Array slicing

The Fortran 66 programmers were only able to take advantage of slicing matrices by row, and then only when passing that row to a subroutine:

      SUBROUTINE PRINT V(VEC, LEN)
        REAL VEC(*)
        PRINT *, (VEC(I), I = 1, LEN)
      END

      PROGRAM MAIN
        PARAMETER(LEN = 3)
        REAL MATRIX(LEN, LEN)
        DATA MATRIX/1, 1, 1, 2, 4, 8, 3, 9, 27/
        CALL PRINT V(MATRIX(1, 2), LEN)
      END

Result:

   2.000000       4.000000       8.000000

Note that there is no dope vector in FORTRAN 66 hence the length of the slice must also be passed as an argument - or some other means - to the SUBROUTINE. 1970s Pascal and C had similar restrictions.

Algol68 final report contains an early example of slicing, slices are specified in the form:

[lower bound:upper bound] ¢ for computers with extended character sets ¢

or:

(LOWER BOUND..UPPER BOUND) # FOR COMPUTERS WITH ONLY 6 BIT CHARACTERS. #

Both bounds are inclusive and can be omitted, in which case they default to the declared array bounds. Neither the stride facility, nor diagonal slice aliases are part of the revised report.

Examples:

[3, 3]real a := ((1, 1, 1), (2, 4, 8), (3, 9, 27)); # declaration of a variable matrix #
[,]  real c = ((1, 1, 1), (2, 4, 8), (3, 9, 27));   # constant matrix, the size is implied #

ref[]real row := a[2,];                    # alias/ref to a row slice #
ref[]real col2 = a[, 2];                   # permanent alias/ref to second column #

print ((a[:, 2], newline));                # second column slice #
print ((a[1⌈a, :], newline));              # last row slice #
print ((a[:, 2⌈a], newline));              # last column slice #
print ((a[:2, :2], newline));              # leading 2-by-2 submatrix "slice" #

+1.000010+0 +4.000010+0 +9.000010+0
+3.000010+0 +9.000010+0 +2.700010+1
+1.000010+0 +8.000010+0 +2.700010+1
+1.000010+0 +1.000010+0 +2.000010+0 +4.000010+0

Arrays in S and GNU R are always one-based, thus the indices of a new slice will begin with one for each dimension, regardless of the previous indices. Dimensions with length of one will be dropped (unless drop = FALSE). Dimension names (where present) will be preserved.

> A <- array(1:60, dim = c(3, 4, 5)) # 3x4x5 three-dimensional or cubic array
> A[, , 3] # 3x4 two-dimensional array along first and second dimensions
     [, 1] [, 2] [, 3] [, 4]
[1,]   25   28   31   34
[2,]   26   29   32   35
[3,]   27   30   33   36
> A[, 2:3, 3, drop = FALSE] # 3x2x1 cubic array subset (preserved dimensions)
, , 1

     [, 1] [, 2]
[1,]   28   31
[2,]   29   32
[3,]   30   33
> A[, 2, 3]  # single-dimension array along first dimension
[1] 28 29 30
> A[1, 2, 3] # single value
[1] 28

The Fortran 77 standard introduced the ability to slice and concatenate strings:

PROGRAM MAIN
  PRINT *, 'ABCDE'(2:4)
END

Produces:

BCD

Such strings could be passed by reference to another subroutine, the length would also be passed transparently to the subroutine as a kind of short dope vector.

SUBROUTINE PRINT S(STR)
  CHARACTER *(*)STR
  PRINT *, STR
END

PROGRAM MAIN
  CALL PRINT S('ABCDE'(2:4))
END

Again produces:

BCD

The standard ROM of the ZX80/81/Spectrum offers BASIC with the ability to slice and concatenate strings:

in the command part (x TO y) which points out the needed array slice the x and y value can be omitted giving the meaning to use all chained array cells (FROM x TO end ) or (begin TO y). With multidimensional arrays, the slicing is only possible with the last level dimension.

