Triangulation and embedding using small sets of beacons

Concurrent with recent theoretical interest in the problem of metric embedding, a growing body of research in the networking community has studied the distance matrix defined by node-to-node latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance matrix into low-dimensional Euclidean space. Here we give algorithms with provable performance guarantees for beacon-based triangulation and embedding. We show that in addition to multiplicative error in the distances, performance guarantees for beacon-based algorithms typically must include a notion of slack - a certain fraction of all distances may be arbitrarily distorted. For metrics of bounded doubling dimension (which have been proposed as a reasonable abstraction of Internet latencies), we show that triangulation-based reconstruction with a constant number of beacons can achieve multiplicative error 1 + δ on a 1 - ε fraction of distances, for arbitrarily small constants δ and ε. For this same class of metrics, we give a beacon-based embedding algorithm that achieves constant distortion on a 1 - ε fraction of distances; this provides some theoretical justification for the success of the recent global network positioning algorithm of Ng and Zhang, and it forms an interesting contrast with lower bounds showing that it is not possible to embed all distances in a doubling metric with constant distortion. We also give results for other classes of metrics, as well as distributed algorithms that require only a sparse set of distances but do not place too much measurement load on any one node.