Is the "*apply" family really not vectorized?

So we are used to say to every R new user that "apply isn't vectorized, check out the Patrick Burns R Inferno Circle 4" which says (I quote):

> A common reflex is to use a function in the apply family. This is not > vectorization, it is loop-hiding. The apply function has a for loop in > its definition. The lapply function buries the loop, but execution > times tend to be roughly equal to an explicit for loop.

Indeed, a quick look on the apply source code reveals the loop:

grep("for", capture.output(getAnywhere("apply")), value = TRUE)
## [1] "        for (i in 1L:d2) {"  "    else for (i in 1L:d2) {"

Ok so far, but a look at lapply or vapply actually reveals a completely different picture:

lapply
## function (X, FUN, ...) 
## {
##     FUN <- match.fun(FUN)
##     if (!is.vector(X) || is.object(X)) 
##        X <- as.list(X)
##     .Internal(lapply(X, FUN))
## }
## <bytecode: 0x000000000284b618>
## <environment: namespace:base>

So apparently there is no R for loop hiding there, rather they are calling internal C written function.

A quick look in the rabbit hole reveals pretty much the same picture

Moreover, let's take the colMeans function for example, which was never accused in not being vectorised

colMeans
# function (x, na.rm = FALSE, dims = 1L) 
# {
#   if (is.data.frame(x)) 
#     x <- as.matrix(x)
#   if (!is.array(x) || length(dn <- dim(x)) < 2L) 
#     stop("'x' must be an array of at least two dimensions")
#   if (dims < 1L || dims > length(dn) - 1L) 
#     stop("invalid 'dims'")
#   n <- prod(dn[1L:dims])
#   dn <- dn[-(1L:dims)]
#   z <- if (is.complex(x)) 
#     .Internal(colMeans(Re(x), n, prod(dn), na.rm)) + (0+1i) * 
#     .Internal(colMeans(Im(x), n, prod(dn), na.rm))
#   else .Internal(colMeans(x, n, prod(dn), na.rm))
#   if (length(dn) > 1L) {
#     dim(z) <- dn
#     dimnames(z) <- dimnames(x)[-(1L:dims)]
#   }
#   else names(z) <- dimnames(x)[[dims + 1]]
#   z
# }
# <bytecode: 0x0000000008f89d20>
#   <environment: namespace:base>

Huh? It also just calls .Internal(colMeans(... which we can also find in the rabbit hole. So how is this different from .Internal(lapply(..?

Actually a quick benchmark reveals that sapply performs no worse than colMeans and much better than a for loop for a big data set

m <- as.data.frame(matrix(1:1e7, ncol = 1e5))
system.time(colMeans(m))
# user  system elapsed 
# 1.69    0.03    1.73 
system.time(sapply(m, mean))
# user  system elapsed 
# 1.50    0.03    1.60 
system.time(apply(m, 2, mean))
# user  system elapsed 
# 3.84    0.03    3.90 
system.time(for(i in 1:ncol(m)) mean(m[, i]))
# user  system elapsed 
# 13.78    0.01   13.93 

In other words, is it correct to say that lapply and vapply are actually vectorised (compared to apply which is a for loop that also calls lapply) and what did Patrick Burns really mean to say?

First of all, in your example you make tests on a "data.frame" which is not fair for colMeans, apply and "[.data.frame" since they have an overhead:

system.time(as.matrix(m))  #called by `colMeans` and `apply`
#   user  system elapsed 
#   1.03    0.00    1.05
system.time(for(i in 1:ncol(m)) m[, i])  #in the `for` loop
#   user  system elapsed 
#  12.93    0.01   13.07

On a matrix, the picture is a bit different:

mm = as.matrix(m)
system.time(colMeans(mm))
#   user  system elapsed 
#   0.01    0.00    0.01 
system.time(apply(mm, 2, mean))
#   user  system elapsed 
#   1.48    0.03    1.53 
system.time(for(i in 1:ncol(mm)) mean(mm[, i]))
#   user  system elapsed 
#   1.22    0.00    1.21

Regading the main part of the question, the main difference between lapply/mapply/etc and straightforward R-loops is where the looping is done. As Roland notes, both C and R loops need to evaluate an R function in each iteration which is the most costly. The really fast C functions are those that do everything in C, so, I guess, this should be what "vectorised" is about?

