### Advent of Code 2020 review

A look back on the event

Cellular automata and typeclasses

For day 17, I drew on last year's Advent of Code, mainly day 12 (for typeclasses) and day 24 (for running the cellular automaton).

First up, data structures. I chose a `V3`

vector from the `Linear`

library to represent a position, and a `Set`

of them to represent the active cubes.

Reading the grid is iterating over all cells in the grid, filling the set of active cubes as we go.

```
type Coord = V3 Int -- x, y, z
type Grid = S.Set Coord
readGrid :: String -> IO Grid
readGrid filename =
do gs <- readFile filename
let grid = lines gs
let isActive x y = (grid!!y)!!x == '#'
let maxX = length (head grid) - 1
let maxY = length grid - 1
return $ S.fromList [ V3 x y 0 | x <- [0..maxX], y <- [0..maxY], isActive x y]
```

To update the automaton one step, I check every member of the grid to see if it survives. I also find the "empties", the inactive cubes adjacent to an active cube, and check if any of them become active.

```
update :: Grid -> Grid
update grid = S.union (S.filter (cubeSurvives grid) grid)
(S.filter (cubeBorn grid) empties)
where empties = (S.foldr mergeEmpties S.empty grid) `S.difference` grid
mergeEmpties cell acc = S.union acc $ neighbourSpaces cell
```

An inactive cube becomes active (is born) if it has three neighbours; an active cube remains active if it has two or three neighbours.

```
cubeSurvives :: Grid -> Coord -> Bool
cubeSurvives grid cell = alive && (nNbrs == 2 || nNbrs == 3)
where alive = cell `S.member` grid
nNbrs = countOccupiedNeighbours cell grid
cubeBorn :: Grid -> Coord -> Bool
cubeBorn grid cell = dead && (nNbrs == 3)
where dead = cell `S.notMember` grid
nNbrs = countOccupiedNeighbours cell grid
```

Both the rule, and the indentification of the empties, require finding the neighbours of a cell, and possibly counting how many neighbours are active.

```
neighbourSpaces :: Coord -> Grid
neighbourSpaces here = S.map (here ^+^) nbrs
where nbrs = S.fromList [ V3 dx dy dz
| dx <- [-1, 0, 1]
, dy <- [-1, 0, 1]
, dz <- [-1, 0, 1]
, (dx, dy, dz) /= (0, 0, 0)]
countOccupiedNeighbours :: Coord -> Grid -> Int
countOccupiedNeighbours cell grid =
S.size $ S.intersection grid $ neighbourSpaces cell
```

Finally, the simulation runs by iterating `update`

and dropping the unneeded states.

```
main =
do grid0 <- readGrid "data/advent17.txt"
let finalGrid = head $ drop 6 $ iterate update grid0
print $ S.size finalGrid
```

This is exactly the same as part 1, except on a 4-d grid rather than a 3-d grid. The only parts of the code that explicitly refer to the dimensions are the grid construction, and the generation of the neighbours. It seems very wasteful to replicate all the code just to use different functions at the bottom of the call stack.

Polymorphism to the rescue!

Rather than having `Coord`

as a concrete type, I use it as typeclass, with instances for the 3-d and 4-d cases. I split the `neighbourSpaces`

function into two parts: the mapping of `(here ^+^)`

and the creation of the set of `neighbourCells`

of the origin.(I also need to define a new operator, `^+^^`

, to add `Coord`

s.)

```
class (Num a, Ord a) => Coord a where
(^+^^) :: a -> a -> a
neighbourCells :: S.Set a
instance Coord (V3 Int) where
x ^+^^ y = x ^+^ y
neighbourCells = S.fromList [ V3 dx dy dz
| dx <- [-1, 0, 1]
, dy <- [-1, 0, 1]
, dz <- [-1, 0, 1]
, (dx, dy, dz) /= (0, 0, 0)
]
instance Coord (V4 Int) where
x ^+^^ y = x ^+^ y
neighbourCells = S.fromList [ V4 dx dy dz dw
| dx <- [-1, 0, 1]
, dy <- [-1, 0, 1]
, dz <- [-1, 0, 1]
, dw <- [-1, 0, 1]
, (dx, dy, dz, dw) /= (0, 0, 0, 0)
```

The rest of the code stays the same, apart from the use of the new operator in `neighbourSpaces`

and a few changes in types to use the new class.

```
neighbourSpaces :: Coord a => a -> Grid a
neighbourSpaces here = S.map (here ^+^^) neighbourCells
countOccupiedNeighbours :: Coord a => a -> Grid a -> Int
cubeSurvives :: Coord a => Grid a -> a -> Bool
```

Finally, I need to convert the 3-d grid originally read into a 4-d grid.

```
conv34 :: Grid (V3 Int) -> Grid (V4 Int)
conv34 grid = S.map conv34Cell grid
conv34Cell (V3 x y z) = V4 x y z 0
```

The simulations are the same, apart from the dimension conversion.

```
main =
do grid0 <- readGrid "data/advent17.txt"
print $ part1 grid0
print $ part2 grid0
part1 grid0 = S.size finalGrid
where finalGrid = head $ drop 6 $ iterate update grid0
part2 grid0 = S.size finalGrid
where grid4 = conv34 grid0
finalGrid = head $ drop 6 $ iterate update grid4
```

You can find the code here or on GitLab. Part 1 is in `advent17a.hs`

, part 2 in `advent17.hs`

.