# Radix sort

Radix sort is a non-comparative sorting algorithm for sorting things that “behave” like numbers, in that they can be decomposed into digits. This makes it asymptotically more efficient than comparison-based sorts, and it also permits efficient GPU implementation.

The radix sort we will show here is very simple, and not as fast as it could be. The main reason is that it only processes a single digit at a time. Most high-performance implementations consider multiple bits at once. Also, we only sort 32-bit unsigned integers. Most of the work is done by a function `radix_sort_step`

, where `radix_sort_step xs b`

returns `xs`

with the elements sorted with respect to bit `b`

.

`def radix_sort_step [n] (xs: [n]u32) (b: i32): [n]u32 =`

To demonstrate how it works, suppose

```
-- xs = [2, 0, 2, 4, 2, 1, 5, 9]
-- b = 1
```

First, for each element, we compute whether bit `b`

is set.

```
let bits = map (\x -> (i32.u32 (x >> u32.i32 b)) & 1) xs
-- bits = [1, 0, 1, 0, 1, 0, 0, 0]
```

We will also need the negation (swapping 0s and 1s).

```
let bits_neg = map (1-) bits
-- bits_neg = [0, 1, 0, 1, 0, 1, 1, 1]
```

Compute how many elements do not have bit `b`

set.

```
let offs = reduce (+) 0 bits_neg
-- offs = 5
```

For those elements that do *not* have bit `b`

set, we use a prefix sum (`scan (+) 0`

) to compute their *1-indexed* positions in the final result. The elements that *do* have bit `b`

set are set to 0 by the `map`

.

```
let idxs0 = map2 (*) bits_neg (scan (+) 0 bits_neg)
-- idxs0 = [0, 1, 0, 2, 0, 3, 4, 5]
```

Similarly, compute the final positions for the elements that *do* have bit `b`

set - note that we also offset these by `offs`

.

```
let idxs1 = map2 (*) bits (map (+offs) (scan (+) 0 bits))
-- idxs1 = [6, 0, 7, 0, 8, 0, 0, 0]
```

Add `idxs0`

and `idxs1`

together. This will give a sensible result because they are never nonzero in the same position.

```
let idxs2 = map2 (+) idxs0 idxs1
-- idxs2 = [6, 1, 7, 2, 8, 3, 4, 5]
```

Our calculations have produced 1-indexed offsets, but Futhark arrays are 0-indexed, so decrement each element.

```
let idxs = map (\x->x-1) idxs2
-- idxs = [5, 0, 6, 1, 7, 2, 3, 4]
```

Finally, copy `xs`

(just to have an array of the right size and type) and scatter the elements of `xs`

with the indexes we computed. We also convert `idxs`

to `i64`

, as required by the type of `scatter`

.

```
let xs' = scatter (copy xs) (map i64.i32 idxs) xs
-- xs' = [0, 4, 1, 5, 9, 2, 2, 2]
in xs'
```

A full radix sort is then just sequentially looping through each bit position and apply the step function for each.

```
def radix_sort [n] (xs: [n]u32): [n]u32 =
loop xs for i < 32 do radix_sort_step xs i
```

A useful optimisation is to first check the position of the most significant bit in each array element, and cap the number of iterations to that.