An example where we find the mean in each of a "list"s elements:

(EDIT May 11 '16 : I believe the example with finding the "mean" is not a good setup for the differences between evaluating an R function iteratively and compiled code, (1) because of the particularity of R's mean algorithm on "numeric"s over a simple sum(x) / length(x) and (2) it should make more sense to test on "list"s with length(x) >> lengths(x). So, the "mean" example is moved to the end and replaced with another.)

As a simple example we could consider the finding of the opposite of each length == 1 element of a "list":

In a tmp.c file:

#include <R.h>
#define USE_RINTERNALS 
#include <Rinternals.h>
#include <Rdefines.h>

/* call a C function inside another */
double oppC(double x) { return(ISNAN(x) ? NA_REAL : -x); }
SEXP sapply_oppC(SEXP x)
{
    SEXP ans = PROTECT(allocVector(REALSXP, LENGTH(x)));
    for(int i = 0; i < LENGTH(x); i++) 
        REAL(ans)[i] = oppC(REAL(VECTOR_ELT(x, i))[0]);
    
    UNPROTECT(1);
    return(ans);
}

/* call an R function inside a C function;
 * will be used with 'f' as a closure and as a builtin */    
SEXP sapply_oppR(SEXP x, SEXP f)
{
    SEXP call = PROTECT(allocVector(LANGSXP, 2));
    SETCAR(call, install(CHAR(STRING_ELT(f, 0))));
     
    SEXP ans = PROTECT(allocVector(REALSXP, LENGTH(x)));	 
    for(int i = 0; i < LENGTH(x); i++) { 
        SETCADR(call, VECTOR_ELT(x, i));
        REAL(ans)[i] = REAL(eval(call, R_GlobalEnv))[0];
    }
    
    UNPROTECT(2);
    return(ans);
}

And in R side:

system("R CMD SHLIB /home/~/tmp.c")
dyn.load("/home/~/tmp.so")

with data:

set.seed(007)
myls = rep_len(as.list(c(NA, runif(3))), 1e7)

#a closure wrapper of `-`
oppR = function(x) -x

for_oppR = compiler::cmpfun(function(x, f)
{
    f = match.fun(f)  
    ans = numeric(length(x))
    for(i in seq_along(x)) ans[[i]] = f(x[[i]])
    return(ans)
})

Benchmarking:

#call a C function iteratively
system.time({ sapplyC =  .Call("sapply_oppC", myls) }) 
#   user  system elapsed 
#  0.048   0.000   0.047 

#evaluate an R closure iteratively
system.time({ sapplyRC =  .Call("sapply_oppR", myls, "oppR") }) 
#   user  system elapsed 
#  3.348   0.000   3.358 

#evaluate an R builtin iteratively
system.time({ sapplyRCprim =  .Call("sapply_oppR", myls, "-") }) 
#   user  system elapsed 
#  0.652   0.000   0.653 

#loop with a R closure
system.time({ forR = for_oppR(myls, "oppR") })
#   user  system elapsed 
#  4.396   0.000   4.409 

#loop with an R builtin
system.time({ forRprim = for_oppR(myls, "-") })
#   user  system elapsed 
#  1.908   0.000   1.913 

#for reference and testing 
system.time({ sapplyR = unlist(lapply(myls, oppR)) })
#   user  system elapsed 
#  7.080   0.068   7.170 
system.time({ sapplyRprim = unlist(lapply(myls, `-`)) }) 
#   user  system elapsed 
#  3.524   0.064   3.598 

all.equal(sapplyR, sapplyRprim)
#[1] TRUE 
all.equal(sapplyR, sapplyC)
#[1] TRUE
all.equal(sapplyR, sapplyRC)
#[1] TRUE
all.equal(sapplyR, sapplyRCprim)
#[1] TRUE
all.equal(sapplyR, forR)
#[1] TRUE
all.equal(sapplyR, forRprim)
#[1] TRUE

(Follows the original example of mean finding):

#all computations in C
all_C = inline::cfunction(sig = c(R_ls = "list"), body = '
    SEXP tmp, ans;
	PROTECT(ans = allocVector(REALSXP, LENGTH(R_ls)));
	
	double *ptmp, *pans = REAL(ans);
	
	for(int i = 0; i < LENGTH(R_ls); i++) {
	    pans[i] = 0.0;
		
