Fun with Algorithms, 4 conf., FUN 2007(LNCS4475, Springer, 2007)(ISBN 9783540729136)(281s)_CsLn_

Fun with Sub-linear Time AlgorithmsLuca Trevisan ⋆Computer Science Division, U.C. Berkeley679 Soda Hall, Berkeley, CA 94720luca@cs.berkeley.eduAbstract. Provided that one is willing to use randomness and to toleratean approximate answer, many computational problems admit ultrafastalgorithms that run in less than linear time in the length of theinput. In many interesting cases, even algorithms that run in constanttime are known, whose efficiency depends only on the accuracy of theapproximation and not on the length of the inputs.Algorithms for graph problems on dense graphs are especially efficientand simple. I will describe an algorithm that estimates the size of themaximum cut in a dense graph, and its specialization to the task ofdistinguishing bipartite dense graphs from dense graphs that are “farfrom bipartite.” Results “explaining” the simplicity of such algorithmswill also be discussed.Some sublinear-time algorithms are also known for graph problemsin sparse graphs, but they are typically more elaborate. I will describea simple but very clever algorithm that approximates the number ofconnected components of a given graph, and its generalization to theproblem of approximating the weight of the minimum spanning tree ofa given weighted graph. The algorithm runs in time dependent only onthe maximum degree, the required quality of approximation, and therange of weights, but the running time is independent of the number ofvertices.⋆ This material is based upon work supported by the National Science Foundationunder grant CCF 0515231 and by the US-Israel Binational Science Foundation Grant2002246.P. Crescenzi, G. Prencipe, and G. Pucci (Eds.): FUN 2007, LNCS 4475, p. 15, 2007.c○ Springer-Verlag Berlin Heidelberg 2007