# TL;DR

What do psychedelic triangles and chimeras have in common? Oddly the answer is “list data structures”. I discovered this while procrastinating from the side project that was my procrastination from my actual projects. Much to my surprise, shading triangles from vertex colors is faster with list data structures than with matrices and arrays. This despite the calculations involved being simple additions and multiplications on large atomic vectors.

As with everything in life there are caveats to this finding. It is primarily applicable to tall and skinny data sets, i.e. those with a very high ratio of observations to variables. It also requires some additional work over standard methods. Given that this type of data set is fairly common, it seems worth showing methods for working with list data structures. In my particular use case I saw speed improvements on the order of 2x to 4x.

If you are bored easily you might want to skip straight to the case study section of this post. If you like benchmarks and want an extensive review of list data structure manipulations, read on.

# Matrices Are Slow?

## Column Subsetting

Generally speaking matrices are amongst the R data structures that are fastest to operate on. After all, they are contiguous chunks of memory with almost no overhead. Unfortunately, they do come with a major limitation: any subset operation requires copying the data:

```
set.seed(1023)
n <- 1e5
M <- cbind(runif(n), runif(n), runif(n), runif(n))
D <- as.data.frame(M)
microbenchmark(M[,1:2], D[,1:2], times=25)
```

```
Unit: microseconds
expr min lq mean median uq max neval
M[, 1:2] 635.4 1664.3 2399.6 1747.1 2410.0 12696.6 25
D[, 1:2] 14.6 24.8 50.2 48.4 74.7 93.3 25
```

The data frame subset is much faster because R treats data frame columns as
independent objects, so referencing an entire column does not require a copy.
But data frames have big performance problems of their own. Let’s see what
happens if we select columns **and** rows:

```
idx <- sample(seq_len(nrow(M)), n/2)
microbenchmark(M[idx,1:2], D[idx,1:2], times=25)
```

```
Unit: microseconds
expr min lq mean median uq max neval
M[idx, 1:2] 834 1357 1673 1526 1863 3189 25
D[idx, 1:2] 9829 10403 11434 11184 12056 16043 25
```

And its even worse if we have duplicate row indices:

```
idx2 <- c(idx, idx[1])
microbenchmark(M[idx2,1:2], D[idx2,1:2], times=25)
```

```
Unit: microseconds
expr min lq mean median uq max neval
M[idx2, 1:2] 476 1151 1297 1418 1483 1668 25
D[idx2, 1:2] 27788 29606 30911 30123 31302 44892 25
```

Data frames have row names that are required to be unique and most of the row subsetting overhead comes from managing them.

## Lists to the Rescue

With a little extra work you can do most of what you would normally do with a
matrix or data frame with a list. For example, to select “rows” in a list of
vectors, we use `lapply`

to apply `[`

to each component vector:

```
L <- as.list(D)
str(L)
```

```
List of 4
$ V1: num [1:100000] 0.249 0.253 0.95 0.489 0.406 ...
$ V2: num [1:100000] 0.7133 0.4531 0.6676 0.651 0.0663 ...
$ V3: num [1:100000] 0.87 0.783 0.714 0.419 0.495 ...
$ V4: num [1:100000] 0.44475 0.90667 0.12428 0.00587 0.77874 ...
```

```
sub.L <- lapply(L[1:2], '[', idx)
str(sub.L)
```

```
List of 2
$ V1: num [1:50000] 0.997 0.43 0.945 0.798 0.252 ...
$ V2: num [1:50000] 0.539 0.287 0.567 0.848 0.56 ...
```

`col_eq(sub.L, D[idx,1:2]) # col_eq compares values column by column`

`[1] TRUE`

This results in the same elements selected as with data frame or matrix
subsetting, except the result remains a list. You can see how `col_eq`

compares
objects in the appendix.