		PROTECT(tmp = coerceVector(VECTOR_ELT(R_ls, i), REALSXP));
		ptmp = REAL(tmp);
		
		for(int j = 0; j < LENGTH(tmp); j++) pans[i] += ptmp[j];
		
		pans[i] /= LENGTH(tmp);
		
		UNPROTECT(1);
	}
	
	UNPROTECT(1);
	return(ans);
')

#a very simple `lapply(x, mean)`
C_and_R = inline::cfunction(sig = c(R_ls = "list"), body = '
    SEXP call, ans, ret;
	
	PROTECT(call = allocList(2));
	SET_TYPEOF(call, LANGSXP);
	SETCAR(call, install("mean"));
	
	PROTECT(ans = allocVector(VECSXP, LENGTH(R_ls)));
	PROTECT(ret = allocVector(REALSXP, LENGTH(ans)));
	
	for(int i = 0; i < LENGTH(R_ls); i++) {
	    SETCADR(call, VECTOR_ELT(R_ls, i));
	    SET_VECTOR_ELT(ans, i, eval(call, R_GlobalEnv));
	}
	
	double *pret = REAL(ret);
	for(int i = 0; i < LENGTH(ans); i++) pret[i] = REAL(VECTOR_ELT(ans, i))[0];
	
	UNPROTECT(3);
	return(ret);
')					  

R_lapply = function(x) unlist(lapply(x, mean))                       

R_loop = function(x) 
{
    ans = numeric(length(x))
	for(i in seq_along(x)) ans[i] = mean(x[[i]])
	return(ans)
} 

R_loopcmp = compiler::cmpfun(R_loop)
                    

set.seed(007); myls = replicate(1e4, runif(1e3), simplify = FALSE)
all.equal(all_C(myls), C_and_R(myls))
#[1] TRUE
all.equal(all_C(myls), R_lapply(myls))
#[1] TRUE
all.equal(all_C(myls), R_loop(myls))
#[1] TRUE
all.equal(all_C(myls), R_loopcmp(myls))
#[1] TRUE

microbenchmark::microbenchmark(all_C(myls), 
                               C_and_R(myls), 
                               R_lapply(myls), 
                               R_loop(myls), 
                               R_loopcmp(myls), 
                               times = 15)
#Unit: milliseconds
#            expr       min        lq    median        uq      max neval
#     all_C(myls)  37.29183  38.19107  38.69359  39.58083  41.3861    15
#   C_and_R(myls) 117.21457 123.22044 124.58148 130.85513 169.6822    15
#  R_lapply(myls)  98.48009 103.80717 106.55519 109.54890 116.3150    15
#    R_loop(myls) 122.40367 130.85061 132.61378 138.53664 178.5128    15
# R_loopcmp(myls) 105.63228 111.38340 112.16781 115.68909 128.1976    15

To me, vectorisation is primarily about making your code easier to write and easier to understand.

The goal of a vectorised function is to eliminate the book-keeping associated with a for loop. For example, instead of:

means <- numeric(length(mtcars))
for (i in seq_along(mtcars)) {
  means[i] <- mean(mtcars[[i]])
}
sds <- numeric(length(mtcars))
for (i in seq_along(mtcars)) {
  sds[i] <- sd(mtcars[[i]])
}

You can write:

means <- vapply(mtcars, mean, numeric(1))
sds   <- vapply(mtcars, sd, numeric(1))

That makes it easier to see what's the same (the input data) and what's different (the function you're applying).

A secondary advantage of vectorisation is that the for-loop is often written in C, rather than in R. This has substantial performance benefits, but I don't think it's the key property of vectorisation. Vectorisation is fundamentally about saving your brain, not saving the computer work.

I agree with Patrick Burns' view that it is rather loop hiding and not code vectorisation. Here's why:

Consider this C code snippet:

for (int i=0; i<n; i++)
  c[i] = a[i] + b[i]

What we would like to do is quite clear. But how the task is performed or how it could be performed isn't really. A for-loop by default is a serial construct. It doesn't inform if or how things can be done in parallel.

