Parallel Distance Matrix Calculation with RcppParallel

The RcppParallel package includes high level functions for doing parallel programming with Rcpp. For example, the parallelFor function can be used to convert the work of a standard serial “for” loop into a parallel one.

This article describes using RcppParallel to compute pairwise distances for each row in an input data matrix and return an n x n lower-triangular distance matrix which can be used with clustering tools from within R, e.g., hclust.

Jensen-Shannon Distance

In this example, we compute the Jensen-Shannon distance (JSD); a metric not a part of base R. Calculating distance matrices is a common practice in clustering applications (unsupervised learning). Certain clustering methods, such as partitioning around medoids (PAM) and hierarchical clustering, operate directly on this matrix.

A distance matrix stores the n*(n-1)/2 pairwise distances/similarities between observations in an n x p matrix where n correspond to the independent observational units and p represent the covariates measured on each individual. As a result we are typically limited by the size of n as the algorithm scales quadratically in both time and space in n.

Implementation in R

As a baseline we’ll start with the implementation of Jenson-Shannon distance in plain R:

js_distance <- function(mat) {
  kld = function(p,q) sum(ifelse(p == 0 | q == 0, 0, log(p/q)*p))
  res = matrix(0, nrow(mat), nrow(mat))
  for (i in 1:(nrow(mat) - 1)) {
    for (j in (i+1):nrow(mat)) {
      m = (mat[i,] + mat[j,])/2
      d1 = kld(mat[i,], m)
      d2 = kld(mat[j,], m)
      res[j,i] = sqrt(.5*(d1 + d2))
    }
  }
  res
}

Implementation using Rcpp

Here is a re-implementation of js_distance using Rcpp. Note that this doesn’t yet take advantage of parallel processing, but still yields an approximately 50x speedup over the original R version on a 2.6GHz Haswell MacBook Pro.

Abstractly, a Distance function will take two vectors in R^J and return a value in R⁺. In this implementation, we don’t support arbitrary distance metrics, i.e., the JSD code computes the values from within the parallel kernel.

Our distance function kl_divergence is defined below and takes three parameters: iterators to the beginning and end of vector 1 and an iterator to the beginning of vector 2 (the end position of vector2 is implied by the end position of vector1).

#include <Rcpp.h>
using namespace Rcpp;

#include <cmath>
#include <algorithm>

// generic function for kl_divergence
template <typename InputIterator1, typename InputIterator2>
inline double kl_divergence(InputIterator1 begin1, InputIterator1 end1, 
                            InputIterator2 begin2) {
  
   // value to return
   double rval = 0;
   
   // set iterators to beginning of ranges
   InputIterator1 it1 = begin1;
   InputIterator2 it2 = begin2;
   
   // for each input item
   while (it1 != end1) {
      
      // take the value and increment the iterator
      double d1 = *it1++;
      double d2 = *it2++;
      
      // accumulate if appropirate
      if (d1 > 0 && d2 > 0)
         rval += std::log(d1 / d2) * d1;
   }
   return rval;  
}

With the kl_distance function defined we can now iteratively apply it to the rows of the input matrix to generate the distance matrix:

// helper function for taking the average of two numbers
inline double average(double val1, double val2) {
   return (val1 + val2) / 2;
}

// [[Rcpp::export]]
NumericMatrix rcpp_js_distance(NumericMatrix mat) {
  
   // allocate the matrix we will return
   NumericMatrix rmat(mat.nrow(), mat.nrow());
   
   for (int i = 0; i < rmat.nrow(); i++) {
      for (int j = 0; j < i; j++) {
      
         // rows we will operate on
         NumericMatrix::Row row1 = mat.row(i);
         NumericMatrix::Row row2 = mat.row(j);
         
         // compute the average using std::tranform from the STL
         std::vector<double> avg(row1.size());
         std::transform(row1.begin(), row1.end(), // input range 1
                        row2.begin(),             // input range 2
                        avg.begin(),              // output range 
                        average);                 // function to apply
      
         // calculate divergences
         double d1 = kl_divergence(row1.begin(), row1.end(), avg.begin());
         double d2 = kl_divergence(row2.begin(), row2.end(), avg.begin());
        
         // write to output matrix
         rmat(i,j) = std::sqrt(.5 * (d1 + d2));
      }
   }
   
   return rmat;
}

Parallel Version using RcppParallel

Adapting the serial version to run in parallel is straightforward. A few notes about the implementation:

• To implement a parallel version we need to create a function object that can process discrete chunks of work (i.e. ranges of input).