Despite the explicit looping over columns this list based subsetting is comparable to matrix subsetting in speed:

`microbenchmark(lapply(L[1:2], '[', idx), M[idx,1:2], D[idx,1:2], times=25)`

```
Unit: microseconds
expr min lq mean median uq max neval
lapply(L[1:2], "[", idx) 800 1180 1355 1378 1443 1903 25
M[idx, 1:2] 1087 1289 1405 1456 1558 1712 25
D[idx, 1:2] 9327 9904 11013 10335 11502 19525 25
```

And if we only index columns then the list is even faster than the data frame:

`microbenchmark(L[1:2], M[,1:2], D[,1:2], times=25)`

```
Unit: nanoseconds
expr min lq mean median uq max neval
L[1:2] 685 900 2815 1009 3937 10522 25
M[, 1:2] 1597754 1773076 1875640 1871310 1911109 2305469 25
D[, 1:2] 14167 15049 36128 23122 64984 87234 25
```

# Patterns in List Based Analysis

## Row Subsetting

As we saw previously the idiom for subsetting “rows” from a list of equal length vectors is:

`lapply(L, '[', idx)`

The `lapply`

expression is equivalent to:

```
fun <- `[`
result <- vector("list", length(L))
for(i in seq_along(L)) result[[i]] <- fun(L[[i]], idx)
```

We use `lapply`

to apply a function to each element of our list. In our case,
each element is a vector, and the function is the subset operator `[`

. In R,
`L[[i]][idx]`

and `'['(L[[i]], idx)`

are equivalent as operators are really just
functions that R recognizes as functions at parse time.

## Column Operations

We can extend the `lapply`

principle to other operators and functions. To
multiply every column by two we can use:

```
microbenchmark(times=25,
Matrix= M * 2,
List= lapply(L, '*', 2)
)
```

```
Unit: microseconds
expr min lq mean median uq max neval
Matrix 949 2438 3425 2626 2900 16561 25
List 494 1018 1957 2470 2542 3254 25
```

This will work for any binary function. For unary functions we just omit the additional argument:

`lapply(L, sqrt)`

We can also extend this to column aggregating functions:

```
microbenchmark(times=25,
Matrix_sum= colSums(M),
List_sum= lapply(L, sum),
Matrix_min= apply(M, 2, min),
List_min= lapply(L, min)
)
```

```
Unit: microseconds
expr min lq mean median uq max neval
Matrix_sum 517 555 606 569 598 1353 25
List_sum 527 577 628 603 647 990 25
Matrix_min 5509 7775 10067 10140 10531 20025 25
List_min 985 1033 1078 1064 1114 1223 25
```

In the matrix case we have the special `colSums`

function that does all the
calculations in internal C code. Despite that it is no faster than the list
approach. And for operations without special column functions such as finding
the minimum by column we need to use `apply`

, at which point the list approach
is much faster.

If we want to use different scalars for each column things get a bit more
complicated, particularly for the matrix approach. The most natural way to do
this with the matrix is to transpose and take advantage of column recycling.
For lists we can use `Map`

:

```
vec <- 2:5
microbenchmark(times=25,
Matrix= t(t(M) * vec),
List= Map('*', L, vec)
)
```

```
Unit: microseconds
expr min lq mean median uq max neval
Matrix 3253 5909 7497 7458 7801 15652 25
List 906 1196 2754 3165 3272 10977 25
```

`col_eq(t(t(M) * vec), Map('*', L, vec))`

`[1] TRUE`

`Map`

is a close cousin of `mapply`

. It calls its first argument, in this case
the function `*`

, with one element each from the subsequent arguments, looping
through the argument elements. We use `Map`

instead of `mapply`

because it is
guaranteed to always return a list, whereas `mapply`

will simplify the result to
a matrix. It is equivalent to:

```
f <- `*`
result <- vector('list', length(vec))
for(i in seq_along(L)) result[[i]] <- f(L[[i]], vec[[i]])
```

There are some faster ways to do this with matrices, such as:

```
microbenchmark(times=25,
Matrix= M %*% diag(vec),
List= Map('*', L, vec)
)
```

```
Unit: microseconds
expr min lq mean median uq max neval
Matrix 2783 4353 4796 4500 4673 14899 25
List 859 2309 2990 3095 3200 11095 25
```

`col_eq(M %*% diag(vec), Map('*', L, vec))`

`[1] TRUE`

But even that is slower than the list approach, with the additional infirmities that it cannot be extended to other operators, and it works incorrectly if the data contains infinite values.

