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

We present better random sampling algorithms for maximum flows in undirected graphs. Our algorithms apply to capacitated or uncapacitated graphs, and find a maximum flow of value v in ~ O( p mnv) time. This improves on a previous bound of ~ O(m 2=3 n 1=3 v) given by the author recently, which in turn improved on the O(mv) time bound for a typical augmenting path algorithm. In uncapacitated graphs without parallel edges, the bound is no worse than ~ O(n 5=2 ). We give another algorithm that finds a (1 \Gamma ffl) times maximum flow in time ~ O(m p n=ffl), regardless of v. 1 Introduction. Random sampling has been a useful tool for solving cut problems in undirected graphs. In previous work [Kar97a], this author showed that randomly choosing edges from a graph yields a sampled graph in which every cut is close to its expected value with high probability. This led to algorithms for approximating [Kar97a] and exactly finding [Kar96] global min-cuts in near-linear time with hig...