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APL Optimization Techniques Eugene Ying Senior Software Developer Fiserv, Inc. September 14, 2012 1

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Topics Component File Fragmentation The Match Function The Inner Product Storing Numbers in a Native File The Outer Product File I/O Optimization CPU Optimization 2

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A Component File where each Component Contains 100 Rows of Data Updating component 2 with 150 rows of data comp 2 file is fragmented Updating component 2 with 50 rows of data 3

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Suppose your data will not have more than 500 rows of data. To minimize the chance of fragmentation, you allocate 500 rows of data for each component. Initializing a Component File (500 10⍴' ')⎕FAPPEND TIE ⍝ Component 1 (500 4⍴0)⎕FAPPEND TIE ⍝ Component 2 (500 20⍴' ')⎕FAPPEND TIE ⍝ Component 3 (500 5⍴0)⎕FAPPEND TIE ⍝ Component 4 (500 15⍴' ')⎕FAPPEND TIE ⍝ Component 5 4

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Initializing a Component File Intended Initialization Actual Initialization comp 1comp 2 comp 1 comp 3 comp 4 comp 5 comp 3comp 2 comp 4 characters numbers Numeric Components are greatly under-allocated in size 5

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Storage Sizes of APL Numbers BOOLEAN←1000⍴1 ⎕SIZE 'BOOLEAN' 144 INTEGER1←1000⍴2 ⎕SIZE 'INTEGER1' 1016 INTEGER2←1000⍴128 ⎕SIZE 'INTEGER2 ' 2016 INTEGER4←1000⍴32768 ⎕SIZE 'INTEGER4' 4016 FLOAT8←1000⍴0.1 ⎕SIZE 'FLOAT8' 8016 6

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The Default APL Number 0 X←1000⍴0 ⎕SIZE 'X' 144 X←1000⍴0.1-0.1 ⎕SIZE 'X' 144 X←1000⍴0×0.1 ⎕SIZE 'X' 144 X←1000↑0⍴0.1 ⎕SIZE 'X' 144 X←0×1000⍴0.1 ⎕SIZE 'X' 144 7

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F64_0←1⊃11 645 ⎕DR 1000⍴0 ⍝ Floating pt # 0 ⎕SIZE 'F64_0' 8016 B32_999←1⊃163 323 ⎕DR 1000⍴999 ⍝ Binary-32 # 999 ⎕SIZE 'B32_999' 4032 B16_2←1⊃83 163 ⎕DR 1000⍴2 ⍝ Binary-16 # 2 ⎕SIZE 'B16_2' 2032 B8_0←1⊃11 83 ⎕DR 1000⍴0 ⍝ Binary-8 # 0 ⎕SIZE 'B8_0' 1016 How Do You Create A Vector of Integer Zeros or A Vector of Floating Point Zeros? 8

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Declaring Numbers Using a Defined Function to Preserve Numeric Type F64←64 DCL 1000⍴0 ⍝ Floating pt # 0 ⎕SIZE 'F64' 8016 I32←32 DCL 1000⍴999 ⍝ Binary-32 # 999 ⎕SIZE 'I32' 4032 I16←16 DCL 1000⍴2 ⍝ Binary-16 # 2 ⎕SIZE 'I16' 2032 I8←8 DCL 1000⍴0 ⍝ Binary-8 # 0 ⎕SIZE 'I8' 1016 9

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The DCL (Declare) Function [0] Z←X DCL Y;D;R [1] ⍝ Declare a floating point or integer array so that each [2] ⍝ item occupies the number of bits requested by the X argument [3] ⍝ X: # of bits that each number in the array will occupy [4] ⍝ 8 for 8-bit (1-byte) integer (¯128 to 127) [5] ⍝ 16 for 16-bit (2-byte) integer (¯32768 to 32767) [6] ⍝ 32 for 32-bit (4-byte) integer (¯2147483648 to 2147483647) [7] ⍝ 64 for 64-bit (8-byte) floating point # [8] ⍝ Y: Numeric array declared [9] ⍝ Z: Numeric array that occupies the space you requested [10] [11] D←⎕DR Y ⍝ Current data type of Y [12] :Select ⍬⍴X [13] :Case 8 ⋄ R←83 [14] :Case 16 ⋄ R←163 [15] :Case 32 ⋄ R←323 [16] :Case 64 ⋄ R←645 [17] :Else ⋄ ∘ ⍝ Stop if requested data type not supported [18] :EndSelect [19] →(D>R)↑'∘' ⍝ Stop if numeric overflow [20] Z←1⊃(D,R)⎕DR Y ⍝ Convert to requested data type 10

