## Abstract

Various Second-Order Array Combinators that are operationally parallel in a way that can be exploited by the compiler.

The functions here are all recognised specially by the compiler (or
built on those that are). The asymptotic work and
span
is provided for each function, but note that this easily hides very
substantial constant factors. For example, `scan`

is *much*
slower than `reduce`

, although they have the same asymptotic
complexity.

**Higher-order complexity**

Specifying the time complexity of higher-order functions is tricky
because it depends on the functional argument. We use the informal
convention that *W(f)* denotes the largest (asymptotic) *work* of
function *f*, for the values it may be applied to. Similarly,
*S(f)* denotes the largest span. See this Wikipedia
article
for a general introduction to these constructs.

**Reminder on terminology**

A function `op`

is said to be *associative* if

```
(x `op` y) `op` z == x `op` (y `op` z)
```

for all `x`

, `y`

, `z`

. Similarly, it is *commutative* if

```
x `op` y == y `op` x
```

The value `o`

is a *neutral element* if

```
x `op` o == o `op` x == x
```

for any `x`

.

## Synopsis

val map | 'a [n] 'x : | (f: a -> x) -> (as: [n]a) -> *[n]x |

val map1 | 'a [n] 'x : | (f: a -> x) -> (as: [n]a) -> *[n]x |

val map2 | 'a 'b [n] 'x : | (f: a -> b -> x) -> (as: [n]a) -> (bs: [n]b) -> *[n]x |

val map3 | 'a 'b 'c [n] 'x : | (f: a -> b -> c -> x) -> (as: [n]a) -> (bs: [n]b) -> (cs: [n]c) -> *[n]x |

val map4 | 'a 'b 'c 'd [n] 'x : | (f: a -> b -> c -> d -> x) -> (as: [n]a) -> (bs: [n]b) -> (cs: [n]c) -> (ds: [n]d) -> *[n]x |

val map5 | 'a 'b 'c 'd 'e [n] 'x : | (f: a -> b -> c -> d -> e -> x) -> (as: [n]a) -> (bs: [n]b) -> (cs: [n]c) -> (ds: [n]d) -> (es: [n]e) -> *[n]x |

val reduce | [n] 'a : | (op: a -> a -> a) -> (ne: a) -> (as: [n]a) -> a |

val reduce_comm | [n] 'a : | (op: a -> a -> a) -> (ne: a) -> (as: [n]a) -> a |

val reduce_by_index | 'a [n] [m] : | (dest: *[m]a) -> (f: a -> a -> a) -> (ne: a) -> (is: [n]i64) -> (as: [n]a) -> *[m]a |

val reduce_by_index_2d | 'a [n] [m] [k] : | (dest: *[m][k]a) -> (f: a -> a -> a) -> (ne: a) -> (is: [n](i64, i64)) -> (as: [n]a) -> *[m][k]a |

val reduce_by_index_3d | 'a [n] [m] [k] [l] : | (dest: *[m][k][l]a) -> (f: a -> a -> a) -> (ne: a) -> (is: [n](i64, i64, i64)) -> (as: [n]a) -> *[m][k][l]a |

val scan | [n] 'a : | (op: a -> a -> a) -> (ne: a) -> (as: [n]a) -> *[n]a |

val filter | [n] 'a : | (p: a -> bool) -> (as: [n]a) -> *[]a |

val partition | [n] 'a : | (p: a -> bool) -> (as: [n]a) -> ([]a, []a) |

val partition2 | [n] 'a : | (p1: a -> bool) -> (p2: a -> bool) -> (as: [n]a) -> ([]a, []a, []a) |

val reduce_stream | [n] 'a 'b : | (op: b -> b -> b) -> (f: (k: i64) -> [k]a -> b) -> (as: [n]a) -> b |

val reduce_stream_per | [n] 'a 'b : | (op: b -> b -> b) -> (f: (k: i64) -> [k]a -> b) -> (as: [n]a) -> b |

val map_stream | [n] 'a 'b : | (f: (k: i64) -> [k]a -> [k]b) -> (as: [n]a) -> *[n]b |

val map_stream_per | [n] 'a 'b : | (f: (k: i64) -> [k]a -> [k]b) -> (as: [n]a) -> *[n]b |

val all | [n] 'a : | (f: a -> bool) -> (as: [n]a) -> bool |

val any | [n] 'a : | (f: a -> bool) -> (as: [n]a) -> bool |

val scatter | 't [m] [n] : | (dest: *[m]t) -> (is: [n]i64) -> (vs: [n]t) -> *[m]t |

val scatter_2d | 't [m] [n] [l] : | (dest: *[m][n]t) -> (is: [l](i64, i64)) -> (vs: [l]t) -> *[m][n]t |

val scatter_3d | 't [m] [n] [o] [l] : | (dest: *[m][n][o]t) -> (is: [l](i64, i64, i64)) -> (vs: [l]t) -> *[m][n][o]t |

## Description

- ↑val map 'a [n] 'x: (f: a -> x) -> (as: [n]a) -> *[n]x
Apply the given function to each element of an array.

