# 7. Regular Flattening¶

In this chapter, we introduce the concept of regular *moderate
flattening* [HSE+17], which is the
essential technique used for making regular nested parallel Futhark
programs run efficiently in practice on parallel hardware such as
GPUs.

We first introduce a number of parallel segmented operations, which
are essential for dealing with nested parallelism. The segmented
operations, it turns out, can be implemented using Futhark’s standard
SOAC parallel array combinators. In particular, it turns out that the
`scan`

operator is of critical importance in that it can be used to
develop the notion of a *segmented scan* operation, an operation that,
in its own right, is essential to many parallel algorithms. Based on
the segmented scan operation and the other Futhark SOAC operations, we
present a set of utility functions as well as their parallel
implementations. The functions are used by the moderate flattening
transformation presented in Section 7.6, but are also
useful, as we shall see in Section 9, for the
programmer to manage irregular parallelism through flattening
transformations, performed manually by the programmer.

## 7.1. Segmented Scan¶

As mentioned, the segmented scan operation is quite essential for Futhark to flatten nested regular parallelism and for the programmer to flatten irregular nested parallel problems. The operation can be implemented with a simple scan using an associative function that operates on pairs of values [Sch80][Ble90]. Here is the definition of the segmented scan operation, hardcoded to work with addition:

```
-- Segmented scan with integer addition
let segmented_scan_add [n] (flags:[n]bool) (vals:[n]i32) : [n]i32 =
let pairs = scan ( \(v1,f1) (v2,f2) ->
let f = f1 || f2
let v = if f2 then v2 else v1+v2
in (v,f) ) (0,false) (zip vals flags)
let (res,_) = unzip pairs
in res
```

We can make use of Futhark’s support for higher-order functions and
polymorphism to define a generic version of segmented scan that will
work for other monoidal structures than addition on `i32`

values:

```
let segmented_scan 't [n] (g:t->t->t) (ne: t) (flags: [n]bool) (vals: [n]t): [n]t =
let pairs = scan ( \ (v1,f1) (v2,f2) ->
let f = f1 || f2
let v = if f2 then v2 else g v1 v2
in (v,f) ) (ne,false) (zip vals flags)
let (res,_) = unzip pairs
in res
```

We leave it up to the reader to prove that, given an associative
function `g`

, (1) the operator passed to `scan`

is associative
and (2) `(ne, false)`

is a neutral element for the operator.

### 7.1.1. Finding the Longest Streak Using Segmented Scan¶

In this section we revisit the problem of Section 6.4 for finding the longest streak of increasing numbers. We show how we can make direct use of a segmented scan operation for solving the problem:

```
-- Longest streak of increasing numbers
let segmented_streak [n] (xs: [n]i32) : i32 =
let ys = rotate 1 xs
let is = (map2 (\x y -> if x < y then 1 else 0) xs ys)[0:n-1]
let fs = map (==0) is
let ss = segmented_scan_add fs is
let res = reduce max 0 ss
in res
```

The algorithm first constructs the `is`

array, as in the previous
algorithm, and then uses a segmented scan over a negation of this array
over the unit-array to create the `ss3`

vector directly. Here is a
derivation of how the segmented-scan based algorithm works:

Variable | |||||||||
---|---|---|---|---|---|---|---|---|---|

`xs` |
= | 1 | 5 | 3 | 4 | 2 | 6 | 7 | 8 |

`ys` |
= | 5 | 3 | 4 | 2 | 6 | 7 | 8 | 1 |

`is` |
= | 1 | 0 | 1 | 0 | 1 | 1 | 1 | |

`fs` |
= | 0 | 1 | 0 | 1 | 0 | 0 | 0 | |

`ss` |
= | 1 | 0 | 1 | 0 | 1 | 2 | 3 | |

`res` |
= | 3 |

The morale here is that the segmented scan operation provides us with a great abstraction.

