Showing papers by "Richard Cole published in 2009"

Proximity problems for point sets with low doubling dimension

Abstract: In this thesis we consider proximity problems on point sets. Proximity problems arise in many fields of computer science, with broad application to computational geometry, machine learning, computational biology, data mining, and the like. In particular, we consider the problems of approximate nearest neighbor search and dynamic maintenance of a spanner for a point set. It has been conjectured that all algorithms for these two problems suffer from the “curse of dimensionality,” which means that their run times grow exponentially with the dimension of the point set. To avoid this undesirable growth, we consider point sets that possess a doubling dimension λ. We first present a dynamic data structure that uses linear space and supports a (1+e)-approximate nearest neighbor search of the point set. We then extend this data structure to allow the dynamic maintenance of a low degree (1+e)- spanner for the point set. The query and update time of these structures are logarithmic in the size of the point set and exponential in λ (as opposed to exponential in the dimension); when λ is small, this provides a significant speed-up over known algorithms, and when λ and e are constant these run times are optimal up to a constant. The spanner degree is optimal, while the spanner update times improve on all previously known algorithms.