#Partitioned Distances
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(p^3(m^2 + 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!