Sparse Arrays in J

Introduction
Representation
Assertions
The Verb $.
Further Examples
Sparse Linear Algebra
Implementation Status

Copyright © 1998, 1999, Iverson Software Inc.


 Introduction

We describe a sparse array extension to J using a representation that “does not store zeros”. One new verb $. is defined to create and manipulate sparse arrays, and existing primitives are extended to operate on such arrays. These ideas are illustrated in following examples:

   ] d=: (?. 3 4$2) * ?. 3 4$100
 0 75  0 53
 0  0 67 67
93  0 51 83

   ] s=: $. d
0 1 | 75
0 3 | 53
1 2 | 67
1 3 | 67
2 0 | 93
2 2 | 51
2 3 | 83
convert d to sparse and assign to s

the display of s gives the indices of the
“non-zero” cells and the corresponding values

   d -: s
1
d and s match

   o. s
0 1 | 235.619
0 3 | 166.504
1 2 | 210.487
1 3 | 210.487
2 0 | 292.168
2 2 | 160.221
2 3 | 260.752
π times s

   o. d
      0 235.619       0 166.504
      0       0 210.487 210.487
292.168       0 160.221 260.752
π times d

   (o. s) -: o. d
1
function results independent of representation

   0.5 + o. s
0 1 | 236.119
0 3 | 167.004
1 2 | 210.987
1 3 | 210.987
2 0 | 292.668
2 2 | 160.721
2 3 | 261.252

   <. 0.5 + o. s
0 1 | 236
0 3 | 167
1 2 | 210
1 3 | 210
2 0 | 292
2 2 | 160
2 3 | 261

   (<. 0.5 + o. s) -: <. 0.5 + o. d
1

   d + s
0 1 | 150
0 3 | 106
1 2 | 134
1 3 | 134
2 0 | 186
2 2 | 102
2 3 | 166
function arguments can be dense or sparse

   (d + s) -: 2*s
1

   (d + s) -: 2*d
1

   +/ s
0 |  93
1 |  75
2 | 118
3 | 203

   +/"1 s
0 | 128
1 | 134
2 | 227
familiar algebraic properties are preserved

   |. s
0 0 | 93
0 2 | 51
0 3 | 83
1 2 | 67
1 3 | 67
2 1 | 75
2 3 | 53

   |."1 s
0 0 | 53
0 2 | 75
1 0 | 67
1 1 | 67
2 0 | 83
2 1 | 51
2 3 | 93
reverse

   |: s
0 2 | 93
1 0 | 75
2 1 | 67
2 2 | 51
3 0 | 53
3 1 | 67
3 2 | 83

   $ |: s
4 3

transpose
   $.^:_1 |: s
 0  0 93
75  0  0
 0 67 51
53 67 83

   (|:s) -: |:d
1

$.^:_1 converts a sparse array to dense
   , s
 1 | 75
 3 | 53
 6 | 67
 7 | 67
 8 | 93
10 | 51
11 | 83

   $ , s
12
ravel; a sparse vector

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  Representation

A sparse array y may be boolean, integer, floating point, complex, literal, or boxed, and has the (internal) parts sh;a;e;i;x;flag where:

sh Shape, $y . Elements of the shape must be less than 2^31 , but the product over the shape may be larger than 2^31 .
sh Axe(s), a vector of the sorted sparse (indexed) axes.
e Sparse element (“zero”). e is also used as the fill in any overtake of the array.
i Indices, an integer matrix of indices for the sparse axes.
x Values, a (dense) array of usually non-zero cells for the non-sparse axes corresponding to the index matrix i .
flag   Various bit flags.

For the sparse matrix s used in the introduction,

   ] d=: (?. 3 4$2) * ?. 3 4$100
 0 75  0 53
 0  0 67 67
93  0 51 83

   ] s=: $. d
0 1 | 75
0 3 | 53
1 2 | 67
1 3 | 67
2 0 | 93
2 2 | 51
2 3 | 83

The shape is 3 4 ; the sparse axes are 0 1 ; the sparse element is 0 ; the indices are the first two columns of numbers in the display of s ; and the values are the last column.

