05291 Abstracts Collection Sublinear Algorithms Dagstuhl Seminar

From 17.07.05 to 22.07.05, the Dagstuhl Seminar 05291 Sublinear Algorithms was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available. 05291 Executive Summary Sublinear Algorithms This paper summarizes the content and structure of the Dagstuhl seminar Sublinear Algorithms, which was held from 17.7.2005 to 22.7.2005 in Schloss Dagstuhl, Germany. The purpose of the Dagstuhl seminar 'Sublinear Algorithms' was to bring together researchers working on the development of algorithms for very large data sets. Over the last few years data sets have become increasingly massive and the need to design special algorithms and data structures that deal with such amounts of data has emerged. For example, the set of all credit card transactions in the world for a month would have been considered a massive data set some time ago. That is comparable to the number of packet transactions a single router processes in one hour on an interface and we are now facing problems of analyzing the tra c at a large network of such routers, each with many interfaces! Internet tra c logs, clickstreams, web data are all examples of modern data sets that show unprecedented scale. Managing and analyzing such data sets forces us to revisit the traditional notions of e cient algorithms. The long-held golden Dagstuhl Seminar Proceedings 05291 Sublinear Algorithms http://drops.dagstuhl.de/opus/volltexte/2006/681 2 A. Czumaj, S. M. Muthukrishnan, R. Rubinfeld and C. Sohler standard of linear algorithms algorithms that take time proportional to the input and store no more space than it takes to archive the input is no longer as e cient as one needs or can a ord. Thus, there is now a need for sublinear algorithms, that is algorithms that use resources (time and space) signi cantly less than the input size. The main areas addressed in the workshop were property testing, sublinear time approximation algorithms, and data streaming algorithms. These areas are not only connected by the fact that they require algorithms with sublinear resources but also that they heavily rely on randomization and random sampling. Therefore, we hoped that this workshop helped to exchange ideas between these di erent areas. During the seminar one could obtain a good overview of the current state of sublinear algorithms. In many interesting talks new algorithms and models as well as solutions to well-known open problems were presented. To give an idea about the topics of the seminar we present a few examples of topics that were discussed in a number of talks at the seminar. These examples are not meant to be exhaustive. Testing graph properties Property testing deals with a certain notion of approximation of decision problems. One tries to distinguish the case that an object has a certain property from the case that it is far from the property. One topic of the workshop was the classi cation of testable graph properties. In this context we say that a property is testable, if it can be tested in time/query complexity independent of the graph. This classi cation problem was recently solved and a talk about these recent achievements was given at the seminar. Although we know property testers for many graph properties their running is often extraordinary high, i.e. a tower of towers of 1/ . It has been suggested during the seminar that existing results should be further re ned to be able to distinguish between properties that are, say, testable with query complexity polynomial in 1/ and those testable in superpolynomial time. Testing and approximating of distributions Another interesting topic of the workshop was the question how to approximate and/or determine properties of an unknown distributions. Typically, an unknown distribution, for example over the numbers 1, ..., n, is given as a black box. The only access to the distribution is by randomly sampling elements from it. A typical question considered for this model would be to determine whether the unknown distribution is (close to) uniform or di ers signi canly from the uniform distribution. A number of other approximation and testing algorithms for fundamental properties of distributions were presented in a number of talks. Sublinear Algorithms 3 Sublinear geometric algorithms Geometric problems have been considered in the context of sublinear approximation algorithms and data streaming. A survey talk about the recent developments in geometric data streams has been given. Other contributions included algorithms for clustering problems, sublinear time intersection detection of convex polygons and polyhedra and online data reconstruction. Connections between the areas One focus of the workshop was to nd connections between the di erent areas of sublinear algorithms. One such connection is the notion of tolerant property testing and distance approximation. In tolerant property testing one wants to accept all objects that have a property or that are very close to it and reject all objects that are far from the property. This is in contrast to classical property testing where one must only accept objects that have the property. One may think of tolerant testing as a hybrid between testing and sublinear time approximation. In distance approximation we are asked to approximate the distance of an object to a given property, which can be interpreted as a sublinear approximation algorithm. Another result discussed at the workshop showed that certain property testers can be turned into streaming algorithms. Also a connection between data streaming and the area of compressed sensing, which recently received some amount of attention in mathematics, was presented at the workshop. Open problem session On Wednesday evening an open problem session was held, where many researchers presented interesting open problems.