## 7.2. Replicated Iota¶

The first utility function that we will present is called
`replicated_iota`

. Given an array of natural numbers specifying
repetitions, the function returns an array of weakly increasing
indices (starting from 0) and with each index repeated according to
the repetition array. As an example, `replicated_iota [2,3,1,1]`

returns the array `[0,0,1,1,1,2,3]`

. The function is defined in
terms of other parallel operations, including `scan`

, `map`

,
`scatter`

, and `segmented_scan`

:

```
let replicated_iota [n] (reps:[n]i32) : []i32 =
let s1 = scan (+) 0 reps
let s2 = map (\i -> if i==0 then 0 else s1[i-1]) (iota n)
let tmp = scatter (replicate (reduce (+) 0 reps) 0) s2 (iota n)
let flags = map (>0) tmp
in segmented_scan (+) 0 flags tmp
```

An example evaluation of a call to the function `replicated_iota`

is
provided below.

Args/Result | ||||||||
---|---|---|---|---|---|---|---|---|

`reps` |
= | 2 | 3 | 1 | 1 | |||

`s1` |
= | 2 | 5 | 6 | 7 | |||

`s2` |
= | 0 | 2 | 5 | 6 | |||

`replicate` |
= | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

`tmp` |
= | 0 | 0 | 1 | 0 | 0 | 2 | 3 |

`flags` |
= | 0 | 0 | 1 | 0 | 0 | 1 | 1 |

`segmented_scan` |
= | 0 | 0 | 1 | 1 | 1 | 2 | 3 |

## 7.3. Segmented Replicate¶

Another useful utility function is called
`segmented_replicate`

. Given a one-dimensional replication array
containing natural numbers and a data array of the same shape,
`segmented_replicate`

returns an array of size equal to the sum of
the values in the replication array with values from the data array
replicated according to the corresponding replication values. As an
example, a call `segmented_replicate [2,1,0,3,0] [5,6,9,8,4]`

result
in the array `[5,5,6,8,8,8]`

. Here is the code that implements the
function `segmented_replicate`

:

```
let segmented_replicate [n] (reps:[n]i32) (vs:[n]i32) : []i32 =
let idxs = replicated_iota reps
in map (\i -> vs[i]) idxs
```

The `segmented_replicate`

function makes use of the previously
defined function `replicated_iota`

.

## 7.4. Segmented Iota¶

Another useful utility function is the function `segmented_iota`

that, given a array of flags (i.e., booleans), returns an array of
index sequences, each of which is reset according to the booleans in
the array of flags. As an example, the expression:

```
segmented_iota [false,false,false,true,false,false,false]
```

returns the array `[0,1,2,0,1,2,3]`

. The `segmented_iota`

function
can be implemented with the use of a simple call to `segmented_scan`

followed by a call to `map`

:

```
let segmented_iota [n] (flags:[n]bool) : [n]i32 =
let iotas = segmented_scan (+) 0 flags (replicate n 1)
in map (\x -> x-1) iotas
```

## 7.5. Indexes to Flags¶

Many segmented operations, such as `segmented_scan`

takes as
argument an array of boolean flags for specifying when new segments
start. Often, only the sizes of segments are known, which means that
it may come in useful to be able to transform an array of segment
sizes to a corresponding array of boolean flags. Here is one possible
parallel implementation of such an `idxs_to_flags`

function:

```
let idxs_to_flags [n] (is : [n]i32) : []bool =
let vs = segmented_replicate is (iota n)
let m = length vs
```

As an example use of the function, the expression ```
idxs_to_flags
[2,1,3]
```

evaluates to the flag array
`[false,false,true,true,false,false]`

. Notice that the
implementation also works in case some segments are of size zero.

## 7.6. Moderate Flattening¶

The flattening rules that we shall introduce here allow the Futhark
compiler to generate parallel kernels for various code block
patterns. In contrast to the general concept of flattening as
introduced by Blelloch [BHS+94], Futhark
applies a technique called *moderate flattening*
[HSE+17], which does not cover
arbitrary nested parallelism, but does cover well many regular
nested parallel patterns. We shall come back to the issue of
flattening irregular nested parallelism in
Section 9.

In essence, moderate flattening works by matching compositions of fused constructs against a number of flattening rules. The aim is to merge (i.e., flatten) nested parallel operations into sequences of parallel operations. Although, such flattening is often possible, in particular due to an integrated transformation called vectorisation, there are situations where choices needs to be made. In particular, when a map is nested on top of a loop, we may choose to parallelise the outer map and sequentialise the inner loop, which on the GPU will amount to all threads running sequential loops in parallel. An alternative, when possible, will be to interchange the outer map and the loop and then sequentialise the outer loop (on the host) and parallelise the inner map, which will then be executed multiple times. It turns out that Futhark can make some guesses about which strategy to pursue based on possible information about the sizes of the arrays. An extension to the static concept moderate flattening, Futhark also supports a notion of flattening that generates multiple versions of flattened code, guarded by parameters that may be autotuned to achieve good performance for a range of different data sets [HTEO19].

In the following we shall focus on the transformations performed by moderate flattening.

### 7.6.1. Vectorisation¶

Assuming `e'`

contains SOACs, transform the expression