10 LET a$="ABCDE"(2 to 4)
20 PRINT a$

Produces:

BCD

10 LET a$="ABCDE"
20 LET b$=a$(4 TO)+a$(2 TO 3)+a$(1)
30 PRINT b$

Produces:

DEBCA

The Wikibook Ada_Programming has a page on the topic of: Types/array

Ada 83 supports slices for all array types. Like Fortran 77 such arrays could be passed by reference to another subroutine, the length would also be passed transparently to the subroutine as a kind of short dope vector.

with Text_IO;
 
procedure Main is
   Text : String := "ABCDE";
begin
   Text_IO.Put_Line (Text (2 .. 4));
end Main;

Produces:

BCD

Note: Since in Ada indices are n-based the term Text (2 .. 4) will result in an Array with the base index of 2.

The definition for Text_IO.Put_Line is:

package Ada.Text_IO is
   
   procedure Put_Line(Item : in  String);

The definition for String is:

package Standard is

   subtype Positive is Integer range 1 .. Integer'Last;

   type String is array(Positive range <>) of Character;
   pragma Pack(String);

As Ada supports true negative indices as in type History_Data_Array is array (-6000 .. 2010) of History_Data; it places no special meaning on negative indices. In the example above the term Some_History_Data (-30 .. 30) would slice the History_Data from 31 BC to 30 AD (since there was no year zero, the year number 0 actually refers to 1 BC).

If we have

@a = (2, 5, 7, 3, 8, 6, 4);

as above, then the first 3 elements, middle 3 elements and last 3 elements would be:

@a[0..2];   # (2, 5, 7)
@a[2..4];   # (7, 3, 8)
@a[-3..-1]; # (8, 6, 4)

Perl supports negative list indices. The -1 index is the last element, -2 the penultimate element, etc. In addition, Perl supports slicing based on expressions, for example:

@a[ 3.. $#a ];   # 4th element until the end (3, 8, 6, 4)
@a[ grep { !($_ % 3) } (0...$#a) ];    # 1st, 4th and 7th element (2,3,4)
@a[ grep { !(($_+1) % 3) } (0..$#a) ]; # every 3rd element (7,6)

If you have the following list:

>>> nums = [1, 3, 5, 7, 8, 13, 20]

Then it is possible to slice by using a notation similar to element retrieval:

>>> nums[3]   # no slicing
7
>>> nums[:3]  # from index 0 (inclusive) until index 3 (exclusive)
[1, 3, 5]
>>> nums[1:5]
[3, 5, 7, 8]
>>> nums[-3:]
[8, 13, 20]

Note that Python allows negative list indices. The index -1 represents the last element, -2 the penultimate element, etc. Python also allows a step property by appending an extra colon and a value. For example:

>>> nums[3:]
[7, 8, 13, 20]
>>> nums[3::] # == nums[3:]
[7, 8, 13, 20]
>>> nums[::3] # starting at index 0 and getting every third element
[1, 7, 20]
>>> nums[1:5:2] # from index 1 until index 5 and getting every second element
[3, 7]

The stride syntax (nums[1:5:2]) was introduced in the second half of the 1990s, as a result of requests put forward by scientific users in the Python "matrix-SIG" (special interest group).[2]

Slice semantics potentially differ per object; new semantics can be introduced when operator overloading the indexing operator. With Python standard lists (which are dynamic arrays), every slice is a copy. Slices of NumPy arrays, by contrast, are views onto the same underlying buffer.

