Pairwise distances in R

For a recent project I needed to calculate the pairwise distances of a set of observations to a set of cluster centers. In MATLAB you can use the pdist function for this. As far as I know, there is no equivalent in the R standard packages. So I looked into writing a fast implementation for R. Turns out that vectorizing makes it about 40x faster. Using Rcpp is another 5-6x faster, ending up with a 225x speed-up over the naive implementation.

At the start I wrote a naive (and very slow) implementation that look liked this:

naive_pdist <- function(A,B) {
    # A: matrix with obersvation vectors 
    #         (nrow = number of observations)
    # B: matrix with another set of vectors
    #          (e.g. cluster centers)
    result = matrix(ncol=nrow(B), nrow=nrow(A))
    for (i in 1:nrow(A))
        for (j in 1:nrow(B))
            result[i,j] = sqrt(sum( (A[i,] - B[j,])^2 ))


When I realized that this is too slow, I started looking for an implementation and I found the pdist CRAN package, which is way faster:

Comparing a naive R pdist with pdist::pdist

pdist::pdist is about 12x faster than the naive R implementation

The speed up made me curious about how pdist was implemented in this package. To my disappointment it is the same naive method only written in C (and using float, not double precision) — no vectorization and no tricks involved. So I was pretty sure there was room for improvement.

In search for tricks on computing the pairwise distance a blog post from Alex Smola turned up. He suggest to “use the second binomial formula to decompose the distance into norms of vectors in A and B and an inner product between them”. Translated into R code this solution looks like this:

vectorized_pdist <- function(A,B) 
    an = apply(A, 1, function(rvec) crossprod(rvec,rvec))
    bn = apply(B, 1, function(rvec) crossprod(rvec,rvec))

    m = nrow(A)
    n = nrow(B)

    tmp = matrix(rep(an, n), nrow=m) 
    tmp = tmp +  matrix(rep(bn, m), nrow=m, byrow=TRUE)
    sqrt( tmp - 2 * tcrossprod(A,B) )
Vectorized R pdist is 3x faster than naive C pdist

Vectorized R pdist is 3x faster than naive C pdist

Now that I knew how to implement pdist with a couple of simple operations, I wanted to know how much faster a C (or C++) implementation would be. Thanks to the excellent Rcpp and RcppArmadillo package, it is easy to translate the above R code into C++:

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

using namespace Rcpp;

// [[Rcpp::export]]
NumericMatrix fastPdist2(NumericMatrix Ar, NumericMatrix Br) {
    int m = Ar.nrow(), 
        n = Br.nrow(),
        k = Ar.ncol();
    arma::mat A = arma::mat(Ar.begin(), m, k, false); 
    arma::mat B = arma::mat(Br.begin(), n, k, false); 

    arma::colvec An =  sum(square(A),1);
    arma::colvec Bn =  sum(square(B),1);

    arma::mat C = -2 * (A * B.t());
    C.each_col() += An;
    C.each_row() += Bn.t();

    return wrap(sqrt(C)); 

This C++ implementation turns out to be another 6x faster than the vectorized R implementation:

RcppArmadillo achieves another 6x speed-up over the R implementaion

RcppArmadillo achieves another 6x speed-up over the R implementaion

All implementations compared

The time measurements for all implementations:

Unit: milliseconds
                   expr         min          lq      median          uq        max  neval
 vectorized_pdist(A, B)   26.667005   30.299216   32.945532   34.548596   134.8368    100
        fastPdist(A, B)    5.357734    5.581193    5.693534    5.798465   109.9736    100
      naive_pdist(A, B) 1259.290444 1280.897937 1290.150653 1320.467180  1425.3864    100
     pdist::pdist(A, B)   98.825835  101.955146  103.719962  105.843313   205.7123    100 

and the speed up among all implementations:

vectorized C++ vectorized R naive C naive R
vectorized C++ 1.00 5.79 18.23 226.74
vectorized R 0.17 1. 00 3.15 39.15
naive C 0.05 0.32 1.00 12.44
naive R 0.00 0.03 0.08 1.00


In my example the (naive) C implementation only acheived a 12x speed up, while the improved R implementation was about 40x faster. These findings agree with what is preached in various blog posts and guides about R: first try to vectorize code, then try to find a faster method (algorithm), and only as last step consider using a faster language.

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  1. `pdist` takes into account NAs. I wonder if your fast implementation allows this feature.

  2. You can speed up the R vectorized_pdist even more by not using apply, because apply will do many function calls, which are slow in R. For example:

    Instead of
        an = apply(A, 1, function(rvec) crossprod(rvec,rvec))
        bn = apply(B, 1, function(rvec) crossprod(rvec,rvec))
    just do
    an = rowSums(A^2)
    bn = rowSums(B^2)

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