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How to thread a list



I have data in format


data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}}

Tableform:


enter image description here


I want to thread it to :


tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}}

Tableform:


enter image description here




And I would like to do better then


pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}];
SetAttributes[pseudofunction, Listable];
Range[2, 4] // pseudofunction


Here is my benchmark data, where data3 is normal sample of real data.


data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1];
data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5];

data = RandomReal[{0, 1}, {10^6, 4}];

Here is my benchmark code


kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list
kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list
OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}]
paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1]
RM[list_] := FoldList[Transpose[{First@list, #2}] &, Null, Rest[list]] // Rest
rcollyer[list_] := With[{fst = First@#, rst = Rest@#}, Thread[{fst, #}] & /@ rst] &@list


Drop[Timing[paradox2[#];] & /@ {data, data2, data3}, None, -1]
Drop[Timing[OleksandrR[#];] & /@ {data, data2, data3}, None, -1]
Drop[Timing[kptnw[#];] & /@ {data, data2, data3}, None, -1]
Drop[Timing[kptnw2[#];] & /@ {data, data2, data3}, None, -1]
Drop[Timing[RM[#];] & /@ {data, data2, data3}, None, -1]
Drop[Timing[rcollyer[#];] & /@ {data, data2, data3}, None, -1]

Results


{{7.503}, {0.968}, {0.031}}
{{0.983}, {0.296}, {0.031}}

{{0.312}, {1.67}, {0.031}}
{{0.094}, {0.218}, {0.031}}
{{3.759}, {0.546}, {0.032}}
{{3.073}, {0.733}, {0.031}}

Answer



If your lists are long, there are faster approaches using high-level functions and structural operations. Here are two alternatives.


First we try Outer and Flatten:


data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}};
Flatten[Outer[List, List@First[data], Rest[data], 1], {{2}, {1, 4}}]



{{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}}

And now Distribute and Transpose:


Transpose[Distribute[{List@First[data], Rest[data]}, List], {1, 3, 2}]


{{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}}

Evidently, they give the correct result. Now for a Timing comparison:



data = RandomReal[{0, 1}, {10^6, 2}];

The timings, in rank order, are:



  1. kptnw's Table/Transpose method: 0.297 seconds

  2. Outer/Flatten: 0.812 seconds

  3. Distribute/Transpose: 0.891 seconds

  4. rcollyer's Thread/Map approach: 2.907 seconds

  5. R.M's Transpose/FoldList method: 3.844 seconds

  6. paradox2's solution with Riffle and Partition: 7.407 seconds



The Outer/Flatten and Distribute/Transpose approaches are quite fast, but clearly Table is much better-optimized than Distribute, since while these two methods are conceptually similar, kptnw's solution using the former is by far the fastest and most memory-efficient. The other solutions, not using structural operations, are considerably slower, which is not unexpected.


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