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For more accurate initialization: Initialization as Intended (500 10⍴' ')⎕FAPPEND TIE ⍝ Component 1 (64 DCL 500 4⍴0)⎕FAPPEND TIE⍝ Component 2 (500 20⍴' ')⎕FAPPEND TIE ⍝ Component 3 (32 DCL 500 5⍴0)⎕FAPPEND TIE ⍝ Component 4 (500 15⍴' ')⎕FAPPEND TIE ⍝ Component 5 11

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Changing the Floating Point 0 Z1000←64 DCL 1000⍴0⍝ 1,000 Floating pt 0 ⎕SIZE 'Z1000' 8016 Z2000←2000↑Z1000⍝ 2,000 Floating pt 0 ⎕SIZE 'Z2000' 268 Z2000←64 DCL 2000⍴0⍝ 2,000 Floating pt 0 ⎕SIZE 'Z2000' 16016 12

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The internal representation of the result R←X ⎕ DR Y is guaranteed to remain unmodified until it is re-assigned (or partially re-assigned) with the result of any function (ref: Dyalog Apl Reference Manual Chapter 6) Precaution Do not change a Declared array and then re-use it. If you need another similar array but of different dimensions, you should declare the new one from scratch. Reason: 13

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Storing Numbers in a Native File 14

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Blanks and commas are the most frequently used separators for numbers stored in a text file. Index Generator is also frequently used. N1←'40001 40002' Storing Numbers as Characters N3←'40000+⍳2' N2←'40001,40002' :For I :In ⍳10000 X←⍎N1 Y←⍎N2 Z←⍎N3 :EndFor ⍝ Elapsed time = 72 ms ⍝ Elapsed time = 89 ms ⍝ Elapsed time = 94 ms The character strings are executed to retrieve the numbers 15

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:For I :In ⍳100 X←⍎N1 Y←⍎N2 Z←⍎N3 :EndFor ⍝ Run Time 96 ms ⍝ Run Time 661 ms Storing 1,000 Numbers as Characters ⍝ Run Time 504 ms N1←⍕N N2←N1 ((N2=' ')/N2)←',' N3←¯1↓,'(',(⍕⍪¯1+(1000⍴1 0)/N),500 5⍴'+⍳2),' N←4000+(1500⍴1 1 0)/⍳1500 ⍝ (4000+⍳2),(4003+⍳2),... Comma separated Index generated ⍝ 4001,4002,4004,4005,... comma separated ⍝ 4001 4002 4004 4005... space separated 16

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Space Wasted by Trailing Blanks Character Matrix with 2 records Record 1 can be compressed a little bit by the Index Generator so that record 2 has less trailing blanks But in a nested vector, record 2 naturally has no trailing blanks 2911029106299111291132911529114 29246 (29109+ ⍳ 2),29106,(29112+ ⍳ 2),29115 29246 2911029106299111291132911529114 29246 17

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File I/O Optimization Suggestions Use the DCL function to Declare arrays to initialize the numeric components of a component file, otherwise the numeric components are under- allocated in size and the component file becomes fragmented too quickly. To store purely numeric data in a native file, do not use commas to separate the numbers, even though CSV format is very popular, because APL commas are being executed as primitive functions. 18

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Outer Product 19

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Replacing Outer Product by Indexing Y←⍳32000 :For I :In ⍳5 L←1≠+/Y∘.=Y M←Y∊((⍳⍴Y)≠Y⍳Y)/Y :EndFor ⎕WA 2656824552 X←1≠+/D∘.=D←⍳33000 LIMIT ERROR ⎕WA 270924 ⍝ 10,000 times smaller WS X←D∊((⍳⍴D)≠D⍳D)/D←⍳33000 ⍝ No LIMIT ERROR ⍝ 1,000 times faster ⍝ 21724 ms ⍝ 20 ms 20

