pick.md

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Pick

Pick (⊑) chooses elements from 𝕩 based on index lists from 𝕨. 𝕨 can be a plain list, or even one number if 𝕩 is a list, in order to get one element from 𝕩. It can also be an array of index lists, or have deeper array structure: each index list will be replaced with the element of 𝕩 at that index, effectively applying to 𝕨 at depth 1.

The one-argument form is called First, and ⊑𝕩 takes the first element of 𝕩 in index order, with an error if 𝕩 is empty.

While sometimes "scatter-point" indexing is necessary, using Pick to select multiple elements from 𝕩 is less array-oriented than Select (⊏), and probably slower. Consider rearranging your data so that you can select along axes instead of picking out elements.

One element

When the left argument is a number, Pick gets an element from a list:

    2 ⊑ 0‿1‿2‿3‿4
    2 ⊑ "abc"
    2 ⊑ ⟨@, 0‿1‿2‿3, "abc"⟩

A negative number 𝕨 behaves like 𝕨+≠𝕩, so that ¯1 will select the last element, and -≠𝕩 the first. A number in 𝕨 must be an integer less than ≠𝕩 but not less than -≠𝕩.

    ¯2 ⊑ 0‿1‿2‿3‿4
    ¯2 ⊑ "abc"

Making 𝕩 a list is only a special case. In general 𝕨 can be a list of numbers whose length is 𝕩's rank. So when =𝕩 is 1, 𝕨 can be length-1 list. The case above where 𝕨 is a number is a simplification, but an enclosed number 𝕨 isn't allowed because it could be confused with the nested case described below.

    ⟨2,0⟩ ⊑ ↕4‿5

Above we see that picking from the result of Range gives the index. For something slightly more interesting, here's a character array:

    ⊢ a ← 'a' + ⥊⟜(↕×´) 4‿5
    2‿0 ⊑ a
    1‿¯1 ⊑ a

𝕩 can even be a unit. By definition it has rank 0, so the only possible value for 𝕨 is the empty list. This extracts an enclosed element, and returns an atom unchanged—the atom is promoted to an array by enclosing it, then the action of Pick undoes this. But there's rarely a reason to use this case, because the monadic form First accomplishes the same thing.

    ⟨⟩ ⊑ <'a'
    ⟨⟩ ⊑ 'a'

First

With no left argument, ⊑ is called First, and is the same as Pick with a default left argument 0¨≢𝕩. For a non-empty array it returns the first element in index order.

    ⊑ <'a'
    ⊑ "First"
    ⊑ ↕4‿2‿5‿1

And if 𝕩 is empty then First results in an error.

    ⊑ ""

    ⊑ ≢π

In APL it's common to get the last element of a list with an idiom that translates to ⊑⌽, or First-Reverse. In BQN the most straightforward way is to select with index ¯1 instead. I also sometimes use Fold with the Right identity function.

    ⊑⌽ "last"
    ¯1⊑ "last"
    ⊢´ "last"

Many elements

Pick also accepts a list of indices:

    a  # Defined above

    ⟨2‿0, 1‿¯1, 3‿1, ¯1‿¯1⟩ ⊑ a

These indices have to be lists, since if they're numbers it just looks like 𝕨 is an index list for one element.

    ⟨2,1,0,¯1⟩ ⊑ "abc"  # 𝕩 doesn't have rank 4!

    ⟨2,1,0,¯1⟩ ⥊¨⊸⊑ "abc"

    ⟨2,1,0,¯1⟩ ⊏ "abc"  # Better way

It's much more general than just a list of indices though. As long as your indices are lists, you can arrange them in any array structure with arbitrary nesting.

    ⟨2‿0, ⟨⟨1‿¯1, 3‿1⟩, ¯1‿¯1⟩⟩ ⊑ a

    (⟨2‿0, 1‿¯1⟩≍⟨3‿1, ¯1‿¯1⟩) ⊑ a

    (⟨2‿0, <1‿¯1⟩≍⟨<3‿1, ¯1‿¯1⟩) ⊑ a

This option is easily described using the Depth modifier. Pick applies to depth-1 components of the left argument and the entire right argument, which corresponds to a depth operand of 1‿∞. The left argument components have to be lists of numbers, or Pick gives an error.

    (⟨2‿0, <1‿¯1⟩≍⟨<3‿1, ¯1‿¯1⟩) ⊑⚇1‿∞ a

    ⟨⟨2,3⟩,1⟩ ⊑ a  # 1 isn't a valid index