```
map (\x -> let y = e in e') xs
```

into the expression

```
let ys = map (\x -> e) xs
in map (\(x,y) -> e') (zip xs ys)
```

This transformation does not itself capture any nested parallelism but
may enable other transformations by eliminating the inner
`let`

-expression.

### 7.6.2. Map-Map Nesting¶

Nested applications of `map`

constructs are in essence transformed
into a single `map`

construct by (1) flattening the argument
array, (2) applying the inner function on the flattened array, and (3)
unflattening the concatenated results. This process can be repeated
for multiple nested `map`

constructs. It turns out that the
administrative operations can be implemented with zero overhead.

### 7.6.3. Map-Scan Nesting¶

In case of an expression made up from a `map`

construct appearing on
top of a `scan`

operation, the expression is transformed into a
regular segmented scan operation. That is, the expression:

```
map (\xs -> scan f ne xs) xss
```

is transformed into the expression:

```
regular_segmented_scan f ne xss
```

Notice here that we assume the availability of a regular segmented scan operation of type:

```
val regular_segmented_scan 't [n] [m]: (t->t->t) -> t -> [n][m]t -> [n][m]t
```

Internally, this function will use the inner size of the
multi-dimensional argument array (i.e., `m`

) to construct an
appropriate flag vector suitable for the segmented scan. Again, for an
in-depth discussion of how to implement a segmented scan operation on
top of an ordinary scan operation, please consult
Section 7.1.

### 7.6.4. Map-Reduce Nesting¶

In case of a `map`

construct appearing on top of a `reduce`

operation, this expression is transformed into a regular segmented
reduction [LH17]. That is, the
expression:

```
map (\xs -> reduce f ne xs) xss
```

is transformed into the expression:

```
regular_segmented_reduce f ne xss
```

Notice here that we assume the availability of a regular segmented reduction operation of type:

```
val regular_segmented_reduce 't [n] : (t->t->t) -> t -> [n][]t -> [n]t
```

Internally, this function can be implemented based on the function
`regular_segmented_scan`

discussed above. Here is a simple definition::

```
let regular_segmented_reduce = map last <-< regular_segmented_scan
```

### 7.6.5. Map-Iota Nesting¶

A `map`

over an `iota`

expression can be transformed to the
composition of the `segmented_iota`

function defined in
Section 7.4 and a function `ìdxs_to_flags`

, which converts
an array of indices to an array `fs`

of boolean flags of size equal
to the sum of the values in `xs`

and with `true`

-values in
indexes specified by the prefix sums of the index values.

As an example, the expression `idxs_to_flags [2,1,3]`

evaluates to
the flag array `[false,false,true,true,false,false]`

. Notice that
the expression `idxs_to_flags [2,0,4]`

evaluates to the same boolean
vector as `idxs_to_flags [2,4]`

. We shall not here give a definition
of the `idxs_to_flags`

function, but refer the reader to
Section 7.5.

All in all, an expression of the form:

```
map iota xs
```

is transformed into:

```
(segmented_iota <-< idxs_to_flags) xs
```

### 7.6.6. Map-Replicate Nesting¶

Recall that `replicate`

has the type:

```
val replicate 't : (n:i32) -> t -> [n]t
```

A `map`

over a `replicate`

expression takes the form:

```
map (\x -> replicate n x) xs
```

where `n`

is invariant to `x`

. Such an expression can be
transformed into the expression:

```
segmented_replicate (replicate (length xs) n) xs
```

As an example, consider the expression ```
map (replicate 2)
[8,5,1]
```

. This expression is transformed into the expression:

```
segmented_replicate (replicate 3 2) [8,5,1]
```

which evaluates to `[8,8,5,5,1,1]`

. Notice that the subexpression
`replicate 3 2`

evaluates to `[2,2,2]`

.