TY - JOUR

T1 - Testing metric properties

AU - Parnas, Michal

AU - Ron, Dana

N1 - Funding Information:
Research supported by the Israel Science Foundation (Grant No. 32/00-1). ∗Corresponding author. E-mail addresses: michalp@mta.ac.il (M. Parnas), danar@eng.tau.ac.il (D. Ron).

PY - 2003/12/15

Y1 - 2003/12/15

N2 - 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 M, 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 M and parameter ∈, we are interested in determining, by probing M, whether M has a particular metric property P, or whether it is ∈-far from having the property. In ∈-far we mean that at least an ∈-fraction of the entries of M must be modified so that it obtains the property. The algorithm may query the matrix on entries M[i, j] 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 M is an approximate ultrametric. In all cases the query complexity and running time are polynomial in 1/∈ 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.

AB - 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 M, 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 M and parameter ∈, we are interested in determining, by probing M, whether M has a particular metric property P, or whether it is ∈-far from having the property. In ∈-far we mean that at least an ∈-fraction of the entries of M must be modified so that it obtains the property. The algorithm may query the matrix on entries M[i, j] 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 M is an approximate ultrametric. In all cases the query complexity and running time are polynomial in 1/∈ 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.

UR - http://www.scopus.com/inward/record.url?scp=0346335711&partnerID=8YFLogxK

U2 - 10.1016/S0890-5401(03)00160-3

DO - 10.1016/S0890-5401(03)00160-3

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AN - SCOPUS:0346335711

SN - 0890-5401

VL - 187

SP - 155

EP - 195

JO - Information and Computation

JF - Information and Computation

IS - 2

ER -