Download - APL Optimization Techniques Eugene Ying Senior Software Developer Fiserv, Inc. September 14, 2012 1
APL Optimization Techniques
Eugene YingSenior Software Developer
Fiserv, Inc.September 14, 2012
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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
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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
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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
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Initializing a Component File
IntendedInitialization
ActualInitialization
comp 1 comp 2
comp 1
comp 3 comp 4 comp 5
comp 5comp 3comp 2 comp 4
characters characters characters
characters characters characters
numbers numbers
numbers numbers
Numeric Components are greatly under-allocated in size
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Storage Sizes of APL Numbers
BOOLEAN←1000⍴1 ⎕SIZE 'BOOLEAN'
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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
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The Default APL Number 0
X←1000⍴0 ⎕SIZE 'X'
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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
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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 Zerosor A Vector of Floating Point Zeros?
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Declaring NumbersUsing 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
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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
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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
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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
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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:
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Storing Numbers in a Native File
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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
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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
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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
2 9 1 1 0 2 9 1 0 6 2 9 9 1 1 1 2 9 1 1 3 2 9 1 1 5 2 9 1 1 4
2 9 2 4 6
( 2 9 1 0 9 + ⍳ 2 ) , 2 9 1 0 6 , ( 2 9 1 1 2 + ⍳ 2 ) , 2 9 1 1 5
2 9 2 4 6
2 9 1 1 0 2 9 1 0 6 2 9 9 1 1 1 2 9 1 1 3 2 9 1 1 5 2 9 1 1 4
2 9 2 4 6
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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.
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Outer Product
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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
⎕WA2656824552 X←1≠+/D∘.=D←⍳33000LIMIT 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
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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 ⎕WA1397828 L←1≠×/×M∘.-1 12WS FULL
⎕WA37832 L←(M≥1)^M≤12
⍝ 40 times smaller WS⍝ No WS FULL
⍝ 9210 ms⍝ 813 ms⍝ 10 times faster
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Replacing Outer Product by a Loop
:For J :In ⍳10 X←+/((⍳⍴A)∘.≥⍳⍴A)^A∘.<B Y←⍬ :For I :In ⍳⍴B Y,←+/A[I]<I↑B :EndFor:EndFor
⎕WA2047735492 X←+/((⍳⍴A)∘.≥⍳⍴A)^A∘.<BLIMIT ERROR
⎕WA405316X←⍬ :For I :In ⍳⍴B X,←+/A[I]<I↑B:EndFor
⍝ 3 times faster
⍝ 5,000 times smaller workspace
A←32800?32800B←20000+32800?32800
⍝ No LIMIT ERROR
⍝ 75810 ms
⍝ 26422 ms
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Inner Product
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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
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“one less pair of parentheses”
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
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vs
Array Comparisons
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Comparing Array Contents with a scalar
^/M^.=' '
or^/^/M=' '
orM≡(⍴M)⍴' '
M←1000 1000⍴⎕AV
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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
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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
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Comparing Vectors
A←10000?10000B←10000?10000
C←A^.=B
:For I :In ⍳10000 {}A^.=B {}A≡B :EndFor
C←A≡B
⍝ 1244 ms⍝ 135 ms
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Comparing Vectors of Unequal Lengths
A←10000?10000 B←9999?9999
C←A^.=BLENGTH ERROR C←A^.=B ^
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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
orC←A≡B
To avoid LENGTH ERROR
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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.
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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
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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
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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.
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The End
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Eugene YingFiserv, Inc.