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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.