Scalars continue to be represented as before (densely). All primitives accept sparse or dense arrays as arguments (e.g. sparse+dense or sparse$sparse). The display of a sparse array is a display of the index matrix (the i part), a blank column, a column of vertical lines, another blank column, and the corresponding value cells (the x part).

Letting the sparse element be variable rather than fixed at zero makes many more functions closed on sparse arrays (e.g. ^y or 10+y or -.y), and familiar results can be produced by familiar phrases (e.g. <.0.5+y for rounding to the nearest integer).

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 Assertions

The following assertions hold for a sparse array, and displaying a sparse array invokes these consistency checks on it.

imax =: _1+2^31 the largest internal integer
rank =: #@$ rank
type =: 3!:0 internal type
 
1 = rank sh vector
sh -: <. sh integral
imax >: #sh at most imax elements
(0<:sh) *. (sh<:imax) bounded by 0 and imax
 
1 = rank a vector
a e. i.#sh bounded by 0 and rank-1
a -: ~. a elements are unique
a -: /:~ a sorted
 
0 = rank e atomic
(type e) = type x has the same internal type as x
 
2 = rank i matrix
4 = type i integral
(#i) = #x as many rows as the number of items in x
({:$i) = #a as many columns as there are sparse axes
(#i) <: */a{sh # rows bounded by product over sparse axes lengths
imax >: */$i # elements is bounded by imax
(0<:i) *. (i <"1 a{sh) bounded by 0 and the lengths of the sparse axes
i -: ~.i rows are unique
i -: /:~ i rows are sorted
 
(rank x) = 1+(#sh)-#a rank equals 1 plus the number of dense axes
imax >: */$x # elements is bounded by imax
(}.$x)-:((i.#sh)-.a){sh item shape is the dimensions of the dense axes
(type x) e. 1 2 4 8 16 32   internal type is boolean, character, integer, real, complex, or boxed

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 The Verb $.

The ranks of $. are infinite. The inverse of n&$. is (-n)&$. .

$.y converts a dense array to sparse, and conversely $.^:_1 y converts a sparse array to dense. The identities f -: f&.$. and f -: f&.($.^:_1) hold for any function f, with the possible exception of those (like overtake {.) which use the sparse element as the fill.

0$.y applies $. or $.^:_1 as appropriate; that is, converts a dense array to sparse and a sparse array to dense.

1$.sh;a;e produces a sparse array. sh specifies the shape. a specifies the sparse axes; negative indexing may be used. e specifies the “zero” element, and its type determines the type of the array. The argument may also be sh;a (e is assumed to be a floating point 0) or just sh (a is assumed to be i.#sh — all axes are sparse — and e a floating point 0).

2$.y gives the sparse axes (the a part);
(2;a)$.y (re-)specifies the sparse axes;
(2 1;a)$.y
gives the number of bytes required for (2;a)$.y ;
(2 2;a)$.y
gives the number of items in the i part or the x part for the specified sparse axes a (that is, #4$.(2;a)$.y).

3$.y gives the sparse element (the e part); (3;e)$.y respecifies the sparse element.

4$.y gives the index matrix (the i part).

5$.y gives the value array (the x part).

6$.y gives the flag (the flag part).

7$.y gives the number of non-sparse entries in array y ; that is, #4$.y or #5$.y .

8$.y removes any completely “zero” value cells and the corresponding rows in the index matrix.

