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# Scans and reductions

The `reduce` function reduces an array to a single value by conceptually inserting a binary operator (or any two-parameter function) between each element.

``````    reduce (+) 0 [1, 2, 3, 4]
== [1 + 2 + 3 + 4]
== 10``````

We can use them to define a function for computing averages:

``````def average (xs: []f64) =
reduce (+) 0.0 xs / f64.i64 (length xs)``````

There are some restrictions to enable parallel execution. In an expression `reduce f ne xs`, the function `f` must be associative and have `ne` as neutral element. Intuitively, associativity means that we can move around the parantheses in an application:

``f (f x y) z == f x (f y z)``

It’s a bit easier to understand if we write the function as an infix operator instead:

``(x + y) + z == x + (y + z)``

`ne` being a neutral element means that it does not affect the result of the function:

``f x ne == f ne x == x``

As a simple example, 0 is the neutral element for addition, and 1 for multiplication.

If we pass `reduce` a function that is not associative, or does not have the provided neutral element, we will get wrong results at run-time. What’s worse, the compiler will not be able to detect that we messed up (it’s actually impossible in general), however techniques exist for testing associativity empirically. You can also invent a neutral element if necessary.

Scans (also called prefix sums) are similar to reductions, but rather than producing a single result, they produce an array of the same size as the input, where each element is conceptually a reduction of a prefix of the array:

``````    scan (+) 0 [1, 2, 3, 4]
== [reduce (+) 0 ,
reduce (+) 0 [1,2],
reduce (+) 0 [1,2,3],
reduce (+) 0 [1,2,3,4]]
== [1, 3, 6, 10]``````

Somewhat surprisingly, these can also be efficiently computed in parallel, and have the same restrictions with respect to associativity and a neutral element as `reduce`. For now, scans may look a bit exotic, and they certainly are, but we’ll return to them in other examples, as they are an important building block in advanced parallel algorithms.