R: Summing up neighboring matrix elements. How to speed up?

Question

I'm working with large matrices of about 2500x2500x50 (lonxlatxtime). The matrix contains only 1 and 0. I need to know for each timestep the sum of the 24 surrounding elements. So far I did it about this way:

xdim <- 2500
ydim <- 2500
tdim <- 50
a <- array(0:1,dim=c(xdim,ydim,tdim))
res <- array(0:1,dim=c(xdim,ydim,tdim))

for (t in 1:tdim){
  for (x in 3:(xdim-2)){
    for (y in 3:(ydim-2)){
      res[x,y,t] <- sum(a[(x-2):(x+2),(y-2):(y+2),t])
    }
  }
}

This works, but it is much too slow for my needs. Has anybody please an advice how to speed up?

Answer 1

Intro

I have to say, there are so many hidden things behind just the setup of the arrays. The remainder of the problem is trivial though. As a result, there are two ways to go about it really:

1. Bruteforce given by @Alex (written in C++)
2. Observing replication patterns

Bruteforce with OpenMP

If we want to 'brute force' it, then we can use the suggestion given by @Alex to employ OpenMP with Armadillo

#include <RcppArmadillo.h>

// [[Rcpp::depends(RcppArmadillo)]]

// Add a flag to enable OpenMP at compile time
// [[Rcpp::plugins(openmp)]]

// Protect against compilers without OpenMP
#ifdef _OPENMP
  #include <omp.h>
#endif

// [[Rcpp::export]]
arma::cube cube_parallel(arma::cube a, arma::cube res, int cores = 1) {

  // Extract the different dimensions
  unsigned int tdim = res.n_slices;

  unsigned int xdim = res.n_rows;

  unsigned int ydim = res.n_cols;

  // Same calculation loop
  #pragma omp parallel for num_threads(cores)
  for (unsigned int t = 0; t < tdim; t++){
    // pop the T
    arma::mat temp_mat = a.slice(t);

    // Subset the rows
    for (unsigned int x = 2; x < xdim-2; x++){

      arma::mat temp_row_sub = temp_mat.rows(x-2, x+2);

      // Iterate over the columns with unit accumulative sum
      for (unsigned int y = 2; y <  ydim-2; y++){
        res(x,y,t) = accu(temp_row_sub.cols(y-2,y+2));
      }
    }
  }

  return res;
}

Replication Patterns

However, the smarter approach is understanding how the array(0:1, dims) is being constructed.

Most notably:

• Case 1: If xdim is even, then only the rows of a matrix alternate.
• Case 2: If xdim is odd and ydim is odd, then rows alternate as well as the matrices alternate.
• Case 3: If xdim is odd and ydim is even, then only the rows alternate

Examples

Let's see the cases in action to observe the patterns.

Case 1:

xdim <- 2
ydim <- 3
tdim <- 2
a <- array(0:1,dim=c(xdim,ydim,tdim))

Output:

, , 1

     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    1    1    1

, , 2

     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    1    1    1

Case 2:

xdim <- 3
ydim <- 3
tdim <- 3
a <- array(0:1,dim=c(xdim,ydim,tdim))

Output:

, , 1

     [,1] [,2] [,3]
[1,]    0    1    0
[2,]    1    0    1
[3,]    0    1    0

, , 2

     [,1] [,2] [,3]
[1,]    1    0    1
[2,]    0    1    0
[3,]    1    0    1

, , 3

     [,1] [,2] [,3]
[1,]    0    1    0
[2,]    1    0    1
[3,]    0    1    0

Case 3:

xdim <- 3
ydim <- 4
tdim <- 2
a <- array(0:1,dim=c(xdim,ydim,tdim))

Output:

, , 1

     [,1] [,2] [,3] [,4]
[1,]    0    1    0    1
[2,]    1    0    1    0
[3,]    0    1    0    1

, , 2

     [,1] [,2] [,3] [,4]
[1,]    0    1    0    1
[2,]    1    0    1    0
[3,]    0    1    0    1

Pattern Hacking

Alrighty, based on the above discussion, we opt to make a bit of code the exploits this unique pattern.