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 Further Examples

   ] d=: (0=?. 2 3 4$3) * ?. 2 3 4$100
13 0 0 0
21 4 0 0
 0 0 0 0

 3 5 0 0
 0 0 6 0
 0 0 0 0

   ] s=: $. d
0 0 0 | 13
0 1 0 | 21
0 1 1 |  4
1 0 0 |  3
1 0 1 |  5
1 1 2 |  6
convert d to sparse and assign to s

   d -: s
1
match is independent of representation

   2 $. s
0 1 2
sparse axes

   3 $. s
0
sparse element

   4 $. s
0 0 0
0 1 0
0 1 1
1 0 0
1 0 1
1 1 2
index matrix; columns correspond to the sparse axes

   5 $. s
13 21 4 3 5 6
corresponding values

   ] u=: (2;2)$.s
0 | 13 21 0
  |  3  0 0
  |
1 |  0  4 0
  |  5  0 0
  |
2 |  0  0 0
  |  0  6 0
make 2 be the sparse axis

   4 $. u
0
1
2
index matrix

   5 $. u
13 21 0
 3  0 0

 0  4 0
 5  0 0

 0  0 0
 0  6 0
corresponding values

   ] t=: (2;0 1)$.s
0 0 | 13 0 0 0
0 1 | 21 4 0 0
1 0 |  3 5 0 0
1 1 |  0 0 6 0
make 0 1 be the sparse axes

   7 {. t
0 0 | 13 0 0 0
0 1 | 21 4 0 0
1 0 |  3 5 0 0
1 1 |  0 0 6 0

   $ 7 {. t
7 3 4
take

   7 {."1 t
0 0 | 13 0 0 0 0 0 0
0 1 | 21 4 0 0 0 0 0
1 0 |  3 5 0 0 0 0 0
1 1 |  0 0 6 0 0 0 0

   0 = t
0 0 | 0 1 1 1
0 1 | 0 0 1 1
1 0 | 0 0 1 1
1 1 | 1 1 0 1
take with rank

   3 $. 0 = t                 
1

   +/ , 0 = t
18
the sparse element of 0=t is 1

   +/ , 0 = d
18
answers are independent of representation

   0{t
0 | 13 0 0 0
1 | 21 4 0 0
from

   _2 (<1 2 3)}t
0 0 | 13 0 0  0
0 1 | 21 4 0  0
1 0 |  3 5 0  0
1 1 |  0 0 6  0
1 2 |  0 0 0 _2
amend

   ] p=: (i.!n) A. i.n=: 3
0 1 2
0 2 1
1 0 2
1 2 0
2 0 1
2 1 0
all permutations of order 3

   C.!.2 p
0 1 1 0 0 1
the parity of each permutation

   $.^:_1 (_1^C.!.2 p) (<"1 p)} 1 $. n$n
 0  0  0
 0  0  1
 0 _1  0

 0  0 _1
 0  0  0
 1  0  0

 0  1  0
_1  0  0
 0  0  0

The last expression computes the complete skewed tensor of order 3.

   s=: 1 $. 20 50 1000 75 366

   $s
20 50 1000 75 366
20 countries, 50 regions, 1000 salespersons,
75 products, 366 days in a year

   */ $s
2.745e10
the product over the shape can be greater than 2^31

   r=: ?. 1e5 $ 1e6
   i=: ?. 1e5 5 $ $ s
   s=: r (<"1 i)} s
revenues
corresponding locations
assign revenues to corresponding locations

   7 {. ": s
0 0  5 30 267 | 128133
0 0 26 20 162 | 319804
0 0 31 37 211 | 349445
0 0 37 10 351 | 765935
0 0 56  6  67 | 457449
0 0 66 54 120 |  38186
0 0 71 74 246 | 515473
the first 7 rows in the display of s ;
the first row says that for country 0, region 0,
salesperson 5, product 30, day 267,
the revenue was 128133

   +/ , s
|limit error
| +/    ,s
total revenue
the expression failed on ,s because it would have required a vector of length 2.745e10

   +/@, s                     
5.00289e10                    
total revenue
f/@, is supported by special code