• Since the parallel version will be called from background threads, we can’t use R and Rcpp APIs directly. Rather, we use the threadsafe RMatrix accessor class provided by RcppParallel to read and write to directly the underlying matrix memory.

• Other than organzing the code as a function object and using RMatrix, the parallel code is almost identical to the serial code. The main difference is that the outer loop starts with the begin index passed to the worker function rather than 0.

Parallelizing in this case has a big payoff: we observe performance of about 5.5x the serial version on a 2.6GHz Haswell MacBook Pro with 4 cores (8 with hyperthreading). Here is the definition of the JsDistance function object:

// [[Rcpp::depends(RcppParallel)]]
#include <RcppParallel.h>
using namespace RcppParallel;

struct JsDistance : public Worker {
   
   // input matrix to read from
   const RMatrix<double> mat;
   
   // output matrix to write to
   RMatrix<double> rmat;
   
   // initialize from Rcpp input and output matrixes (the RMatrix class
   // can be automatically converted to from the Rcpp matrix type)
   JsDistance(const NumericMatrix mat, NumericMatrix rmat)
      : mat(mat), rmat(rmat) {}
   
   // function call operator that work for the specified range (begin/end)
   void operator()(std::size_t begin, std::size_t end) {
      for (std::size_t i = begin; i < end; i++) {
         for (std::size_t j = 0; j < i; j++) {
            
            // rows we will operate on
            RMatrix<double>::Row row1 = mat.row(i);
            RMatrix<double>::Row row2 = mat.row(j);
            
            // compute the average using std::tranform from the STL
            std::vector<double> avg(row1.length());
            std::transform(row1.begin(), row1.end(), // input range 1
                           row2.begin(),             // input range 2
                           avg.begin(),              // output range 
                           average);                 // function to apply
              
            // calculate divergences
            double d1 = kl_divergence(row1.begin(), row1.end(), avg.begin());
            double d2 = kl_divergence(row2.begin(), row2.end(), avg.begin());
               
            // write to output matrix
            rmat(i,j) = sqrt(.5 * (d1 + d2));
         }
      }
   }
};

Now that we have the JsDistance function object we can pass it to parallelFor, specifying an iteration range based on the number of rows in the input matrix:

// [[Rcpp::export]]
NumericMatrix rcpp_parallel_js_distance(NumericMatrix mat) {
  
   // allocate the matrix we will return
   NumericMatrix rmat(mat.nrow(), mat.nrow());

   // create the worker
   JsDistance jsDistance(mat, rmat);
     
   // call it with parallelFor
   parallelFor(0, mat.nrow(), jsDistance);

   return rmat;
}

Benchmarks

We now compare the performance of the three different implementations: pure R, serial Rcpp, and parallel Rcpp:

# create a matrix
n  = 1000
m = matrix(runif(n*10), ncol = 10)
m = m/rowSums(m)

# ensure that serial and parallel versions give the same result
r_res <- js_distance(m)
rcpp_res <- rcpp_js_distance(m)
rcpp_parallel_res <- rcpp_parallel_js_distance(m)
stopifnot(all(rcpp_res == rcpp_parallel_res))
stopifnot(all(rcpp_parallel_res - r_res < 1e-10)) ## precision differences

# compare performance
library(rbenchmark)
res <- benchmark(js_distance(m),
                 rcpp_js_distance(m),
                 rcpp_parallel_js_distance(m),
                 replications = 3,
                 order="relative")
res[,1:4]

                          test replications elapsed relative
3 rcpp_parallel_js_distance(m)            3   0.110    1.000
2          rcpp_js_distance(m)            3   0.618    5.618
1               js_distance(m)            3  35.560  323.273

The serial Rcpp version yields a more than 50x speedup over straight R code. The parallel Rcpp version provides another 5.5x speedup, amounting to a total gain of over 300x compared to the original R version.

Note that performance gains will typically be 30-50% less on Windows systems as a result of less sophisticated thread scheduling (RcppParallel does not currently use TBB on Windows whereas it does on the Mac and Linux).

You can learn more about using RcppParallel at https://github.com/RcppCore/RcppParallel.