`Map`

also allows us to operate pairwise on columns. Another way to compute the
square of our data might be:

```
microbenchmark(times=25,
Matrix= M * M,
List= Map('*', L, L)
)
```

```
Unit: microseconds
expr min lq mean median uq max neval
Matrix 980 2446 3107 2510 2642 12048 25
List 499 1126 2095 2523 2625 2846 25
```

`col_eq(M * M, Map('*', L, L))`

`[1] TRUE`

## Row-wise Operations

A classic row-wise operation is to sum values by row. This is easily done with
`rowSums`

for matrices. For lists, we can use `Reduce`

to collapse every column
into one:

```
microbenchmark(times=25,
Matrix= rowSums(M),
List= Reduce('+', L)
)
```

```
Unit: microseconds
expr min lq mean median uq max neval
Matrix 1824 2567 3047 3092 3486 4211 25
List 684 1484 2271 1888 2119 14733 25
```

`col_eq(rowSums(M), Reduce('+', L))`

`[1] TRUE`

Much to my surprise `Reduce`

is substantially faster, despite explicitly looping
through the columns. From informal testing `Reduce`

has an edge up to ten
columns or so. This may be because `rowSums`

will use `long double`

to store
the interim results of the row sums, whereas `+`

will just add double values
directly, with some loss of precision as a result.

`Reduce`

collapses the elements of the input list by repeatedly applying a
binary function to turn two list elements into one. `Reduce`

is equivalent to:

```
result <- numeric(length(L[[1]]))
for(l in L) result <- result + l
```

An additional advantage of the `Reduce`

approach is you can use any binary
function for the row-wise aggregation.

## List to Matrix and Back

R is awesome because it adopts some of the better features of functional languages and list based languages. For example, the following two statements are equivalent:

```
ML1 <- cbind(x1=L[[1]], y1=L[[2]], x2=L[[3]], y2=L[[4]])
ML2 <- do.call(cbind, L)
```

Proof:

`col_eq(ML1, ML2)`

`[1] TRUE`

`col_eq(ML1, M)`

`[1] TRUE`

The blurring of calls, lists, data, and functions allows for very powerful
manipulations of list based data. In this case, `do.call(cbind, LIST)`

will
transform any list containing equal length vectors into a matrix, irrespective
of how many columns it has.

To go in the other direction:

```
LM <- lapply(seq_len(ncol(M)), function(x) M[,x])
col_eq(L, LM)
```

`[1] TRUE`

You *could* use `split`

but it is much slower.

## More on `do.call`

List based analysis works particularly well with `do.call`

and functions that
use `...`

parameters:

```
microbenchmark(times=25,
List= do.call(pmin, L),
Matrix= pmin(M[,1],M[,2],M[,3],M[,4])
)
```

```
Unit: milliseconds
expr min lq mean median uq max neval
List 3.00 3.38 3.93 3.47 3.62 14.4 25
Matrix 5.17 7.61 7.93 7.77 8.00 16.0 25
```

The expression is both easier to type, and faster!

There are a couple of “gotchas” with `do.call`

, but they only crop up if you try
to pass quoted language as part of the argument list (e.g. `quote(a + b)`

). If
you are just using normal R data objects then the one drawback that I am
aware of is that the call stack can get messy if you ever invoke `traceback()`

or `sys.calls()`

from a function called by `do.call`

.

Remember that data frames are lists, so you can do things like:

`do.call(pmin, iris[-5])`

## High Dimensional Data

Matrices naturally extend to arrays in high dimensional data. Imagine we wish to track triangle vertex coordinates in 3D space. We can easily set up an array structure to do this:

```
verts <- 3
dims <- 3
V <- runif(n * verts * dims)
tri.A <- array(
V, c(n, verts, dims),
dimnames=list(NULL, vertex=paste0('v', seq_len(verts)), dim=c('x','y','z'))
)
tri.A[1:2,,]
```

```
, , dim = x
vertex
v1 v2 v3
[1,] 0.982 0.411 0.936
[2,] 0.798 0.675 0.345
, , dim = y
vertex
v1 v2 v3
[1,] 0.561 0.156 0.782
[2,] 0.188 0.572 0.854
, , dim = z
vertex
v1 v2 v3
[1,] 0.435 0.361 0.892
[2,] 0.146 0.109 0.922
```

