AoC 2021 Day 09: Smoke Basin

Dated Mar 16, 2022; last modified on Wed, 16 Mar 2022

Day 9 - Advent of Code 2021. . Accessed Mar 16, 2022.

Part I

These caves seem to be lava tubes. Parts are even still volcanically active; small hydrothermal vents release smoke into the caves that slowly settles like like rain.

If you can model how the smoke flows through the caves, you might be able to avoid it and be that much safer. The submarine generates a height-map of the floor of the nearby caves for you (your puzzle input).

Smoke flows to the lowest point of the area it’s in. Your first goal is to find the low points - the locations that are lower than any of its adjacent locations. Most locations have four adjacent locations (up, down, left, and right); locations on the edge or corner of the map have three or two adjacent locations, respectively. (Diagonal locations do not count as adjacent.)

The risk level of a low point is 1 plus its height. Find all of the low points on your height-map. What is the sum of the risk levels of all low points on your height-map?

The modeling of the risk level as 1 + height is unintuitive. I’d think that the lowest point on the map has the greatest risk, e.g. if there is a lot of smoke flow, the global low point will get the greatest depth of smoke. So I’d have modeled the risk level as 1 / (1 + height).

{-#  OPTIONS_GHC -Wall  #-}

module AoC2021.SmokeBasin

import Data.Massiv.Core.Index (Ix2(..), Sz(..), Border(..), unSz)
import qualified Data.Massiv.Array as A
        Array, P(..), computeAs, sum, (!?), (!), flatten, size, zeroIndex,
        (..:), toList
import Data.Massiv.Array.Stencil (Stencil, makeStencil, mapStencil)
import Data.List (foldl', sort)
import Data.Maybe (isJust, fromJust, maybeToList)
import qualified Data.Set as Set
        Set, fromList, union, empty, singleton, null, deleteFindMin, insert,
        difference, size, member

Input Representation

The data is an \(n \times n\) grid. In another language, using a one-dimensional array might have reaped me the benefits of cache locality, but lists in Haskell are linked lists that lack guaranteed locality.

In Day 04: Giant Squid , Hidding used the array library, whose tagline is “multi-dimensional arrays with fusion , stencils and parallel computation”. It’s at least worth checking out in the problem.

Notable Design Decisions in massiv

For example, foo = length . filter p . map g . map f . concat would be pretty inefficient if executed literally. Fusion refers to program transformations aimed at removing intermediate data structures. GHC has transformation rules that enable fusion.

Also encountered stream fusion in the Data.Vector library . A stream represents the traversal of a list-like structure. All the list functions become stream functions – but crucially. stream operations are non-recursive, and can therefore be glued together.

Like Data.Vector , Massiv also supports boxed elements (may point to a thunk), unboxed elements, and storable types. However, Massiv has optimized containers for instances of the Prim class .

A bit surprised that Boolean is not an instance of the Prim type. Other languages tend to have Boolean as one of primitive types.

The size of a bool in C++ is defined from the implementation and may be greater than 1 (byte) . However, the sizeof for char, signed char, unsigned char, std::byte (C++17), and char8_t (C++20) will always evaluate to 1 (byte). . A tad counter-intuitive that a bool is at least 8 bits in C++. attributes this to every C++ data type needing to be addressable, and most CPU architectures are designed with a 8-bit chunks as the smallest addressable memory.

While std::vector<bool> is allowed to be space-efficient, it loses some guarantees of std::vector e.g. safely modifying elements concurrently in a multi-threaded context, contiguous storage (that allows pointer arithmetic), etc.

Massiv has delayed arrays which do not exist in memory, and are instead defined as a function or a composition of functions. This allows to operate on a massive array in constant memory.

Massiv has a wrapping data type for indices, e.g. makeArrayR D Seq (Sz (3 :. 5)) (\ (i :. j) -> i * j), which creates a 2D array with 3 arrays, each with 5 elements, where the element at index i :. j is computed as i * j. There is a constructor, IxN that supports N-dimensional arrays, e.g. 10 :> 20 :> 30 :. 40 is an index into a 4D array.

A stencil is a function that can read the neighboring elements of the stencil’s center (the zero index), and only those, and then outputs a new value for the center element.

mentions “Moore neighborhood”, which led to me to the Von Neumann neighborhood (the cell itself and cells at a Manhattan distance of 1) which happens to be the kind of neighborhood being evaluated in this problem.

Input Representation (Cont’d)

--  `A.P` because the underlying representation (Int) is an instance of the
--  `Prim` type class.
type HeightMap = A.Array A.P Ix2 Int

For a while, I was stuck thinking of \(a = [[0,1,2], [1,2,5], …, [3,2,6]]\) as an N-dimensional array, where \(N\) can only be determined after reading the whole file. Gleaning at made me see \(a\) as the 2D array that it is.

