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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`

:

```
def histogram [n] (k: i64) (is: [n]i64): [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`

.