Creating Alternating Vectors

An alternating vector in this case switches between two different values.

#include <RcppArmadillo.h>
// [[Rcpp::depends(RcppArmadillo)]]

// ------- Make Alternating Vectors

arma::vec odd_vec(unsigned int xdim){

  // make a temporary vector to create alternating 0-1 effect by row.
  arma::vec temp_vec(xdim);

  // Alternating vector (anyone have a better solution? )
  for (unsigned int i = 0; i < xdim; i++) {
    temp_vec(i) = (i % 2 ? 0 : 1);
  }

  return temp_vec;
}

arma::vec even_vec(unsigned int xdim){

  // make a temporary vector to create alternating 0-1 effect by row.
  arma::vec temp_vec(xdim);

  // Alternating vector (anyone have a better solution? )
  for (unsigned int i = 0; i < xdim; i++) {
    temp_vec(i) = (i % 2 ? 1 : 0); // changed
  }

  return temp_vec;
}

Creating the three cases of matrix

As mentioned above, there are three cases of matrix. The even, first odd, and second odd cases.

// --- Handle the different cases 

// [[Rcpp::export]]
arma::mat make_even_matrix(unsigned int xdim, unsigned int ydim){

  arma::mat temp_mat(xdim,ydim);

  temp_mat.each_col() = even_vec(xdim);

  return temp_mat;
}

// xdim is odd and ydim is even
// [[Rcpp::export]]
arma::mat make_odd_matrix_case1(unsigned int xdim, unsigned int ydim){

  arma::mat temp_mat(xdim,ydim);

  arma::vec e_vec = even_vec(xdim);
  arma::vec o_vec = odd_vec(xdim);

  // Alternating column 
  for (unsigned int i = 0; i < ydim; i++) {
    temp_mat.col(i) = (i % 2 ? o_vec : e_vec);
  }

  return temp_mat;
}

// xdim is odd and ydim is odd    
// [[Rcpp::export]]
arma::mat make_odd_matrix_case2(unsigned int xdim, unsigned int ydim){

  arma::mat temp_mat(xdim,ydim);

  arma::vec e_vec = even_vec(xdim);
  arma::vec o_vec = odd_vec(xdim);

  // Alternating column 
  for (unsigned int i = 0; i < ydim; i++) {
    temp_mat.col(i) = (i % 2 ? e_vec : o_vec); // slight change
  }

  return temp_mat;
}

Calculation Engine

Same as the previous solution, just without the t as we no longer need to repeat calculations.

// --- Calculation engine

// [[Rcpp::export]]
arma::mat calc_matrix(arma::mat temp_mat){

  unsigned int xdim = temp_mat.n_rows;

  unsigned int ydim = temp_mat.n_cols;

  arma::mat res = temp_mat;

  // Subset the rows
  for (unsigned int x = 2; x < xdim-2; x++){

    arma::mat temp_row_sub = temp_mat.rows(x-2, x+2);

    // Iterate over the columns with unit accumulative sum
    for (unsigned int y = 2; y <  ydim-2; y++){
      res(x,y) = accu(temp_row_sub.cols(y-2,y+2));
    }
  }

  return res;
}

Call Main Function

Here is the core function that pieces everything together. This gives us the desired distance arrays.

// --- Main Engine

// Create the desired cube information
// [[Rcpp::export]]
arma::cube dim_to_cube(unsigned int xdim = 4, unsigned int ydim = 4, unsigned int tdim = 3) {

  // Initialize values in A
  arma::cube res(xdim,ydim,tdim);

  if(xdim % 2 == 0){
    res.each_slice() = calc_matrix(make_even_matrix(xdim, ydim));
  }else{

    if(ydim % 2 == 0){

      res.each_slice() = calc_matrix(make_odd_matrix_case1(xdim, ydim));

    }else{

      arma::mat first_odd_mat = calc_matrix(make_odd_matrix_case1(xdim, ydim));

      arma::mat sec_odd_mat = calc_matrix(make_odd_matrix_case2(xdim, ydim));

      for(unsigned int t = 0; t < tdim; t++){
        res.slice(t) = (t % 2 ? sec_odd_mat : first_odd_mat);
      }