How do we do this with lists? Well, it turns out that you can make list-matrices and list-arrays. First we need to manipulate our data little as we created it originally for use with an array:

```
MV <- matrix(V, nrow=n)
tri.L <- lapply(seq_len(verts * dims), function(x) MV[,x])
```

Then, we just add dimensions:

```
dim(tri.L) <- tail(dim(tri.A), -1)
dimnames(tri.L) <- tail(dimnames(tri.A), -1)
tri.L
```

```
dim
vertex x y z
v1 Numeric,100000 Numeric,100000 Numeric,100000
v2 Numeric,100000 Numeric,100000 Numeric,100000
v3 Numeric,100000 Numeric,100000 Numeric,100000
```

`class(tri.L)`

`[1] "matrix"`

`typeof(tri.L)`

`[1] "list"`

A chimera! `tri.L`

is a list-matrix. While this may seem strange if you haven’t
come across one previously, the list matrix and 3D array are essentially
equivalent:

`head(tri.A[,'v1','x'])`

`[1] 0.9825 0.7980 0.5111 0.0385 0.6194 0.1001`

`head(tri.L[['v1','x']])`

`[1] 0.9825 0.7980 0.5111 0.0385 0.6194 0.1001`

An example operation might be to find the mean value of each coordinate for each vertex:

`colMeans(tri.A)`

```
dim
vertex x y z
v1 0.499 0.501 0.5
v2 0.500 0.500 0.5
v3 0.500 0.500 0.5
```

`apply(tri.L, 1:2, do.call, what=mean)`

```
dim
vertex x y z
v1 0.499 0.501 0.5
v2 0.500 0.500 0.5
v3 0.500 0.500 0.5
```

`apply`

with the second argument set to `1:2`

is equivalent to:

```
MARGIN <- 1:2
result <- vector('list', prod(dim(tri.L)[MARGIN]))
dim(result) <- dim(tri.L)[MARGIN]
fun <- do.call
for(i in seq_len(nrow(tri.L)))
for(j in seq_len(ncol(tri.L)))
result[i, j] <- fun(what=mean, tri.L[i, j])
```

We use `do.call`

because `tri.L[i, j]`

is a list, and we wish to call `mean`

with the contents of the list, not the list itself. We could also have used a
function like `function(x) mean(x[[1]])`

instead of `do.call`

.

We’ll be using list matrices in the next section.

# A Case Study

One of my side projects required me to implement barycentric coordinate conversions. You can think of barycentric coordinates of a point in a triangle as the weights of the vertices that cause the triangle to balance on that point.

We show here three different points (red crosses) with the barycentric coordinates associated with each vertex. The size of the vertex helps visualize the “weights”:

This is useful if we want to interpolate values from the vertices to points in the triangle. Here we interpolate a color for a point in a triangle from the colors of the vertices of that triangle:

The formula for converting cartesian coordinates to Barycentric is:

\[\lambda_1 = \frac{(y_2 - y_3)(x - x_3) + (x_3 - x_2)(y - y_3)}{(y_2 - y_3)(x_1 - x_3) + (x_3 - x_2)(y_1 - y_3)}\\ \lambda_2 = \frac{(y_3 - y_1)(x - x_3) + (x_1 - x_3)(y - y_3)}{(y_2 - y_3)(x_1 - x_3) + (x_3 - x_2)(y_1 - y_3)}\\ \lambda_3 = 1 - \lambda_1 - \lambda_2\]

You can see that whatever the data structure is, we will be using a lot of
column subsetting for this calculation. We implement `bary_A`

and `bary_L`

to
convert Cartesian coordinates to barycentric using array and list-matrix data
structures respectively. Since both of these are direct translations of
the above formulas, we relegate them to the code
appendix. We will use them to shade every point of
our triangle with the weighted color of the vertices.

