Schelling's model

gistfile1.txt

For the Schelling model, there is one file of interest, https://github.com/jxxcarlson/schelling/blob/master/src/Schelling.elm.

The main data structures are Cell, an opaque type, and Array Cell.  
I use 1D arrays, but process them as if they were 2D arrays using 
the functions

  location :(Int,  Int) -> Int
  location (row, col) =
    nRows * row + col

  matrixIndex : Int -> (Int, Int)
  matrixIndex n =
    (n // nCols, modBy nCols n)
 
 Seems retro, but actually works well for this problem.  I don't 
 know if it has bad performance implications.

The main function is 

    updateCells : List (Int, Int) ->  Array Cell -> Array Cell
    updateCells tupleList cellArray =
       List.foldl updateCell cellArray tupleLis
       
The functon of the tupleList : List (Int, Int) to (a) determine the order in which
cells are updated and (b) to furnish a random number which is used
in the update process for the given cell.  In practie, (via Main.elm),
the tupleList is a list or pairs where the projection to the first
coordinate is a random permutation of the list 0, 1, ... (N^2 - 1),
where the array is NxN. The projection to the second coordinate is a
list of random integers.

The function upDateCells is just a driver for
 
  updateCell : (Int, Int) -> Array Cell -> Array Cell
  updateCell (idx, rand) cellArray =
    let
        (row, col) = matrixIndex idx
        updatedCell = updateEmotionalStateOfCellAtIndex  (row, col) cellArray
    in
      if emotionalState updatedCell == Unsatisfied then
        swapWithUnoccupiedCell rand updatedCell cellArray
      else
        replace updatedCell cellArray
        
While the app works just fine for 40x40 arrays with 500 ms delay between updates,
I would like for it perform as well as possible. In general, I am interested
in places where the code can be improved.

going through the code from the repository, I think a simple optimization can be made at

numberOccupied : Array Cell -> Int
numberOccupied cellArray =
   cellArray
     |> Array.filter (\cell -> occupied cell)
     |> Array.length

This will

• allocate a new array
• walk over cellArray
• put the element into the new array if it satisfies the predicate
• determine the length of the new array (this is stored at the root, so O(1))

So, it will allocate a whole new array that is possibly large, only to determine its length. A fold is more efficient

numberOccupied : Array Cell -> Int
numberOccupied cellArray =
    let folder cell count = 
            if occupied cell then 
                1 + count
            else
                count
    in
        Array.foldl folder 0 cellArray

When writing fast code, perhaps more so in garbage collected languages than otherwise, allocation is something to avoid.

for completeness:

fractionSatisfied : Array Cell -> Float
fractionSatisfied cellArray =
    let folder cell (occupied, satisfied) = 
            case cell of 
                Unoccupied -> 
                    -- no changes
                    ( occupied, satisfied )

                Occupied _ _ _ emotion -> 
                    ( occupied + 1
                    , if emotion == Satisfied then satisfied + 1 else satisfied 
                    )
    in
        Array.foldl folder (0, 0) cellArray
            |> (\(occupied, satisfied) -> toFloat satisfied / toFloat occupied)

What I've done here is some manual inlining of the helpers. I think the readability doesn't suffer too much here but in general that is something to watch out for. On the other hand, the elm compiler doesn't do much inlining (I think arithmetic and bitwise operators are the only things) and function calls are expensive, so inlining helpers can make code go faster sometimes.

I see (I think) --- is it Array.filter that is allocating. new array? Indeed the arrays can be large.

The profiling revealed the culprit -- or a culprit. I found that the function indexAux was being called more than any other of "my" functions -- 3963 times and 4158 ms, accounting for 8.68% of the execution time, compared to the most called function, F9 at 4722 ms and 9.09%. The code is below. The index function is O(N), and is used all the time. The problem is that cells can move in the array (like chess pieces on the board), so the implementation in the code below uses a linear search. I found a good solution which is O(1) and a new data structure. I'll describe it in the next post.

index : Cell -> Array Cell -> Int
index cell cellArray =
    let
      (status, idx) = Array.foldl (indexAux cell) (False, -1) cellArray
    in
      if status == True then
        idx
      else
         -1

indexAux : Cell -> Cell -> (Bool, Int) -> (Bool, Int)
indexAux givenCell cell state =
    if (id givenCell) == (id cell) then
      (True, (Tuple.second state) + 1)
    else if Tuple.first state == False then
      (False, (Tuple.second state) + 1)
    else
      state

Here is the relevant part of the old data structure:

type Cell
    = Occupied Id Threshold Identity EmotionalState
    | Unoccupied Id

type Id = Id Int

I changed it to this:

type Cell
    = Occupied CellIndex Id Threshold Identity EmotionalState
    | Unoccupied CellIndex Id

type CellIndex = CellIndex Int

type Id = Id Int

Then the index function is

idx : Cell -> Int
idx cell =
    case cell of
        Unoccupied (CellIndex k) _ -> k
        Occupied (CellIndex k) _ _ _ _ -> k

Of course one now has to be VERY careful about how cells are updated. If they are not "moved", all is OK. In this project, the only way cells move is by swapping two cells. So we run all changes through this function,
which swaps the indices when the cells are swapped. Then cells always
carry a valid index.

With Evan's amazing compiler, it took exactly one hour to conceive and implement
all the changes. The result (using the benchmark program) is a 47% decrease in
in execution time. This carries over the app. I had reduced the update interval
from 500 ms to 250 ms. But it was somewhat jerky. I've reduced the update
interval to 50ms and it is still very smooth. This is better than can be explained
by the above.

swapCells : Cell -> Cell -> Array Cell -> Array Cell
swapCells cell1 cell2 cellArray =
    let
       idx1 = idx cell1
       idx2 = idx cell2
    in
    cellArray
      |> setWithIndex idx2 (setCellIndex idx2 cell1)
      |> setWithIndex idx1 (setCellIndex idx1 cell2)