Fun with Sublinear Time Algorithms
 Luca Trevisan
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Abstract
Provided that one is willing to use randomness and to tolerate an approximate answer, many computational problems admit ultrafast algorithms that run in less than linear time in the length of the input. In many interesting cases, even algorithms that run in constant time are known, whose efficiency depends only on the accuracy of the approximation and not on the length of the inputs.
Algorithms for graph problems on dense graphs are especially efficient and simple. I will describe an algorithm that estimates the size of the maximum cut in a dense graph, and its specialization to the task of distinguishing bipartite dense graphs from dense graphs that are “far from bipartite.” Results “explaining” the simplicity of such algorithms will also be discussed.
Some sublineartime algorithms are also known for graph problems in sparse graphs, but they are typically more elaborate. I will describe a simple but very clever algorithm that approximates the number of connected components of a given graph, and its generalization to the problem of approximating the weight of the minimum spanning tree of a given weighted graph. The algorithm runs in time dependent only on the maximum degree, the required quality of approximation, and the range of weights, but the running time is independent of the number of vertices.
 Title
 Fun with Sublinear Time Algorithms
 Book Title
 Fun with Algorithms
 Book Subtitle
 4th International Conference, FUN 2007, Castiglioncello, Italy, June 35, 2007. Proceedings
 Pages
 p 15
 Copyright
 2007
 DOI
 10.1007/9783540729143_3
 Print ISBN
 9783540729136
 Online ISBN
 9783540729143
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 4475
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors
 Authors

 Luca Trevisan ^{(1)}
 Author Affiliations

 1. Computer Science Division, U.C. Berkeley, 679 Soda Hall, Berkeley, CA 94720,
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