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 [m] [n] : (dest: *[m]a) -> (f: a -> a -> a) -> (ne: a) -> (is: [n]i64) -> (as: [n]a) -> *[m]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 [m] [n]: (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 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 index in 0..length is-1:
i = is[index]
v = vs[index]
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. If is contains duplicates (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)