**Work:***O(n ✕ W(f))***Span:***O(S(f))*- ↑val map1 'a [n] 'x: (f: a -> x) -> (as: [n]a) -> *[n]x
Apply the given function to each element of a single array.

**Work:***O(n ✕ W(f))***Span:***O(S(f))*- ↑val map2 'a 'b [n] 'x: (f: a -> b -> x) -> (as: [n]a) -> (bs: [n]b) -> *[n]x
As

`map1`

, but with one more array.**Work:***O(n ✕ W(f))***Span:***O(S(f))*- ↑val map3 'a 'b 'c [n] 'x: (f: a -> b -> c -> x) -> (as: [n]a) -> (bs: [n]b) -> (cs: [n]c) -> *[n]x
As

`map2`

, but with one more array.**Work:***O(n ✕ W(f))***Span:***O(S(f))*- ↑val map4 'a 'b 'c 'd [n] 'x: (f: a -> b -> c -> d -> x) -> (as: [n]a) -> (bs: [n]b) -> (cs: [n]c) -> (ds: [n]d) -> *[n]x
As

`map3`

, but with one more array.**Work:***O(n ✕ W(f))***Span:***O(S(f))*- ↑val map5 'a 'b 'c 'd 'e [n] 'x: (f: a -> b -> c -> d -> e -> x) -> (as: [n]a) -> (bs: [n]b) -> (cs: [n]c) -> (ds: [n]d) -> (es: [n]e) -> *[n]x
As

`map3`

, but with one more array.**Work:***O(n ✕ W(f))***Span:***O(S(f))*- ↑val reduce [n] 'a: (op: a -> a -> a) -> (ne: a) -> (as: [n]a) -> a
Reduce the array

`as`

with`op`

, with`ne`

as the neutral element for`op`

. The function`op`

must be associative. If it is not, the return value is unspecified. If the value returned by the operator is an array, it must have the exact same size as the neutral element, and that must again have the same size as the elements of the input array.**Work:***O(n ✕ W(op))***Span:***O(log(n) ✕ W(op))*Note that the complexity implies that parallelism in the combining operator will

*not*be exploited.- ↑val reduce_comm [n] 'a: (op: a -> a -> a) -> (ne: a) -> (as: [n]a) -> a
As

`reduce`

, but the operator must also be commutative. This is potentially faster than`reduce`

. For simple built-in operators, like addition, the compiler already knows that the operator is commutative, so plain`reduce`

will work just as well.**Work:***O(n ✕ W(op))***Span:***O(log(n) ✕ W(op))*- ↑val reduce_by_index 'a [n] [m]: (dest: *[m]a) -> (f: a -> a -> a) -> (ne: a) -> (is: [n]i64) -> (as: [n]a) -> *[m]a
`reduce_by_index dest f ne is as`

returns`dest`

, but with each element given by the indices of`is`

updated by applying`f`

to the current value in`dest`

and the corresponding value in`as`

. The`ne`

value must be a neutral element for`f`

. If`is`

has duplicates,`f`

may be applied multiple times, and hence must be associative and commutative. Out-of-bounds indices in`is`

are ignored.**Work:***O(n ✕ W(op))***Span:***O(n ✕ W(op))*in the worst case (all updates to same position), but*O(W(op))*in the best case.In practice, the

*O(n)*behaviour only occurs if*m*is also very large.- ↑val reduce_by_index_2d 'a [n] [m] [k]: (dest: *[m][k]a) -> (f: a -> a -> a) -> (ne: a) -> (is: [n](i64, i64)) -> (as: [n]a) -> *[m][k]a
As

`reduce_by_index`

, but with two-dimensional indexes.- ↑val reduce_by_index_3d 'a [n] [m] [k] [l]: (dest: *[m][k][l]a) -> (f: a -> a -> a) -> (ne: a) -> (is: [n](i64, i64, i64)) -> (as: [n]a) -> *[m][k][l]a
As

`reduce_by_index`

, but with three-dimensional indexes.- ↑val scan [n] 'a: (op: a -> a -> a) -> (ne: a) -> (as: [n]a) -> *[n]a
Inclusive prefix scan. Has the same caveats with respect to associativity and complexity as

`reduce`

.**Work:***O(n ✕ W(op))***Span:***O(log(n) ✕ W(op))*- ↑val filter [n] 'a: (p: a -> bool) -> (as: [n]a) -> *[]a
Remove all those elements of

`as`

that do not satisfy the predicate`p`

.**Work:***O(n ✕ W(p))***Span:***O(log(n) ✕ W(p))*- ↑val partition [n] 'a: (p: a -> bool) -> (as: [n]a) -> ([]a, []a)
Split an array into those elements that satisfy the given predicate, and those that do not.

