BK tree

Abstract: In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion Ω(log n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.

Abstract: Approximate matching of strings is reviewed with the aim of surveying techniques suitable for finding an item in a database when there may be a spelling mistake or other error in the keyword. The methods found are classified as either equivalence or similarity problems. Equivalence problems are seen to be readily solved using canonical forms. For sinuiarity problems difference measures are surveyed, with a full description of the wellestablmhed dynamic programming method relating this to the approach using probabilities and likelihoods. Searches for approximate matches in large sets using a difference function are seen to be an open problem still, though several promising ideas have been suggested. Approximate matching (error correction) during parsing is briefly reviewed.

Abstract: We design a generation algorithm for a problem which a&es in computational chemistry: the random generation of models of amorphous strutures. Such structures can be modeled as graphs which see embedded in d-space. The algorithm uses the well known approach of simulating a rapidly-mixing Markov chain. Gur analysis of the Mixing rate is based on Dobrushin uniqueness. The structure of the problem forces us to extend the basic method and find an alternative to Dobrushin’s condition which is more appropriate for our problem. This extension appears to be of more general interest.

Abstract: In this paper we give approximation algorithms for the following problems on metric spaces: Furthest Pair, kmedian, Minimum Routing Cost Spanning Tree, Multiple Sequence Alignment, Maximum Traveling Salesman Problem, Maximum Spanning Tree and Average Distance. The key property of our algorithms is that their running time is linear in the number of metric space points. As the full specification o‘f an n-point metric space is of size Q(n’), the complexity of our algorithms is sublinear with respect to the input size. All previous algorithms (exact or approximate) for the problems we consider have running time n(n’). We believe that OUT techniques can be applied to get similar bounds for other problems.