A Speed-Up Story

The first e-mail of the work week came from Nicolas Delcros. He wondered whether anything clever can be done with ∘.≡ on enclosed character strings. I immediately thought of using “magic functions“, an implementation technique whereby interpreter facilities are coded as dfns. I thought of magic functions because the APL expressions involved in this case are particularly terse:

   t ←' zero one two three four five six seven eight nine'
   t,←' zéro un deux trois quatre cinq six sept huit neuf'
   t,←' zero eins zwei drei vier fünf sechs sieben acht neun'
   b←⌽a←1↓¨(' '=t)⊂t

   cmpx 'a∘.≡a' '∘.=⍨⍳⍨a'
  a∘.≡a   → 9.69E¯5 |   0% ⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕
  ∘.=⍨⍳⍨a → 8.21E¯6 | -92% ⎕⎕
   cmpx 'a∘.≡b' '(a⍳a)∘.=(a⍳b)'
  a∘.≡b         → 9.70E¯5 |   0% ⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕
  (a⍳a)∘.=(a⍳b) → 1.43E¯5 | -86% ⎕⎕⎕⎕


   cmpx 'x∘.≡x' '∘.=⍨⍳⍨x'
  x∘.≡x   → 9.53E¯3 |   0% ⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕
  ∘.=⍨⍳⍨x → 1.52E¯4 | -99% ⎕
   cmpx 'x∘.≡y' '(x⍳x)∘.=(x⍳y)'
  x∘.≡y         → 9.55E¯3 |   0% ⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕
  (x⍳x)∘.=(x⍳y) → 1.95E¯4 | -98% ⎕

The advantage will be even greater if/when we speed up . (And, obviously, the idea is also applicable to ∘.≢ : just replace with and = with .)

Jay Foad objected that the comparisons above aren’t quite fair as the more verbose expressions should check that either ⎕ct←0 or that the arguments do not contain any elements subject to tolerant comparison. I countered that the checking in C would not greatly affect the benchmarks as the time to do the checking is O(m+n) but the benefits are O(m×n) .

Since the factors are so promising and the coding relatively easy, I went ahead and did the work, with the following results:

14.1 14.0 ratio cost of checking
a∘.≡a 9.16e¯6 9.69e¯5 10.58 1.09
a∘.≡b 1.53e¯5 9.70e¯5 6.34 1.08
x∘.≡x 1.48e¯4 9.53e¯3 64.39 1.00
x∘.≡y 1.94e¯4 9.55e¯3 49.23 1.01

The last column (the cost of checking that tolerant comparison is not involved) is under 10% and decreases as the argument sizes increase.

This work is another illustration of the ubiquity and practical usefulness of the “selfie” concept — in the new (14.1) implementation, x∘.≡y is faster when the left and right arguments are the same than when they are not. In particular, the selfie x⍳x or ⍳⍨x occurs twice, and bolsters the point made in item 3 of Sixteen APL Amuse-Bouches:

x⍳x are like ID numbers; questions of identity on x can often be answered more efficiently on x⍳x than on x itself.

Finally, after all that, I recommend that one should consider using x⍳y before using x∘.≡y . The latter requires O(m×n) space and O(m×n) time, and is inherently inefficient.

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