**Work:***O(n ✕ W(p))***Span:***O(log(n) ✕ W(p))*- ↑val partition2 [n] 'a: (p1: a -> bool) -> (p2: a -> bool) -> (as: [n]a) -> ([]a, []a, []a)
Split an array by two predicates, producing three arrays.

**Work:***O(n ✕ (W(p1) + W(p2)))***Span:***O(log(n) ✕ (W(p1) + W(p2)))*- ↑val reduce_stream [n] 'a 'b: (op: b -> b -> b) -> (f: (k: i64) -> [k]a -> b) -> (as: [n]a) -> b
`reduce_stream op f as`

splits`as`

into chunks, applies`f`

to each of these in parallel, and uses`op`

(which must be associative) to combine the per-chunk results into a final result. The`i64`

passed to`f`

is the size of the chunk. This SOAC is useful when`f`

can be given a particularly work-efficient sequential implementation. Operationally, we can imagine that`as`

is divided among as many threads as necessary to saturate the machine, with each thread operating sequentially.A chunk may be empty, and

`f 0 []`

must produce the neutral element for`op`

.**Work:***O(n ✕ W(op) + W(f))***Span:***O(log(n) ✕ W(op))*- ↑val reduce_stream_per [n] 'a 'b: (op: b -> b -> b) -> (f: (k: i64) -> [k]a -> b) -> (as: [n]a) -> b
As

`reduce_stream`

, but the chunks do not necessarily correspond to subsequences of the original array (they may be interleaved).**Work:***O(n ✕ W(op) + W(f))***Span:***O(log(n) ✕ W(op))*- ↑val map_stream [n] 'a 'b: (f: (k: i64) -> [k]a -> [k]b) -> (as: [n]a) -> *[n]b
Similar to

`reduce_stream`

, except that each chunk must produce an array*of the same size*. The per-chunk results are concatenated.**Work:***O(n ✕ W(f))***Span:***O(S(f))*- ↑val map_stream_per [n] 'a 'b: (f: (k: i64) -> [k]a -> [k]b) -> (as: [n]a) -> *[n]b
Similar to

`map_stream`

, but the chunks do not necessarily correspond to subsequences of the original array (they may be interleaved).**Work:***O(n ✕ W(f))***Span:***O(S(f))*- ↑val all [n] 'a: (f: a -> bool) -> (as: [n]a) -> bool
Return

`true`

if the given function returns`true`

for all elements in the array.**Work:***O(n ✕ W(f))***Span:***O(log(n) + S(f))*- ↑val any [n] 'a: (f: a -> bool) -> (as: [n]a) -> bool
Return

`true`

if the given function returns`true`

for any elements in the array.**Work:***O(n ✕ W(f))***Span:***O(log(n) + S(f))*- ↑val scatter 't [m] [n]: (dest: *[m]t) -> (is: [n]i64) -> (vs: [n]t) -> *[m]t
`scatter as is vs`

calculates the equivalent of this imperative code:`for j in 0...length is-1: i = is[j] v = vs[j] if i >= 0 && i < length as: as[i] = v`

The

`is`

and`vs`

arrays must have the same outer size.`scatter`

acts in-place and consumes the`as`

array, returning a new array that has the same type and elements as`as`

, except for the indices in`is`

. Notice that writing outside the index domain of the target array has no effect. If`is`

contains duplicates for valid indexes into the target array (i.e., several writes are performed to the same location), the result is unspecified. It is not guaranteed that one of the duplicate writes will complete atomically - they may be interleaved. See`reduce_by_index`

for a function that can handle this case deterministically.This is technically not a second-order operation, but it is defined here because it is closely related to the SOACs.

**Work:***O(n)***Span:***O(1)*- ↑val scatter_2d 't [m] [n] [l]: (dest: *[m][n]t) -> (is: [l](i64, i64)) -> (vs: [l]t) -> *[m][n]t
`scatter_2d as is vs`

is the equivalent of a`scatter`

on a 2-dimensional array.**Work:***O(n)***Span:***O(1)*- ↑val scatter_3d 't [m] [n] [o] [l]: (dest: *[m][n][o]t) -> (is: [l](i64, i64, i64)) -> (vs: [l]t) -> *[m][n][o]t
`scatter_3d as is vs`

is the equivalent of a`scatter`

on a 3-dimensional array.**Work:***O(n)***Span:***O(1)*