a couple of implementations I have posted before for u/ when x items and
initial y are of different shape.
reduce =: 1 : '<"_1@[ ([: u (&.>)/(>@:) ,) <@:]'
reducE =: 1 : (':'; 'o=. y for_i. x do. o =. i u o end.')
reducE is explicit, and left to right. reduce boxes each side, and uses the
same right to
left order as /. reduce is 2-3x faster for +
There's an even faster method, but before I mention it, I'd like to go over how
the rank 0 operators work in J
(1 + 1 2 3 4) -: 1 1 1 1 + 1 2 3 4
`a scalar u"0 a_larger_shape` gets expanded to match the larger shape, and its
result is identical to the operation with the "pre-expanded" shape.
1 2 + 1 2 3 4 is rank error.
But there are a couple of obvious expansions that would not create an error
1 2 1 2 + 1 2 3 4 or
1 2 0 0 + 1 2 3 4
note that this is not an error
1 2 + i.2 2 but the result (with shape ommitted) is identical to
1 1 2 2 + 0 1 2 3
The new reduce adverb is at its core a conjunction. Where one of its verbs is
how to expand the x items such that they are similar (or compatible) shape to y.
reduceC =: 2 : 'u/@:(v , ])'
reduceS =: reduceC ($@] $"1 _1 [)
reduceS is an adverb that generically expands (i.2 2) + reduceS 1 2 3 4 to
+/ 0 1 0 1 , 2 3 2 3 ,: 1 2 3 4
which may be better than an error, but is unusual, and so the conjunction
version is useful to taylor the expansion as you prefer.
some custom applied conjunctions
reduceC ($@] {.!._"1 _1 [) NB. take with fill set to _ to match shape of y.
reduceC ($@] {. 2 # [) NB. expand 1 2 to 1 1 2 2,
NB. but make sure there are 4 items (end matches shape of y)
5 timespacex ' +/ (i. 250000 4), 1 2 3 4'
0.0200582 2.51692e7
5 timespacex '(i. 250000 2) + reduceS 1 2 3 4'
0.0205952 2.51699e7
5 timespacex '(i. 250000 2) + reduceC (2 #"1 [) 1 2 3 4'
0.0231547 2.51707e7
5 timespacex '(i. 250000 2) + reduce 2 2 $ 1 2 3 4'
0.134078 4.66376e7
5 timespacex '(i. 1000000) + reduce 1 2'
0.563169 1.61364e8
5 timespacex '(i. 1000000) + reduceS 1 2'
0.0407191 4.1947e7
another technique that is pretty funny how its fast, but works well especially
when y is a string, and x is not, is to convert everything to strings (using
linear representations of x, lets you have x items with widely varying shapes)
NB. swaps 2 items described by x in y.
amV =: (0 {:: [)`(1 {:: [)`]}
swap =: (((0 { [) ;~ ] {~ 1 { [) amV ] amV~ (1 { [) ;~ ] {~ 0 { [)
timespacex '(52 | i. 2000 2) swap reduce Alpha_j_'
0.0180793 412160
timespacex '(":("1) 52 | i. 2000 2) (".@[ swap ])/ Alpha_j_'
0.00243008 2.32998e6
timespacex '(52 | i. 2000 2) swap reduceC ([) i.52'
0.0193757 2.2007e6
a [ for v argument to reduceC is same as prepending to y and letting it fill.
The swap function accesses x parameters by index and so it is harmless to
expand it.
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