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# Computing histograms

Mathematically, a histogram is a discretised representation of a probability distribution. A histogram computation takes as input a collection of elements, maps each to one of k bins, and counts the number of elements that fall into each bin (discarding invalid bins). In Futhark, histogram-like computations can be done with `reduce_by_index`:

``````let histogram [n] (k: i32) (is: [n]i32): [k]i32 =
let bins = replicate k 0
in reduce_by_index bins (+) 0 is (replicate n 1)``````

For example, `histogram 3 [0, 1, 3, 2, 1, 0, 0, 1]` produces `[3, 3, 1]`. Note that out-of-bounds bins (the `3`) are ignored.

`reduce_by_index` is a surprisingly flexible function. In imperative pseudocode, we can describe the behaviour of `reduce_by_index dest f ne is as` as:

``````for j < length is:
i = is[j]
a = as[j]
if i >= 0 && i < length dest:
dest[i] = f(dest[i], a)``````

The `f` function must be associative and have `ne` as its neutral element (like with scans and reductions). Furthermore, it must also be commutative, which simply means that `f x y == f y x`.