Algorithmica (2005) 42: 199–201 DOI: 10.1007/s00453-005-1164-z
Algorithmica © 2005 Springer Science+Business Media, Inc.
Preface Rolf M¨ohring1 and Rajeev Raman2
This issue contains seven papers that were presented in preliminary form at the 10th Annual European Symposium on Algorithms (ESA 2002), held at the University of Rome “La Sapienza”, Rome, Italy, September 17–21, 2002. For the first time, ESA had two tracks, which dealt respectively with the design and mathematical analysis of algorithms and real-world applications, engineering, and experimental analysis of algorithms. The authors of the papers in this volume were invited to submit extended versions of their ESA 2002 papers by January 31, 2003. All papers were accepted after going through the common refereeing process for Algorithmica. These papers are representative of several areas of research within ESA: graph and network algorithms, computational geometry, scheduling, with diverse applications such as web multicasting and terrain construction for radiation therapy. The paper by P. K. Agarwal, S. Har-Peled, N. H. Mustafa, and Y. Wang on “NearLinear Time Approximation Algorithms for Curve Simplification” considers the following problem: given a curve that is specified by a sequence of n d-dimensional points, find a subsequence of these points that approximates the curve according to some metric. This problem has many applications: for example, in geographical information systems it is used to represent cartographic features approximately in low-resolution versions of maps. Algorithms that minimize the number of points for a given error tolerance take quadratic time in most cases; the Douglas–Puecker heuristic, which has no guaranteed bounds but runs in O(n log n) time, is commonly used in practice. By considering the Fr´echet metric, rather than the more intensively studied Hausdorff metric, the authors give an O(n log n) time approximation algorithm for this problem. Implementations suggest that it runs faster while producing solutions competive with (and in many cases better than) the Douglas–Puecker heuristic. The paper by P. K. Agarwal, C. M. Procopiuc, and K. R. Varadarajan on “Approximation Algorithms for a k-Line Center” investigates the question how to cover n given points in d-dimensional space with k congruent cylinders such that their radius is as small as possible. This is a problem that occurs in clustering, data mining, and classification. The authors develop a randomized algorithm that computes k cylinders with a radius exceeding the optimal radius only by a factor of 1 + ε. The proof techniques make use of
1
Sekr. MA 6-1, Institut f¨ur Mathematik, Technische Universit¨at Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany.
[email protected]. 2 Department of Computer Science, University of Leicester, Leicester LE1 7RH, England.
[email protected].
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small “certificates” which have the property that if they can be covered by k cylinders, then an expansion by a factor of 1 + ε covers the whole point set. This idea is combined with sampling and iterated reweighting. In their paper “The k-Splittable Flow Problem”, G. Baier, E. K¨ohler, and M. Skutella deal with a natural and interesting generalization of the well-studied unsplittable flow problem. In their flow model a commodity can be split into at most k number of chunks which can than be routed on different path. Such flows have applications in telecommunication and logistics. Already the single commodity case of this problem is NP-hard and even hard to approximate. The authors present approximation algorithms for this problem which are based on an interesting duality between maximum flow and minimum cut for the special case of uniform exactly-k-splittable s − t-flows and a polynomial time algorithm to compute such flows via an augmenting path algorithm. The paper “Constructing Plane Spanners of Bounded Degree and Low Weight” by P. Bose, J. Gudmundsson, and M. Smid deals with the efficient construction of a “good” spanner for a given set of n points in the plane. Here “good” means that the spanner has bounded degree, is itself a planar graph, is a t-spanner for ≈ 10, and has total weight proportional to the weight of a minimum spanning tree of the given point set. The authors present an O(n log n) time algorithm for this task. Previously, no algorithm was known for constructing planar t-spanners of bounded degree. The paper “Optimal Terrain Construction Problems and Applications in IntensityModulated Radiation Therapy” by D. Z. Chen, X. S. Hu, S. Luan, X. Wu, and C. X. Yu considers rectilinear terrain reconstruction problems. These model so-called leaf-sequencing problems in radiation therapy when administered by multileaf collimators. Good solutions to the leaf-sequencing problem have the potential to reduce significantly the time to administer a dose to a patient, improving both the throughput of these (currently expensive) machines and the patient experience. Although the problem formulation is geometric, the authors show by a nontrivial argument that it is reducible to a shortest-paths computation; this gives optimal solutions under reasonable assumptions regarding the input instances. The algorithm is implemented and tested on real medical data against existing (heuristic) software for leaf-sequencing. The new algorithm is slightly slower than existing algorithms, but is sufficiently fast (it runs in a few seconds on moderately large instances) and appears to produce significantly better solutions on real instances. In their paper “A Comparison of Multicast Pull Models”, K. R. Pruhs and P. Uthaisombut study a relatively new area of broadcast scheduling. In this problem clients request documents from a given set of servers at certain points in time. A server answers these requests via one (presumably high bandwidth) channel. All clients are connected to this channel waiting for the broadcast of their document. The main difference to standard job scheduling problems is that several requests, possibly arriving at different times, can be answered by a single broadcast. The goal is to minimize an objective function such as average response time, average stretch, or maximum response time and stretch. Many results of the authors are concerned with a comparison of optimal schedules for the cyclic buffering case with optimal schedules for the case without buffering. Also, tight upper and lower bounds for the ratio of the corresponding objective function values are provided. A quite general result makes it possible to transform a schedule for the case with buffering to a schedule for the case without buffering and twice the bandwidth. The
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presented results are quite relevant for the area of broadcast scheduling, in particular because of their generality. H. Shachnai, T. Tamir, and G. J. Woeginger consider in their paper “Minimizing Makespan and Preemption Costs on a System of Uniform Machines” two variations of the classical problem of scheduling jobs preemptively on uniform machines with the objective to minimize the makespan. Motivated by practical applications, where preempting jobs causes extra costs, the authors analyze two modifications of the classical problem. In one of them, the number of job preemption is restricted by a jobwise bound, or by a total bound on the overall number of preemptions. In the other, only schedules with an optimal makespan are feasible and the objective is to find a feasible schedule that minimizes the number of preemptions. The authors develop approximation algorithms, and derive hardness results and lower and upper bounds for the minimum number of preemptions. Both problems are interesting natural generalizations of the classical problem of minimizing makespan on uniform machines. We thank the referees for their conscientious and timely work, and Professor C. K. Wong, the Founding Editor-in-Chief of Algorithmica, for the opportunity to edit this special issue.