The most obvious way is that the code is run in a sequential manner. Load a[i] and b[i] on to registers, add them, store the result in c[i], and do this for each i.

However, modern processors have vector or SIMD instruction set which is capable of operating on a vector of data during the same instruction when performing the same operation (e.g., adding two vectors as shown above). Depending on the processor/architecture, it might be possible to add, say, four numbers from a and b under the same instruction, instead of one at a time.

> We would like to exploit the Single Instruction Multiple Data and perform data level parallelism, i.e., load 4 things at a time, add 4 things at time, store 4 things at a time, for example. And this is code vectorisation. > > Note that this is different from code parallelisation -- where multiple computations are performed concurrently.

It'd be great if the compiler identifies such blocks of code and automatically vectorises them, which is a difficult task. Automatic code vectorisation is a challenging research topic in Computer Science. But over time, compilers have gotten better at it. You can check the auto vectorisation capabilities of GNU-gcc here. Similarly for LLVM-clang here. And you can also find some benchmarks in the last link compared against gcc and ICC (Intel C++ compiler).

gcc (I'm on v4.9) for example doesn't vectorise code automatically at -O2 level optimisation. So if we were to execute the code shown above, it'd be run sequentially. Here's the timing for adding two integer vectors of length 500 million.

We either need to add the flag -ftree-vectorize or change optimisation to level -O3. (Note that -O3 performs other additional optimisations as well). The flag -fopt-info-vec is useful as it informs when a loop was successfully vectorised).

# compiling with -O2, -ftree-vectorize and  -fopt-info-vec
# test.c:32:5: note: loop vectorized
# test.c:32:5: note: loop versioned for vectorization because of possible aliasing
# test.c:32:5: note: loop peeled for vectorization to enhance alignment    

This tells us that the function is vectorised. Here are the timings comparing both non-vectorised and vectorised versions on integer vectors of length 500 million:

x = sample(100L, 500e6L, TRUE)
y = sample(100L, 500e6L, TRUE)
z = vector("integer", 500e6L) # result vector

# non-vectorised, -O2
system.time(.Call("Csum", x, y, z))
#    user  system elapsed 
#   1.830   0.009   1.852

# vectorised using flags shown above at -O2
system.time(.Call("Csum", x, y, z))
#    user  system elapsed 
#   0.361   0.001   0.362

# both results are checked for identicalness, returns TRUE

---

This part can be safely skipped without losing continuity.

Compilers will not always have sufficient information to vectorise. We could use OpenMP specification for parallel programming, which also provides a simd compiler directive to instruct compilers to vectorise the code. It is essential to ensure that there are no memory overlaps, race conditions etc.. when vectorising code manually, else it'll result in incorrect results.

#pragma omp simd
for (i=0; i<n; i++) 
  c[i] = a[i] + b[i]

By doing this, we specifically ask the compiler to vectorise it no matter what. We'll need to activate OpenMP extensions by using compile time flag -fopenmp. By doing that:

# timing with -O2 + OpenMP with simd
x = sample(100L, 500e6L, TRUE)
y = sample(100L, 500e6L, TRUE)
z = vector("integer", 500e6L) # result vector
system.time(.Call("Cvecsum", x, y, z))
#    user  system elapsed 
#   0.360   0.001   0.360

which is great! This was tested with gcc v6.2.0 and llvm clang v3.9.0 (both installed via homebrew, MacOS 10.12.3), both of which support OpenMP 4.0.

---

In this sense, even though Wikipedia page on Array Programming mentions that languages that operate on entire arrays usually call that as vectorised operations, it really is loop hiding IMO (unless it is actually vectorised).

In case of R, even rowSums() or colSums() code in C don't exploit code vectorisation IIUC; it is just a loop in C. Same goes for lapply(). In case of apply(), it's in R. All of these are therefore loop hiding.

> In short, wrapping an R function by: > > just writing a for-loop in C != vectorising your code.
> just writing a for-loop in R != vectorising your code. > > Intel Math Kernel Library (MKL) for example implements vectorised forms of functions.