First, the data (defined in the appendix):

`str(point.L) # every point we want to compute color`

```
List of 2
$ x: num [1:79946] 0 0.00251 0.00501 0.00752 0.01003 ...
$ y: num [1:79946] 0.067 0.067 0.067 0.067 0.067 ...
```

`vertex.L # list-matrix!`

```
Data
V x y r g b
v1 Numeric,79946 Numeric,79946 Numeric,79946 Numeric,79946 Numeric,79946
v2 Numeric,79946 Numeric,79946 Numeric,79946 Numeric,79946 Numeric,79946
v3 Numeric,79946 Numeric,79946 Numeric,79946 Numeric,79946 Numeric,79946
```

The `vertex.L`

list-matrix has the x, y coordinates and the red, green, and blue
color channel vales of the triangle associated with each point. In this
particular case it happens to be the same triangle for every point, so we could
have used a single instance of the triangle. The generalization needs to
allow for many triangles with a small and varying number of points per triangle,
and to handle that efficiently in R we need to match each point to its own
triangle.

And the shading function:

```
v_shade_L <- function(p, v) {
## 1) compute barycentric coords
bc <- bary_L(p, v)
## 2) for each point, weight vertex colors by bary coords
clr.raw <- apply(v[, c('r','g','b')], 2, Map, f="*", bc)
## 3) for each point-colorchannel, sum the weighted values
lapply(clr.raw, Reduce, f="+")
}
```

Step 1) is carried out by `bary_L`

, which is a simple adaptation of
the barycentric conversion formulas. Step 2) is the most
complicated. To understand what’s going on we need to look at the two data
inputs to `apply`

, which are:

`v[, c('r','g','b')]`

```
Data
V r g b
v1 Numeric,79946 Numeric,79946 Numeric,79946
v2 Numeric,79946 Numeric,79946 Numeric,79946
v3 Numeric,79946 Numeric,79946 Numeric,79946
```

`str(bc)`

```
List of 3
$ : num [1:79946] 1 0.997 0.995 0.992 0.99 ...
$ : num [1:79946] 0 0 0 0 0 0 0 0 0 0 ...
$ : num [1:79946] 0 0.00251 0.00501 0.00752 0.01003 ...
```

In:

`clr.raw <- apply(v[, c('r','g','b')], 2, Map, f="*", bc)`

We ask `apply`

to call `Map`

for each column of `v[, c('r','g','b')]`

. The
additional arguments `f='*'`

and `bc`

are passed on to `Map`

such that, for the
first column, the `Map`

call is:

`Map('*', v[,'r'], bc)`

`v[,'r']`

is a list with the red color channel values for each of the three
vertices for each of the points in our data:

`str(v[,'r'])`

```
List of 3
$ v1: num [1:79946] 1 1 1 1 1 1 1 1 1 1 ...
$ v2: num [1:79946] 1 1 1 1 1 1 1 1 1 1 ...
$ v3: num [1:79946] 0 0 0 0 0 0 0 0 0 0 ...
```

`bc`

is a list of the barycentric coordinates of each point which we will use as
the vertex weights when averaging the vertex colors to point colors:

`str(bc)`

```
List of 3
$ : num [1:79946] 1 0.997 0.995 0.992 0.99 ...
$ : num [1:79946] 0 0 0 0 0 0 0 0 0 0 ...
$ : num [1:79946] 0 0.00251 0.00501 0.00752 0.01003 ...
```

The `Map`

call is just multiplying each color value by its vertex weight.

`apply`

will repeat this over the three color channels giving us the barycentric
coordinate-weighted color channel values for each point:

`str(clr.raw)`

```
List of 3
$ r:List of 3
..$ v1: num [1:79946] 1 0.997 0.995 0.992 0.99 ...
..$ v2: num [1:79946] 0 0 0 0 0 0 0 0 0 0 ...
..$ v3: num [1:79946] 0 0 0 0 0 0 0 0 0 0 ...
$ g:List of 3
..$ v1: num [1:79946] 0 0 0 0 0 0 0 0 0 0 ...
..$ v2: num [1:79946] 0 0 0 0 0 0 0 0 0 0 ...
..$ v3: num [1:79946] 0 0.00251 0.00501 0.00752 0.01003 ...
$ b:List of 3
..$ v1: num [1:79946] 1 0.997 0.995 0.992 0.99 ...
..$ v2: num [1:79946] 0 0 0 0 0 0 0 0 0 0 ...
..$ v3: num [1:79946] 0 0.00251 0.00501 0.00752 0.01003 ...
```

Step 3) collapses the weighted values into (r,g,b) triplets for each point:

`str(lapply(clr.raw, Reduce, f="+"))`

```
List of 3
$ r: num [1:79946] 1 0.997 0.995 0.992 0.99 ...
$ g: num [1:79946] 0 0.00251 0.00501 0.00752 0.01003 ...
$ b: num [1:79946] 1 1 1 1 1 1 1 1 1 1 ...
```

And with the colors we can render our fully shaded triangle (see appendix for
`plot_shaded`

):

```
color.L <- v_shade_L(point.L, vertex.L)
plot_shaded(point.L, do.call(rgb, color.L))
```

The corresponding function in array format looks very similar to the list one:

```
v_shade_A <- function(p, v) {
## 1) compute barycentric coords
bc <- bary_A(p, v)
## 2) for each point, weight vertex colors by bary coords
v.clrs <- v[,,c('r','g','b')]
clr.raw <- array(bc, dim=dim(v.clrs)) * v.clrs
## 3) for each point-colorchannel, sum the weighted values
rowSums(aperm(clr.raw, c(1,3,2)), dims=2)
}
```

For a detailed description of what it is doing see the appendix. The take-away for now is that every function in the array implementation is fast in the sense that all the real work is done in internal C code.

The data are also similar, although they are in matrix and array format instead of list and matrix-list:

`str(point.A)`

```
num [1:79946, 1:2] 0 0.00251 0.00501 0.00752 0.01003 ...
- attr(*, "dimnames")=List of 2
..$ : NULL
..$ : chr [1:2] "x" "y"
```

`str(vertex.A)`

```
num [1:79946, 1:3, 1:5] 0 0 0 0 0 0 0 0 0 0 ...
- attr(*, "dimnames")=List of 3
..$ : NULL
..$ V : chr [1:3] "v1" "v2" "v3"
..$ Data: chr [1:5] "x" "y" "r" "g" ...
```

Both shading implementations produce the same result:

```
color.A <- v_shade_A(point.A, vertex.A)
col_eq(color.L, color.A)
```

`[1] TRUE`

Yet, the list based shading is substantially faster:

```
microbenchmark(times=25,
v_shade_L(point.L, vertex.L),
v_shade_A(point.A, vertex.A)
)
```

```
Unit: milliseconds
expr min lq mean median uq max neval
v_shade_L(point.L, vertex.L) 12.5 16.2 20.1 18.4 21.4 38.4 25
v_shade_A(point.A, vertex.A) 54.8 64.5 77.3 76.6 85.1 126.2 25
```

We can get a better sense of where the speed differences are coming from by
looking at the profile data using `treeprof`

:

`treeprof(v_shade_L(point.L, vertex.L))`

```
Ticks: 848; Iterations: 42; Time Per: 23.24 milliseconds; Time Total: 976.3 milliseconds; Time Ticks: 0.848
milliseconds
v_shade_L -------------------- : 23.24 - 0.00
bary_L ------------------- : 12.01 - 12.01
apply -------------------- : 6.61 - 0.03
| FUN ------------------ : 6.58 - 0.00
| mapply ----------- : 6.58 - 0.00
| <Anonymous> -- : 6.58 - 6.58
lapply ------------------- : 4.63 - 0.00
FUN ------------------ : 4.63 - 0.00
f ---------------- : 4.63 - 4.63
```

`treeprof(v_shade_A(point.A, vertex.A))`

```
Ticks: 775; Iterations: 13; Time Per: 71.10 milliseconds; Time Total: 924.3 milliseconds; Time Ticks: 0.775
milliseconds
v_shade_A ---------------------- : 71.10 - 0.00
bary_A --------------------- : 44.50 - 43.58
| cbind ------------------ : 0.92 - 0.92
rowSums -------------------- : 19.45 - 7.89
| is.data.frame ---------- : 11.56 - 0.00
| aperm -------------- : 11.56 - 0.18
| aperm.default -- : 11.38 - 11.38
array ---------------------- : 7.16 - 7.16
```

Most of the time difference is in the `bary_L`

vs `bary_A`

functions that compute the barycentric coordinates. You can also tell that
because almost all the time in the `bary_*`

functions is “self” time (the second
column in the profile data) that this time is all spent doing primitive /
internal operations.

