Testing metric properties

Finite metric spaces, and in particular tree metrics play an important role in various disciplines such as evolutionary biology and statistics. A natural family of problems concerning metrics is deciding, given a matrix <italic>M</italic>, whether or not it is a distance metric of a certain predetermined type. Here we consider the following relaxed version of such decision problems: For any given matrix <italic>M</italic> and parameter <italic>\eps</italic>, we are interested in determining, by probing <italic>M</italic>, whether <italic>M</italic> has a particular metric property <italic>P</italic>, or whether it is <italic>ε far</italic> from having the property. In <italic>ε far</italic> we mean that more than an ε-fraction of the entries of <italic>M</italic> must be modified so that it obtains the property. The algorithm may query the matrix on entries <italic>M[i,j]</italic> of its choice, and is allowed a constant probability of error. We describe algorithms for testing Euclidean metrics, tree metrics and ultrametrics. Furthermore, we present an algorithm that tests whether a matrix <italic>M<italic> is an approximate ultrametric. In all cases the query complexity and running time are polynomial in <italic>1 &egr<italic> and independent of the size of the matrix. Finally, our algorithms can be used to solve relaxed versions of the corresponding search problems in time that is sub-linear in the size of the matrix.

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