We motivate and develop a natural bicriteria measure for assessing the quality of a clustering which avoids the drawbacks of existing measures. A simple recursive heuristic has poly-logarithmic worst-case guarantees under the new measure. The main result of the paper is the analysis of a popular spectral algorithm. One variant of spectral clustering turns out to have effective worst-case guarantees

In this paper we give approximation algorithms for the following problems on metric spaces: Furthest Pair, k- median, 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 \Theta(n 2 ), 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\Omega\Gamma n 2 ). We believe that our techniques can be applied to get similar bounds for other problems. 1 Introduction In recent years there has been a dramatic growth of interest in algorithms operating on massive data sets. This poses new challenges for algorithm design, as algorithms quite efficient on small inputs (for example, having quadratic running time) ...

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The use of proxies for local error recovery and congestion control is a scalable technique used to overcome a number of wellknown problems in Reliable Multicast (RM). The idea is that the multicast delivery tree is partitioned into subgroups that form a hierarchy rooted at the source, hence the term Hierarchical Reliable Multicast (HRM). For each subgroup, there is a designated node, the proxy, which is responsible for collecting the feedback from the subgroup and for locally re-transmitting the lost packets. The performance of any RM protocol is affected by the underlying multicast routing tree and its loss characteristics. Furthermore, the performance of the HRM approach, in particular, strongly depends on the appropriate partitioning of the tree and the selection of proxies. In this paper, we first model the HRM problem, then define and compute appropriate performance metrics and finally give insights on the optimal location of proxies. Keywords Performance analysis, reliable mult...