Step 2) takes about the same amount of time for both the list-matrix and array
methods (the `apply`

call for `v_shade_L`

and the `array`

call for `v_shade_A`

,
but step 3) is also much faster for the list approach because with the
array approach we need to permute the dimensions so we can use `rowSums`

(more
details in the appendix).

# Conclusion

If your data set is tall and skinny, list data structures can improve performance without compiled code or additional external dependencies. Most operations are comparable in speed whether done with matrices or lists, but key operations are faster and that make some tasks noticeably more efficient.

One trade-off is that the data manipulation can be more difficult with lists,
but for code in a package the extra work may be justified. Additionally, this
is not always true: patterns like `do.call(fun, list)`

simplify tasks that
can be awkward with matrices.

Questions? Comments? @ me on Twitter.

# Appendix

## Session Info

`sessionInfo()`

```
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] treeprof_0.2.3 microbenchmark_1.4-4
loaded via a namespace (and not attached):
[1] Rcpp_0.12.19 bookdown_0.7 later_0.7.3
[4] digest_0.6.15 rprojroot_1.3-2 mime_0.5
[7] R6_2.3.0 xtable_1.8-2 backports_1.1.2
[10] magrittr_1.5 evaluate_0.10.1 blogdown_0.8
[13] stringi_1.2.3 promises_1.0.1 data.table_1.11.4
[16] rmarkdown_1.10 tools_3.5.1 stringr_1.3.1
[19] shiny_1.1.0 httpuv_1.4.4.1 xfun_0.3
[22] yaml_2.2.0 compiler_3.5.1 htmltools_0.3.6
[25] knitr_1.20
```

## Column Equality

```
col_eq <- function(x, y) {
if(is.matrix(x)) x <- lapply(seq_len(ncol(x)), function(z) x[, z])
if(is.matrix(y)) y <- lapply(seq_len(ncol(y)), function(z) y[, z])
# stopifnot(is.list(x), is.list(y))
all.equal(x, y, check.attributes=FALSE)
}
```

## Barycentric Conversions

```
## Conversion to Barycentric Coordinates; List-Matrix Version
bary_L <- function(p, v) {
det <- (v[[2,'y']]-v[[3,'y']])*(v[[1,'x']]-v[[3,'x']]) +
(v[[3,'x']]-v[[2,'x']])*(v[[1,'y']]-v[[3,'y']])
l1 <- (
(v[[2,'y']]-v[[3,'y']])*( p[['x']]-v[[3,'x']]) +
(v[[3,'x']]-v[[2,'x']])*( p[['y']]-v[[3,'y']])
) / det
l2 <- (
(v[[3,'y']]-v[[1,'y']])*( p[['x']]-v[[3,'x']]) +
(v[[1,'x']]-v[[3,'x']])*( p[['y']]-v[[3,'y']])
) / det
l3 <- 1 - l1 - l2
list(l1, l2, l3)
}
```

```
## Conversion to Barycentric Coordinates; Array Version
bary_A <- function(p, v) {
det <- (v[,2,'y']-v[,3,'y'])*(v[,1,'x']-v[,3,'x']) +
(v[,3,'x']-v[,2,'x'])*(v[,1,'y']-v[,3,'y'])
l1 <- (
(v[,2,'y']-v[,3,'y']) * (p[,'x']-v[,3,'x']) +
(v[,3,'x']-v[,2,'x']) * (p[,'y']-v[,3,'y'])
) / det
l2 <- (
(v[,3,'y']-v[,1,'y']) * (p[,'x']-v[,3,'x']) +
(v[,1,'x']-v[,3,'x']) * (p[,'y']-v[,3,'y'])
) / det
l3 <- 1 - l1 - l2
cbind(l1, l2, l3)
}
```

In reality we could narrow some of the performance difference between these two
implementations by avoiding repeated subsets of the same data. For example,
`v[,3,'y']`

is used several times, and we could have saved that subset to
another variable for re-use.