    }

  }

  return res;
}

Timing

Now, the real truth is how well does this perform:

Unit: microseconds
       expr      min        lq       mean    median        uq       max neval
    r_1core 3538.022 3825.8105 4301.84107 3957.3765 4043.0085 16856.865   100
 alex_1core 2790.515 2984.7180 3461.11021 3076.9265 3189.7890 15371.406   100
  cpp_1core  174.508  180.7190  197.29728  194.1480  204.8875   338.510   100
  cpp_2core  111.960  116.0040  126.34508  122.7375  136.2285   162.279   100
  cpp_3core   81.619   88.4485  104.54602   94.8735  108.5515   204.979   100
  cpp_cache   40.637   44.3440   55.08915   52.1030   60.2290   302.306   100

Script used for timing:

cpp_parallel = cube_parallel(a,res, 1)
alex_1core = alex(a,res,xdim,ydim,tdim)
cpp_cache = dim_to_cube(xdim,ydim,tdim)
op_answer = cube_r(a,res,xdim,ydim,tdim)

all.equal(cpp_parallel, op_answer)
all.equal(cpp_cache, op_answer)
all.equal(alex_1core, op_answer)

xdim <- 20
ydim <- 20
tdim <- 5
a <- array(0:1,dim=c(xdim,ydim,tdim))
res <- array(0:1,dim=c(xdim,ydim,tdim))


ga = microbenchmark::microbenchmark(r_1core = cube_r(a,res,xdim,ydim,tdim),
                                    alex_1core = alex(a,res,xdim,ydim,tdim),
                                    cpp_1core = cube_parallel(a,res, 1), 
                                    cpp_2core = cube_parallel(a,res, 2), 
                                    cpp_3core = cube_parallel(a,res, 3),
                                    cpp_cache = dim_to_cube(xdim,ydim,tdim))

Answer 2

Here's one solution that's fast for a large array:

res <- apply(a, 3, function(a) t(filter(t(filter(a, rep(1, 5), circular=TRUE)), rep(1, 5), circular=TRUE)))
dim(res) <- c(xdim, ydim, tdim)

I filtered the array using rep(1,5) as the weights (i.e. sum values within a neighborhood of 2) along each dimension. I then modified the dim attribute since it initially comes out as a matrix.

Note that this wraps the sum around at the edges of the array (which might make sense since you're looking at latitude and longitude; if not, I can modify my answer).

---

---

For a concrete example:

xdim <- 500
ydim <- 500
tdim <- 15
a <- array(0:1,dim=c(xdim,ydim,tdim))

and here's what you're currently using (with NAs at the edges) and how long this example takes on my laptop:

f1 <- function(a, xdim, ydim, tdim){
  res <- array(NA_integer_,dim=c(xdim,ydim,tdim))
  for (t in 1:tdim){
    for (x in 3:(xdim-2)){
      for (y in 3:(ydim-2)){
        res[x,y,t] <- sum(a[(x-2):(x+2),(y-2):(y+2),t])
      }
    }
  }
  return(res)
}

system.time(res1 <- f1(a, xdim, ydim, tdim))
#   user  system elapsed
# 14.813   0.005  14.819

And here's a comparison with the version I described:

f2 <- function(a, xdim, ydim, tdim){
  res <- apply(a, 3, function(a) t(filter(t(filter(a, rep(1, 5), circular=TRUE)), rep(1, 5), circular=TRUE)))
  dim(res) <- c(xdim, ydim, tdim)
  return(res)
}

system.time(res2 <- f2(a, xdim, ydim, tdim))
#  user  system elapsed
# 1.188   0.047   1.236

You can see there's a significant speed boost (for large arrays). And to check that it's giving the correct solution (note that I'm adding NAs so both results match, since the one I gave filters in a circular manner):

## Match NAs
res2NA <- ifelse(is.na(res1), NA, res2)

all.equal(res2NA, res1)
# [1] TRUE

I'll add that your full array (2500x2500x50) took just under a minute (about 55 seconds), although it did use a lot of memory in the process, FYI.