   +/+/+/+/+/ s
5.00289e10

   +/^:5 s
5.00289e10

   +/^:_ s
5.00289e10

   +/ r
5.00289e10
total revenue

  +/"1^:4 s
 0 | 2.48411e9
 1 | 2.55592e9
 2 | 2.55103e9
 3 | 2.52089e9
 4 | 2.49225e9
 5 | 2.45682e9
 6 | 2.52786e9
 7 | 2.45425e9
 8 | 2.48729e9
 9 | 2.50094e9
10 | 2.51109e9
11 | 2.59601e9
12 | 2.49003e9
13 | 2.58199e9
14 | 2.44772e9
15 | 2.47863e9
16 | 2.46455e9
17 | 2.5568e9
18 | 2.43492e9
19 | 2.43582e9
total revenue by country

   t=: +/^:2 +/"1^:2 s

   $t
1000

   7{.t
0 | 4.58962e7
1 | 4.81548e7
2 | 3.97248e7
3 | 4.89981e7
4 | 4.85948e7
5 | 4.69227e7
6 | 4.22094e7
total revenue by salesperson

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 Sparse Linear Algebra

Currently, only sparse matrix multiplication and the solutions of tri-diagonal linear system are implemented. For example:

   f=: }. @ }: @ (,/) @ (,."_1 +/&_1 0 1) @ i.

   f 5
0 1
1 0
1 1
1 2
2 1
2 2
2 3
3 2
3 3
3 4
4 3
4 4
indices for a 5 by 5 tri-diagonal matrix

   s=: (?. 13$100) (<"1 f 5)} 1 $. 5 5;0 1

   $s
5 5

The phrase 1$.5 5;0 1 makes a sparse array with shape 5 5 and sparse axes 0 1 (sparse in both dimensions); <"1 f 5 makes boxed indices; and x (<"1 f 5)}y amends by x the locations in y indicated by the indices (scattered amendment).

   s
0 0 | 13
0 1 | 75
1 0 | 45
1 1 | 53
1 2 | 21
2 1 |  4
2 2 | 67
2 3 | 67
3 2 | 93
3 3 | 38
3 4 | 51
4 3 | 83
4 4 |  3

   ] d=: $.^:_1 s
13 75  0  0  0
45 53 21  0  0
 0  4 67 67  0
 0  0 93 38 51
 0  0  0 83  3
the dense representation of s

   ] y=: ?. 5$80
10 60 36 42 17 3 54 54 74 30 41 66 2

   y %. s
1.27885 _0.0883347 0.339681 0.202906 0.0529263

   y %. d
answers are independent of representation
1.27885 _0.0883347 0.339681 0.202906 0.0529263

   s=: (?. (_2+3*1e5)$1000) (<"1 f 1e5)} 1 $. 1e5 1e5;0 1

   $s
100000 100000

   y=: ?. 1e5$1000
s is a 1e5 by 1e5 matrix

   ts=: 6!:2 , 7!:2@]
time and space for execution

   ts 'y %. s'
0.28 5.2439e6                 

0.28 seconds; 5.2 megabytes (Pentium 266 Mhz)

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 Implementation Status

As of 1999-09-03 12:00 , the following facilities support sparse arrays:

= d =. =:
< <. <:
> >. >:
_ _. _:
 
+ +. +:
* *. *:
- -. m -:
% %. d %:
 
^ ^.
$ m $. $:
~ ~: d
| |. |:
 
.. .:
: :. ::
, ,. ,:
 
#
! !. !:
/ m
\ m \. m
 
[ [. [:
] ].
{ d {. {:
} d }. }:
 
" ". ": m
` `:
@ @. @:
& &. &:
 
j. m
o.
r. m
_9: to 9:
 
3!:0
3!:1
3!:2
3!:3
4!:55

Notes:

•  Sparse literal and boxed arrays not yet implemented.
The dyad %. only implements the case of triadiagonal matrices.
Boxed left arguments for |: (diagonal slices) not yet implemented.
The monads f/ and f/"r are only implemented for + * >. <. +. *. = ~: , (and only boolean arguments for = and ~:); on an axis of length 2, the monads f/ and f/"r are implemented for any function.
{ and } only accept the following index arguments: integer arrays, <"1 on integer arrays, and scalar boxed indices (respectively, item indexing, scattered indexing, and index lists a0;a1;a2;…).

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created:  1998-11-13 07:10
updated: 2019-03-09 11:50