In Fortran 90, slices are specified in the form

lower_bound:upper_bound[:stride]

Both bounds are inclusive and can be omitted, in which case they default to the declared array bounds. Stride defaults to 1. Example:

real, dimension(m, n):: a  ! declaration of a matrix
  
print *, a(:, 2) ! second column
print *, a(m, :) ! last row
print *, a(:10, :10) ! leading 10-by-10 submatrix

Each dimension of an array value in Analytica is identified by an Index variable. When slicing or subscripting, the syntax identifies the dimension(s) over which you are slicing or subscripting by naming the dimension. Such as:

Index I := 1..5   { Definition of a numerical Index }
Index J := ['A', 'B', 'C'] { Definition of a text-valued Index }
Variable X := Array(I, J, [[10, 20, 30], [1, 2, 3], ....]) { Definition of a 2D value }
X[I = 1, J = 'B']  -> 20  { Subscript to obtain a single value }
X[I = 1] ->  Array(J, [10, 20, 30])  { Slice out a 1D array. }
X[J = 2] -> Array(I, [20, 2, ....]) { Slice out a 1D array over the other dimension. }
X[I = 1..3]  {Slice out first four elements over I with all elements over J}

Naming indexes in slicing and subscripting is similar to naming parameters in function calls instead of relying on a fixed sequence of parameters. One advantage of naming indexes in slicing is that the programmer does not have to remember the sequence of Indexes, in a multidimensional array. A deeper advantage is that expressions generalize automatically and safely without requiring a rewrite when the number of dimensions of X changes.

Array slicing was introduced in version 1.0. Earlier versions did not support this feature.

Suppose that A is a 1-d array such as

    A = [1:50];           % A = [1, 2, 3, ...49, 50]

Then an array B of first 5 elements of A may be created using

    B = A[[:4]];

Similarly, B may be assigned to an array of the last 5 elements of A via:

    B = A[[-5:]];

Other examples of 1-d slicing include:

    A[-1]                 % The last element of A
    A[*]                  % All elements of A
    A[[::2]]              % All even elements of A
    A[[1::2]]             % All odd elements of A
    A[[-1::-2]]           % All even elements in the reversed order
    A[[[0:3], [10:14]]]   % Elements 0-3 and 10-14

Slicing of higher-dimensional arrays works similarly:

    A[-1, *]              % The last row of A
    A[[1:5], [2:7]]       % 2d array using rows 1-5 and columns 2-7
    A[[5:1:-1], [2:7]]    % Same as above except the rows are reversed

Array indices can also be arrays of integers. For example, suppose that I = [0:9] is an array of 10 integers. Then A[I] is equivalent to an array of the first 10 elements of A. A practical example of this is a sorting operation such as:

    I = array_sort(A);    % Obtain a list of sort indices
    B = A[I];             % B is the sorted version of A
    C = A[array_sort(A)]; % Same as above but more concise.

Consider the array:

int[] a = [2, 5, 7, 3, 8, 6, 4, 1];

Take a slice out of it:

int[] b = a[2 .. 5];

and the contents of b will be [7, 3, 8]. The first index of the slice is inclusive, the second is exclusive.

auto c = a[$ - 4 .. $ - 2];

means that the dynamic array c now contains [8, 6] because inside the [] the $ symbol refers to the length of the array.

D array slices are aliased to the original array, so:

b[2] = 10;

means that a now has the contents [2, 5, 7, 3, 10, 6, 4, 1]. To create a copy of the array data, instead of only an alias, do:

auto b = a[2 .. 5].dup;

Unlike Python, D slice bounds don't saturate, so code equivalent to this Python code is an error in D:

>>> d = [10, 20, 30]
>>> d[1 : 5]
[20, 30]

The programming language SuperCollider implements some concepts from J/APL. Slicing looks as follows:

a = [3, 1, 5, 7]           // assign an array to the variable a
a[0..1]                    // return the first two elements of a
a[..1]                     // return the first two elements of a: the zero can be omitted
a[2..]                     // return the element 3 till last one
a[[0, 3]]                  // return the first and the fourth element of a

a[[0, 3]] = [100, 200]     // replace the first and the fourth element of a
a[2..] = [100, 200]        // replace the two last elements of a

// assign a multidimensional array to the variable a
a = [[0, 1, 2, 3, 4], [5, 6, 7, 8, 9], [10, 11, 12, 13, 14], [15, 16, 17, 18, 19]]; 
a.slice(2, 3);             // take a slice with coordinates 2 and 3 (returns 13)
a.slice(nil, 3);           // take an orthogonal slice (returns [3, 8, 13, 18])

Arrays in fish are always one-based, thus the indices of a new slice will begin with one, regardless of the previous indices.

