# Partitioned Distances

pdist on CRAN

Given a matrix X with m observations and another matrix Y with n observations, Partitioned Distances computes the m by n distance matrix. A rectangular distance matrix can be more appropriate than a square matrix in many applications; for example, in bipartite graphs we might be concerned with the distance between objects in Graph A with objects in Graph B, but we may not care about the distance between objects within Graph A or Graph B. Currently, R only has a `dist` function which returns square distance matrices.

## Performance

`pdist` is a slightly optimized version of the native `dist` function; distances are not computed between objects that are both in X or both in Y. Using native functions, we could stack X and Y on top of each other using `rbind`, and call `dist` on the result, but this would compute the (m+n) by (m+n) distance matrix, yielding m^2 + mn + n^2 unnecessary distance computations. If the matrices have p columns, and the distance metric is the Euclidean metric, then p(m^2 + mn + n^2) unnecessary flops are made. More complex metrics, such as dynamic time warping, can run in O(p^3), which means a naive dist function would make O(p3(m2 + mn + n^2)) unnecessary flops!

## Timing

Using a matrix X that is 1000 by 100, it took 0.543 seconds to compute the distance matrix based on the Euclidean metric using `dist`. Using pdist, the timing was the same. If we are interested in the subset A taken by the first 100 rows of X, and subset B taken by the next 100 rows of X, we can compute a smaller distance matrix in only 0.006 seconds!