Part I Solution

vonNeumannNeighborhood :: (Ix2 -> Int) -> [Int]
vonNeumannNeighborhood get =
    [get (-1 :. 0), get (1 :. 0), get (0 :. -1), get (0 :. 1)]

riskIfLowPoint :: (Ix2 -> Int) -> Int
riskIfLowPoint get =
    let centerPoint = get (0 :. 0)
        lowPoint = foldl' min (maxBound :: Int) (vonNeumannNeighborhood get)
            | centerPoint < lowPoint = 1 + centerPoint
            | otherwise              = 0
    in  risk

riskIfLowPointStencil :: Stencil Ix2 Int Int
riskIfLowPointStencil = makeStencil (Sz (3 :. 3)) (1 :. 1) riskIfLowPoint
{-#  INLINE riskIfLowPointStencil  #-}

sumOfRiskLevelsOfLowPoints :: HeightMap -> Int
sumOfRiskLevelsOfLowPoints heightMap =
    let borderHandling = Fill (maxBound :: Int)
        risksArrayDW = mapStencil borderHandling riskIfLowPointStencil heightMap
        riskArray = A.computeAs A.P risksArrayDW
    in A.sum riskArray

Most of my time was spent trying to get the syntax and types right. I’m spending a lot of time deciphering massiv’s syntax as opposed to learning Haskell.

Learnings from Others' Part I Solutions

has more succinct solution:

type Array2' r a = A.Array r Ix2 a
type Array2 a = Array2' A.U a

The U is the Unbox type class. The array is usually just as fast as P, but can work with a wider range of data types. Using Array2 allowed to define more types, e.g. Array2 Int for the height map, Array2 Int for the risk values, Array2 (Int, Int) in Part II, and so forth. However, the Array2' type was unncessary IMO.

neighbours :: [Ix2]
neighbours = [-1 :. 0, 1 :. 0, 0 :. -1, 0 :. 1]

My analogous function, vonNeumannNeighborhood :: (Ix2 -> Int) -> [Int], could have been simplified into ’s. After all, I kept on using the get parameter for all 4 elements, and that’s not in tune with the wholemeal programming advocated for in Haskell. My lack of understanding of get as a function that knows how to fetch values made me blind to use of get in findMinStencil.

findMinStencil :: Stencil Ix2 Int Int
findMinStencil = makeStencil (Size (3 :. 3)) (1 :. 1) go
    where go get
            | all ((centerPoint <) . get) neighbors = value + 1
            | otherwise                             = 0
            where centerPoint = get (0 :. 0)

all returns True if all elements of a Foldable structure satisfy the predicate (and interestingly, all (> 3) [] == True) . It’s also possible to nest where definitions. Neat!

sumOfRiskLevelsOfLowPoints :: Array2 Int -> Int
sumOfRiskLevelsOfLowPoints heightMap = A.sum riskArray
    where riskArray :: Array2 Int
          riskArray = A.compute $ mapStencil (Fill 10) findMinStencil heightMap

My use of Fill (maxBound :: Int) for border handling was unncessary. The max height is 9 so 10 suffices. However, I also think that maxBound :: Int is more self explanatory.

Part II

Next, you need to find the largest basins so you know what areas are most important to avoid.

A basin is all locations that eventually flow downward to a single low point. Therefore, every low point has a basin, although some basins are very small. Locations of height 9 do not count as being in any basin, and all other locations will always be part of exactly one basin.

The size of a basin is the number of locations within the basin, including the low point.

What do you get if you multiply together the sizes of the three largest basins?

Part II Solution

maxHeight :: Int
maxHeight = 9

I initially tried to be “efficient” about this problem by only considering right and bottom neighbors to prevent double-counting.

buggyBasinSize :: HeightMap -> Ix2 -> Int
buggyBasinSize heightMap ix@(row :. col) =
    let localBasinSize height
            | height == maxHeight = 0
            | otherwise = 1 + basinSize heightMap (row+1 :. col)
                            + basinSize heightMap (row :. col +1)
    in localBasinSize (A.defaultIndex maxHeight heightMap ix)

Even if the above implementation were correct, the size of a basin would depend on what the initial (i, j) are. If (i, j) are not the top-left cell in the basin, then we might not get the correct answer as the locations to the left and above the starting location are omitted.

This question reduces to the connected components problem in graph theory. Two neighboring locations are considered connected if their heights are both less than maxheight = 9.

is not explicit about what counts as a neighbor. In this height map:


… are 1 and 2 considered to be in the same basin? Maybe not because the two 9s form a wall that prevents water in 1 from flowing to 2. If it does flow, then all 9s are submerged, and we have one huge basin, which is not in the spirit of the problem.