> set A (seq 3 2 11)       # $A is an array with the values 3, 5, 7, 9, 11
 
> echo $A[(seq 2)]         # Print the first two elements of $A 
3 5
 
> set B $A[1 2]            # $B contains the first and second element of $A, i.e. 3, 5
 
> set -e A[$B]; echo $A    # Erase the third and fifth elements of $A, print $A
3 5 9

Cobra supports Python-style slicing. If you have a list

nums = [1, 3, 5, 7, 8, 13, 20]

, then the first 3 elements, middle 3 elements, and last 3 elements would be:

nums[:3]   # equals [1, 3, 5]
nums[2:5]  # equals [5, 7, 8]
nums[-3:]  # equals [8, 13, 20]

Cobra also supports slicing-style syntax for 'numeric for loops':

for i in 2 : 5
    print i
# prints 2, 3, 4

for j in 3
    print j
# prints 0, 1, 2

Arrays are zero-based in PowerShell and can be defined using the comma operator:

PS  > $a = 2, 5, 7, 3, 8, 6, 4, 1
PS  > # Print the first two elements of $a:
PS  > "$($a[0, 1])"
2 5
PS  > # Take a slice out of it using the range operator:
PS  > "$($a[2..5])"
7 3 8 6
PS  > # Get the last 3 elements:
PS  > "$($a[-3..-1])"
6 4 1
PS  > # Return the content of the array in reverse order:
PS  > "$($a[($a.Length - 1)..0])" # Length is a property of System.Object[]
1 4 6 8 3 7 5 2

Go supports Python-style syntax for slicing (except negative indices are not supported). Arrays and slices can be sliced. If you have a slice

nums := []int{1, 3, 5, 7, 8, 13, 20}

then the first 3 elements, middle 3 elements, last 3 elements, and a copy of the entire slice would be:

nums[:3]  // equals []int{1, 3, 5}
nums[2:5] // equals []int{5, 7, 8}
nums[4:]  // equals []int{8, 13, 20}
nums[:]   // equals []int{1, 3, 5, 7, 8, 13, 20}

Slices in Go are reference types, which means that different slices may refer to the same underlying array.

Cilk Plus supports syntax for array slicing as an extension to C and C++.

array_base [lower_bound:length[:stride]]*

Cilk Plus slicing looks as follows:

A[:]     // All of vector A
B[2:6]   // Elements 2 to 7 of vector B
C[:][5]  // Column 5 of matrix C
D[0:3:2] // Elements 0, 2, 4 of vector D

Cilk Plus's array slicing differs from Fortran's in two ways:

• the second parameter is the length (number of elements in the slice) instead of the upper bound, in order to be consistent with standard C libraries;
• slicing never produces a temporary, and thus never needs to allocate memory. Assignments are required to be either non-overlapping or perfectly overlapping, otherwise the result is undefined.

Julia array slicing is like that of Matlab, but uses square brackets. Example:

julia> x = rand(4, 3)
4x3 Array{Float64,2}:
 0.323877  0.186253  0.600605
 0.404664  0.894781  0.0955007
 0.223562  0.18859   0.120011
 0.149316  0.779823  0.0690126

julia> x[:, 2]                # get the second column.
4-element Array{Float64,1}:
 0.186253
 0.894781
 0.18859
 0.779823

julia> x[1, :]                # get the first row.
1x3 Array{Float64,2}:
 0.323877  0.186253  0.600605

julia> x[1:2,2:3]             # get the submatrix spanning rows 1,2 and columns 2,3
2x2 Array{Float64,2}:
 0.186253  0.600605
 0.894781  0.0955007