HTH

---

References:

1. Talk by James Reinders, Intel (this answer is mostly an attempt to summarise this excellent talk)

So to sum the great answers/comments up into some general answer and provide some background: R has 4 types of loops (in from not-vectorized to vectorized order)

1. R for loop that repeatedly calls R functions in each iterations (Not vectorised)
2. C loop that repeatedly calls R functions in each iterations (Not vectorised)
3. C loop that calls R function only once (Somewhat vectorized)
4. A plain C loop that doesn't call any R function at all and uses it own compiled functions (Vectorized)

So the *apply family is the second type. Except apply which is more of the first type

You can understand this from the comment in its source code

> /* .Internal(lapply(X, FUN)) / > > / This is a special .Internal, so has unevaluated arguments. It is
> called from a closure wrapper, so X and FUN are promises. FUN must > be unevaluated for use in e.g. bquote . > */

That means that lapplys C code accepts an unevaluated function from R and later evaluates it within the C code itself. This is basically the difference between lapplys .Internal call

.Internal(lapply(X, FUN))

Which has a FUN argument that holds an R function

And the colMeans .Internal call which does not have a FUN argument

.Internal(colMeans(Re(x), n, prod(dn), na.rm))

colMeans, unlike lapply knows exactly what function it needs to use, thus it calculates the mean internally within the C code.

You can clearly see the evaluation process of the R function in each iteration within lapply C code

 for(R_xlen_t i = 0; i < n; i++) {
	  if (realIndx) REAL(ind)[0] = (double)(i + 1);
	  else INTEGER(ind)[0] = (int)(i + 1);
	  tmp = eval(R_fcall, rho);   // <----------------------------- here it is
	  if (MAYBE_REFERENCED(tmp)) tmp = lazy_duplicate(tmp);
	  SET_VECTOR_ELT(ans, i, tmp);
   }

---

To sum things up, lapply is not vectorized, though it has two possible advantages over the plain R for loop

1. Accessing and assigning in a loop seems to be faster in C (i.e. in lapplying a function) Although the difference seems big, we, still, stay at the microsecond level and the costly thing is the valuation of an R function in each iteration. A simple example:

   ffR = function(x)  {
       ans = vector("list", length(x))
       for(i in seq_along(x)) ans[[i]] = x[[i]]
       ans 
   }
   
   ffC = inline::cfunction(sig = c(R_x = "data.frame"), body = '
       SEXP ans;
       PROTECT(ans = allocVector(VECSXP, LENGTH(R_x)));
       for(int i = 0; i < LENGTH(R_x); i++) 
              SET_VECTOR_ELT(ans, i, VECTOR_ELT(R_x, i));
       UNPROTECT(1);
       return(ans); 
   ')
   
   set.seed(007) 
   myls = replicate(1e3, runif(1e3), simplify = FALSE)     
   mydf = as.data.frame(myls)
   
   all.equal(ffR(myls), ffC(myls))
   #[1] TRUE 
   all.equal(ffR(mydf), ffC(mydf))
   #[1] TRUE
   
   microbenchmark::microbenchmark(ffR(myls), ffC(myls), 
                                  ffR(mydf), ffC(mydf),
                                  times = 30)
   #Unit: microseconds
   #      expr       min        lq    median        uq       max neval
   # ffR(myls)  3933.764  3975.076  4073.540  5121.045 32956.580    30
   # ffC(myls)    12.553    12.934    16.695    18.210    19.481    30
   # ffR(mydf) 14799.340 15095.677 15661.889 16129.689 18439.908    30
   # ffC(mydf)    12.599    13.068    15.835    18.402    20.509    30
   

2. As mentioned by @Roland, it runs a compiled C loop rather an interpreted R loop

---

Though when vectorizing your code, there are some things you need to take into account.

1. If your data set (let's call it df) is of class data.frame, some vectorized functions (such as colMeans, colSums, rowSums, etc.) will have to convert it to a matrix first, simply because this is how they were designed. This means that for a big df this can create a huge overhead. While lapply won't have to do this as it extracts the actual vectors out of df (as data.frame is just a list of vectors) and thus, if you have not so many columns but many rows, lapply(df, mean) can sometimes be better option than colMeans(df).

2. Another thing to remember is that R has a great variety of different function types, such as .Primitive, and generic (S3, S4) see here for some additional information. The generic function have to do a method dispatch which sometimes a costly operation. For example, mean is generic S3 function while sum is Primitive. Thus some times lapply(df, sum) could be very efficient compared colSums from the reasons listed above