Is there a way of finding the connected components using stencils? It feels like there should, given vonNeumannNeighborhood Stencils make sense when we’re applying a function to each element while taking its neighbors into account. In the case of finding neighbors, I don’t think a stencil will be useful.

isInABasin :: Int -> Bool
isInABasin height = height < maxHeight

computeBasinNeighbors :: HeightMap -> Ix2 -> Locations
computeBasinNeighbors heightMap (row :. col) =
    let candidates = [row-1 :. col, row+1 :. col, row :. col-1, row :. col+1]
        heights = map (heightMap A.!?) candidates
        isNeighbor (_, maybeHeight) =
            isJust maybeHeight && isInABasin (fromJust maybeHeight)
    in Set.fromList (map fst $ filter isNeighbor (zip candidates heights))

Is there a more idiomatic way of checking that a given index is within bounds?

type Locations = Set.Set Ix2

getBasin :: HeightMap -> Ix2 -> Maybe Locations
getBasin heightMap location
    | not (isInABasin (heightMap A.! location)) = Nothing
    | otherwise = Just (getLocationsInBasin Set.empty (Set.singleton location)) where
        getLocationsInBasin :: Locations -> Locations -> Locations
        getLocationsInBasin visited visiting
            | Set.null visiting = visited
            | otherwise =
                let (currLocation, otherLocations) = Set.deleteFindMin visiting
                    visited' = Set.insert currLocation visited
                    currNeighbors = computeBasinNeighbors heightMap currLocation
                    unseenLocations = Set.difference currNeighbors visited
                    visiting' = Set.union otherLocations unseenLocations
                in getLocationsInBasin visited' visiting'

collectBasins :: HeightMap -> [Ix2] -> Locations -> [Locations]
collectBasins heightMap (location:locations) visited =
    let maybeBasin
            | Set.member location visited = Nothing
            | otherwise                   = getBasin heightMap location
            | isJust maybeBasin = Set.union visited (fromJust maybeBasin)
            | otherwise         = Set.insert location visited
    in maybeToList maybeBasin ++ collectBasins heightMap locations visited'
collectBasins _ [] _ = []

I’m spending too much time writing down code for collecting the connected components. Two sources of delay:

  • In my head, I have a vague idea of how I’d solve this problem in Python, and I’m trying to translate that into Haskell, which demands a functional style with immutability. I need to learn how to think in functional terms, i.e. going from the algorithm to functional implementation, instead of going from an imperative implementation to a functional one.

  • Not understanding the documentation at . I’m struggling with things that are relatively straightforward in other languages' docs, e.g. given an N-dimensional array, how do I construct a list of all the valid indices? Or am I asking questions that only an imperative thinker would? How do I verify that an index into a given array is valid, without trying to fetch the element at that index?

The Set library does not offer a generic Set.pop API. Instead, it has Set.lookupMin and Set.lookupMax, which both run in \(O(log\ n)\) time. These APIs make it apparent that the Set container is ordered.

C++17 introduced std::set<Key,Compare,Allocator>::extract, which is amortized \(O(1)\) when passing in an iterator. std::unordered_set, which hashes items into buckets, has its extract method taking \(O(1)\) in the average case, and \(O(n)\) in the worst case.

Python’s set.pop specifies that an arbitrary element is removed and returned. There aren’t explicit guidelines, but I assume the underlying implementation has similar complexity.

productOf3LargestBasins :: HeightMap -> Int
productOf3LargestBasins heightMap =
    let indexes = A.toList $ A.flatten $ (A.zeroIndex :: Ix2) A...: unSz (A.size heightMap)
        basinSizes = map Set.size (collectBasins heightMap indexes Set.empty)
    in (product . take 3 . reverse . sort) basinSizes

Learnings from Others' Part II Solutions

’s approach was different: mark the minima, and then repeatedly grow neighborhood around each minimum until two patches meet.

Given an Array2 Int where minima are denoted as non-zero elements, labels each minima with a unique integer, with the help of (which re-exports ).

-- `evalState :: State s a -> s -> a` evaluates a state computation with the
-- initial state, and returns the final value, discarding the final state.
-- `State s a` is a state-passing computation to execute.
-- `s` is the initial value.
-- `a` is the return value of the state computation.

markBasins :: Array2 Int -> Array2 Int
markBasins a = evalState (A.mapM markNonZero a) 0 -- Initial ID is zero.
    where markNonZero :: Int -> State Int Int
          markNonZero x
            -- `modify` maps an old state to a new state inside a state monad;
            -- the old state is thrown away. `modify'` is its strict variant.
            -- `get` returns the state from the internals of the monad.
            -- So whenever x is a minimum, we increase the state, and then
            -- assign the value to that array element. That's why the IDs are
            -- all greater than zero and unique.
            | x /= 0    = modify (+ 1) >> get
            | otherwise = return 0

The (>>) “and then” operator sequentially composes two actions, discarding any value produced by the first. Previous encounter with (>>) operator .

The technique used in markBasins looks promising for future endeavors where I need to process a collection whle keeping track of some state. Previously, I’ve found such problems tricky because I’m used to state being in a mutable variable. The monad state feels mutable, but somehow abstracted from me.

Misc. Learnings from Others' Solutions

used the library, and it looks promising, there are functions in the package that are not “referentially transparent”, e.g. trace :: String -> a -> a whose type indicates purity, but it actually has a side effect of outputting the trace message. I didn’t know this was possible.


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