An algorithm for realizing Euclidean distance matrices

We present an efficient algorithm to find a realization of a (full) n × n squared Euclidean distance matrix in the smallest possible dimension. Most existing algorithms work in a given dimension: most of these can be transformed to an algorithm to find the minimum dimension, but gain a logarithmic factor of n in their worstcase running time. Our algorithm performs cubically in n (and linearly when the dimension is fixed, which happens in most applications).