be either inefficient or resulting in a thin functional facade over the essentially procedural core. The algorithm I posted above has the same underlying idea as yours, but IMO is a functional version of it (although not a direct translation), using recursion and memoization. A similar situation happens e.g. with the longest common sequence algorithm. –  Leonid Shifrin Mar 5 '12 at 21:03
That's beautiful. +1 –  Mr.Wizard Mar 6 '12 at 8:10
Nice solution! +1 –  Pillsy Mar 6 '12 at 16:08
@Pillsy, Mr.Wizard Thanks! –  Leonid Shifrin Mar 6 '12 at 17:11
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