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Replacing Outer Product by Simple Logic M←100000↑50000⍴⍳13 :For I :In ⍳1000 L←1≠×/×M∘.-1 12 N←(M≥1)^M≤12 :EndFor M←100000↑50000⍴⍳13 ⎕WA 1397828 L←1≠×/×M∘.-1 12 WS FULL ⎕WA 37832 L←(M≥1)^M≤12 ⍝ 40 times smaller WS ⍝ No WS FULL ⍝ 9210 ms ⍝ 813 ms ⍝ 10 times faster 21

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Replacing Outer Product by a Loop :For J :In ⍳10 X←+/((⍳⍴A)∘.≥⍳⍴A)^A∘.**
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Inner Product 23

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Matrix on the (wrong) Side of the Expression Requiring a Matrix Transpose 'ABC'^.=⍉((1↑⍴D),3)↑D (((1↑⍴D),3)↑D)^.='ABC' ⍝ Transpose needed ⍝ Transpose not needed 24 “one less pair of parentheses”

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Transposed Inner Product VECTOR^.=⍉MATRIX Y←10000 6⍴⎕A :For I :In ⍳10000 L←'EFGHIJ'^.=⍉Y M←Y^.='EFGHIJ' :EndFor MATRIX^.=VECTOR ⍝ 14561 ms ⍝ 2302 ms 25 vs

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Array Comparisons 26

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Comparing Array Contents with a scalar ^/M^.=' ' or ^/^/M=' ' or M≡(⍴M)⍴' ' M←1000 1000⍴⎕AV 27

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Character Comparison Efficiency M←1000 1000⍴⎕AV :For I :In ⍳10000 {}^/M^.=' ' {}^/^/M=' ' {}M≡(⍴M)⍴' ' :EndFor ⍝ 9108 ms ⍝ 9060 ms ⍝ 587 ms 28

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Numeric Comparison Efficiency M←1000 1000⍴ ⍳10000 :For I :In ⍳10000 {}^/M^.=0 {}^/^/M=0 {}M≡(⍴M)⍴0 :EndFor ⍝ 12254 ms ⍝ 12201 ms ⍝ 52 ms 29

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Comparing Vectors A←10000?10000 B←10000?10000 C←A^.=B :For I :In ⍳10000 {}A^.=B {}A≡B :EndFor C←A≡B ⍝ 1244 ms ⍝ 135 ms 30

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Comparing Vectors of Unequal Lengths A←10000?10000 B←9999?9999 C←A^.=B LENGTH ERROR C←A^.=B ^ 31

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Comparing Vectors of Unequal Lengths L←(⍴A)⌈⍴B C←(L↑A)^.=L↑B or :If C←(⍴A)=⍴B :AndIf C←A^.=B :EndIf or C←A≡B To avoid LENGTH ERROR 32

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Checking the Return Code of a Function →(¯1∊DATA←FUNCTION_1)/ERR But there are still many functions written such that the result returned can be either the data or the return code. Nowadays, many functions are written such that a 2-item nested vector is returned where one item contains the result and another item contains the return code. E.g. if ¯1 returned by a function means an error has occurred; then we need to be very careful with the use of the ∊ membership function. 33

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Example of Function Return Code A popular IBM APL utility function to read text file is called ∆FM (File Matrix I/O). When ∆FM reads a text file and encounters an error, instead of returning the data, it returns an error code of 28. Thus many programmers would write the text file I/O coding in the following way. →(28∊DATA←∆FM 'file.csv')/ERR 34

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Example of Return Code Inefficiency Y←∆FM 'file.csv' ⎕SIZE'Y' 9979076 ⍴Y 72312 138 :For I :In ⍳1000 {}28∊Y {}28≡Y :EndFor ⍝ 3208521 ms ⍝ 4 ms 35

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CPU Optimization Suggestions When an elegant outer product generates a sparse matrix that causes LIMIT ERROR, WS FULL, or computational slow down, replace the outer product by a simpler but not so elegant expression. Example of code elegance: 1≠×/×M∘.-1 12 vs (M≥1)^M≤12 Try to avoid unnecessary transpose of a matrix when you perform an inner product of a matrix with a vector. Remember that in some cases, the match function can run much faster than the inner product or the membership function. 36

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The End 37 Eugene Ying Fiserv, Inc.

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