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Common Lisp library that facilitates working with Common Lisp arrays.

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README.org
This is an alpha release. All the code works and unit tests are expected to run perfectly, but the operations are not optimized and the API change.

Introduction

array-operations is a Common Lisp library that facilitates working with Common Lisp arrays using syntax and semantics that work well with the rest of the language.

The library previously available under this name is deprecated, but you can find it here.

A quick tour of the library

Shorthand for frequently used Common Lisp array functions

The library defines the following short function names that are synomyms for Common Lisp operations:

array-operations Common Lisp
size array-total-size
rank array-rank
dim array-dimension
dims array-dimensions
nrow number of rows in matrix
ncol number of columns in matrix

The array-operations package has the nickname ao, so you can use, for example, (ao:size my-array) without use‘ing the package.

Displaced arrays for fun and profit

displaced array n. an array which has no storage of its own, but which is instead indirected to the storage of another array, called its target, at a specified offset, in such a way that any attempt to access the displaced array implicitly references the target array. (CLHS Glossary)

Displaced arrays are one of the niftiest features of Common Lisp. When an array is displaced to another array, it shares structure with (part of) that array. The two arrays do not need to have the same dimensions, in fact, the dimensions do not be related at all as long as the displaced array fits inside the original one. The row-major index of the former in the latter is called the offset of the displacement.

Displaced arrays are usually constructed using make-array, but this library also provides displace for that purpose:

(defparameter *a* #2A((1 2 3) (4 5 6)))
(ao:displace *a* 2 1) ; => #(2 3)

flatten displaces to a row-major array:

(ao:flatten *a*) ; => #(1 2 3 4 5 6)

The real fun starts with split, which splits off subarrays nested within a given axis:

(ao:split *a* 1) ; => #(#(1 2 3) #(4 5 6))
(defparameter *b* #3A(((0 1) (2 3))
                      ((4 5) (6 7))))
(ao:split *b* 0) ; => #3A(((0 1) (2 3)) ((4 5) (6 7)))
(ao:split *b* 1) ; => #(#2A((0 1) (2 3)) #2A((4 5) (6 7)))
(ao:split *b* 2) ; => #2A((#(0 1) #(2 3)) (#(4 5) #(6 7)))
(ao:split *b* 3) ; => #3A(((0 1) (2 3)) ((4 5) (6 7)))

Note how splitting at 0 and the rank of the array returns the array itself.

Now consider sub, which returns a specific array, composed of the elements that would start with given subscripts:

(ao:sub *b* 0) ; => #2A((0 1) (2 3))
(ao:sub *b* 0 1) ; => #(2 3)
(ao:sub *b* 0 1 0) ; => 2

There is also a (setf sub) function.

partition returns a consecutive chunk of an array separated along its first subscript:

(ao:partition #2A((0 1)
                  (2 3)
                  (4 5)
                  (6 7)
                  (8 9))
              1 3) ; => #2A((2 3) (4 5))

and also has a (setf partition) pair.

combine is the opposite of split:

(ao:combine #(#(0 1) #(2 3))) ; => #2A((0 1) (2 3))

subvec returns a displaced subvector:

(ao:subvec #(0 1 2 3 4) 2 4) ; => #(2 3)

There is also a (setf subvec) function, which is like (setf subseq) except for demanding matching lengths.

Finally, reshape can be used to displace arrays into a different shape:

(ao:reshape *a* '(3 2)) ; => #2A((1 2) (3 4) (5 6))

You can use t for one of the dimensions, to be filled in automatically:

(ao:reshape *b* '(1 t)) ; => #2A((0 1 2 3 4 5 6 7))

reshape-col and reshape-row reshape your array into a column or row matrix, respectively.

Dimension specifications

Functions in the library accept the following in place of dimensions:

  • a list of dimensions (as for make-array),
  • a positive integer, which is used as a single-element list,
  • another array, the dimensions of which are used.

The last one allows you to specify dimensions with other arrays. For example, to reshape an array a1 to look like a2, you can use

(ao:reshape a1 a2)

instead of the longer form

(ao:reshape a1 (ao:dims a2))

Array creation and transformations

When the resulting element type cannot be inferred, functions that create and transform arrays are provided in pairs: one of these will allow you to specify the array-element-type of the result, while the other assumes it is t. The former ends with a *, and the element-type is always its first argument. I give examples for the versions without *, use the other when you are optimizing your code and you are sure you can constrain to a given element-type.

Element traversal order of these functions is unspecified. The reason for this is that the library may use parallel code in the future, so it is unsafe to rely on a particular element traversal order.

generate (and generate*) allow you to generate arrays using functions.

(ao:generate (lambda () (random 10)) 3) ; => #(6 9 5)
(ao:generate #'identity '(2 3) :position) ; => #2A((0 1 2) (3 4 5))
(ao:generate #'identity '(2 2) :subscripts)
;; => #2A(((0 0) (0 1)) ((1 0) (1 1)))
(ao:generate #'cons '(2 2) :position-and-subscripts)
;; => #2A(((0 0 0) (1 0 1)) ((2 1 0) (3 1 1)))

Depending on the last argument, the function will be called with the (row-major) position, the subscripts, both, or no argument.

permute can permutate subscripts (you can also invert, complement, and complete permutations, look at the docstring and the unit tests). Transposing is a special case of permute:

(ao:permute '(0 1) *a*) ; => #2A((1 2 3) (4 5 6))

each applies a function to its (array) arguments elementwise:

(ao:each #'+ #(0 1 2) #(2 3 5)) ; => #(2 4 7)

The semantics of margin are more difficult to explain, so perhaps an example will be more useful. Suppose that you want to calculate column sums in a matrix. You could permute (transpose) the matrix, split its subarrays at rank one (so you get a vector for each row), and apply the function that calculates the sum. margin automates that for you:

(ao:margin (lambda (column)
             (reduce #'+ column))
           #2A((0 1)
               (2 3)
               (5 7)) 0) ; => #(7 11)

But the function is much more general than this: the arguments inner and outer allow arbitrary permutations before splitting.

Finally, recycle allows you to recycle arrays along inner and outer dimensions:

(ao:recycle #(2 3) :inner 2 :outer 4)
; => #3A(((2 2) (3 3)) ((2 2) (3 3)) ((2 2) (3 3)) ((2 2) (3 3)))

Scalars as 0-dimensional arrays

Library functions treat non-array objects as if they were equivalent to 0-dimensional arrays: for example, (ao:split array (rank array)) returns an array that effectively equivalent (eq) to array. Another example is recycle:

(ao:recycle 4 :inner '(2 2)) ; => #2A((4 4) (4 4))

Stacking

You can also stack compatible arrays along any axis:

(defparameter *a1* #(0 1 2))
(defparameter *a2* #(3 5 7))
(ao:stack 0 *a1* *a2*) ; => #(0 1 2 3 5 7)
(ao:stack 1
          (ao:reshape-col *a1*)
          (ao:reshape-col *a2*)) ; => #2A((0 3) (1 5) (2 7))

Shared structure

Rules for that aren’t finalized yet, see the source. Suggestions are welcome.

To-do list

benchmark and optimize walk-subscripts and walk-subscripts-list

  • instead of allocating a new list each time, could map into a preallocated one
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