## Plotting

```
plot_shaded <- function(p, col, v=vinit) {
par(bg='#EEEEEE')
par(xpd=TRUE)
par(mai=c(.25,.25,.25,.25))
plot.new()
points(p, pch=15, col=col, cex=.2)
polygon(v[,c('x', 'y')])
points(
vinit[,c('x', 'y')], pch=21, cex=5,
bg=rgb(apply(v[, c('r','g','b')], 2, unlist)), col='white'
)
}
```

## Array Shading

```
v_shade_A <- function(p, v) {
## 1) compute barycentric coords
bc <- bary_A(p, v)
## 2) for each point, weight vertex colors by bary coords
v.clrs <- v[,,c('r','g','b')]
clr.raw <- array(bc, dim=dim(v.clrs)) * v.clrs
## 3) for each point-colorchannel, sum the weighted values
rowSums(aperm(clr.raw, c(1,3,2)), dims=2)
}
```

There are two items that may require additional explanation in `v_shade_A`

.
First, as part of step 2) we use:

`array(bc, dim=dim(v.clrs)) * v.clrs`

Our problem is that we need to multiply the vertex weights in `bc`

with the
`v.clrs`

color channel values, but they do not conform:

`dim(bc)`

`[1] 79946 3`

`dim(v.clrs)`

`[1] 79946 3 3`

Since we use the same weights for each color channel, it is just a matter of
recycling the `bc`

matrix once for each channel. Also, since the color channel
is the last dimension we can just let `array`

recycle the data to fill out the
product of the vertex data dimensions:

`dim(array(bc, dim=dim(v.clrs)))`

`[1] 79946 3 3`

This is reasonably fast, but not completely ideal since it does mean we need to
allocate a memory chunk the size of `v.clrs`

. Ideally R would internally
recycle the `bc`

matrix as it recycles vectors in situations like `matrix(1:4, 2) * 1:2`

, but that is not how it is.

The other tricky bit is part of step 3):

`rowSums(aperm(clr.raw, c(1,3,2)), dims=2)`

We ask `rowSums`

to preserve the first two dimensions of the array with
`dims=2`

. The other dimensions are collapsed by summing the values that overlap
in the first two. Unfortunately, we can only specify adjacent dimensions to
preserve with `rowSums`

. As a result we permute the data with `aperm`

so that
the point and color channel dimensions can be the first two and thus be
preserved.

There may be clever ways to better structure the data to minimize the number of copies we make.

## Data

```
## Define basic triangle vertex position and colors
height <- sin(pi/3)
v.off <- (1 - height) / 2
vinit <- cbind(
x=list(0, .5, 1), y=as.list(c(0, height, 0) + v.off),
r=list(1, 1, 0), g=list(0,1,1), b=list(1,0,1)
)
## Sample points within the triangles
p1 <- list(x=.5, y=tan(pi/6) * .5 + v.off)
p2 <- list(x=.5, y=sin(pi/3) * .8 + v.off)
p3 <- list(x=cos(pi/6)*sin(pi/3), y=sin(pi/3) * .5 + v.off)
## Generate n x n sample points within triangle; here we also expand the
## vertex matrix so there is one row per point, even though all points are
## inside the same triangle. In our actual use case, there are many
## triangles with a handful of points within each triangle.
n <- 400
rng.x <- range(unlist(vinit[,'x']))
rng.y <- range(unlist(vinit[,'y']))
points.raw <- expand.grid(
x=seq(rng.x[1], rng.x[2], length.out=n),
y=seq(rng.y[1], rng.y[2], length.out=n),
KEEP.OUT.ATTRS=FALSE
)
vp <- lapply(vinit, '[', rep(1, nrow(points.raw)))
dim(vp) <- dim(vinit)
dimnames(vp) <- dimnames(vinit)
## we're going to drop the oob points for the sake of clarity, one
## nice perk of barycentric coordinates is that negative values indicate
## you are out of the triangle.
bc.raw <- bary_L(points.raw, vp)
inbounds <- Reduce('&', lapply(bc.raw, '>=', 0))
## Make a list-matrix version of the data
point.L <- lapply(points.raw, '[', inbounds)
vertex.L <- lapply(vp, '[', inbounds)
dim(vertex.L) <- dim(vp)
dimnames(vertex.L) <- list(V=sprintf("v%d", 1:3), Data=colnames(vp))
## Generate an array version of the same data
point.A <- do.call(cbind, point.L)
vertex.A <- array(
unlist(vertex.L), c(sum(inbounds), nrow(vertex.L), ncol(vertex.L)),
dimnames=c(list(NULL), dimnames(vertex.L))
)
```