Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M Kleinberg Cornell University, Ithaca, NY, USA Friedemann Mattern ETH Zurich, Switzerland John C Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen University of Dortmund, Germany Madhu Sudan Massachusetts Institute of Technology, MA, USA Demetri Terzopoulos New York University, NY, USA Doug Tygar University of California, Berkeley, CA, USA Moshe Y Vardi Rice University, Houston, TX, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany 3503 Sotiris E Nikoletseas (Ed.) Experimental and Efficient Algorithms 4th International Workshop, WEA 2005 Santorini Island, Greece, May 10-13, 2005 Proceedings 13 Volume Editor Sotiris E Nikoletseas University of Patras and Computer Technology Institute (CTI) 61 Riga Fereou Street, 26221 Patras, Greece E-mail: nikole@cti.gr Library of Congress Control Number: 2005925473 CR Subject Classification (1998): F.2.1-2, E.1, G.1-2, I.3.5, I.2.8 ISSN ISBN-10 ISBN-13 0302-9743 3-540-25920-1 Springer Berlin Heidelberg New York 978-3-540-25920-6 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper SPIN: 11427186 06/3142 543210 Preface This proceedings volume contains the accepted papers and invited talks presented at the 4th International Workshop of Efficient and Experimental Algorithms (WEA 2005), that was held May 10–13, on Santorini Island, Greece The WEA events are intended to be an international forum for research on the design, analysis and especially the experimental implementation, evaluation and engineering of algorithms, as well as on combinatorial optimization and its applications The first three workshops in this series were held in Riga (2001), Monte Verita (2003) and Rio de Janeiro (2004) This volume contains invited papers related to corresponding keynote talks: by Prof Christos Papadimitriou (University of California at Berkeley, USA), Prof David Bader (University of New Mexico, USA) and Prof Celso Ribeiro (University of Rio de Janeiro, Brazil) This proceedings includes 54 papers (47 regular and short), selected out of a record number of 176 submissions Each paper was reviewed by at least Program Committee members, while many papers got or reviews A total number of 419 reviews were solicited, with the help of trusted external referees In addition to the 54 papers included in this volume, papers were accepted as poster presentations: these papers were published in a separate poster proceedings volume by CTI Press and a major publisher in Greece, “Ellinika Grammata.” The presentation of these posters at the event was expected to create a fruitful discussion on interesting ideas The 60 papers accepted to WEA 2005 demonstrate the international character of the event: 33 authors are based in Germany, 20 in the USA, 13 in Italy, 12 in Greece, each in Switzerland, France and Brazil, each in Canada, Poland and Belgium, in the Netherlands, to list just the countries with the largest participations Selected papers of WEA 2005 will be considered for a Special Issue of the ACM Journal on Experimental Algorithmics (JEA, http://www.jea.acm.org/) dedicated to the event We would like to thank all authors who submitted papers to WEA 2005 We especially thank the distinguished invited speakers (whose participation honors the event a lot), and the members of the Program Committee, as well as the external referees and the Organizing Committee members We would like to thank the Ministry of National Education and Religious Affairs of Greece for its financial support of the event Also, we gratefully acknowledge the support from the Research Academic Computer Technology Institute (RACTI, Greece, http://www.cti.gr), and the European Union (EU) IST/FET (Future and Emerging Technologies) R&D projects FLAGS (Foundational As- VI Preface pects of Global Computing Systems) and DELIS (Dynamically Evolving, LargeScale Information Systems) I wish to personally acknowledge the great job of the WEA 2005 Publicity Chair Dr Ioannis Chatzigiannakis, and Athanasios Kinalis for maintaining the Web page and processing this volume with efficiency and professionalism I am grateful to the WEA Steering Committee Chairs Prof Jose Rolim and Prof Klaus Jansen for their trust and support Finally, we wish to thank Springer Lecture Notes in Computer Science (LNCS), and in particular Alfred Hofmann and his team, for a very nice and efficient cooperation in preparing this volume May 2005 Sotiris Nikoletseas Organization Program Committee Chair Sotiris Nikoletseas University of Patras and CTI, Greece Program Committee Edoardo Amaldi Evripidis Bampis David A Bader Cynthia Barnhart Azzedine Boukerche Gerth Brodal Rainer Burkard Giuseppe Di Battista Rudolf Fleischer Pierre Fraigniaud Mark Goldberg Juraj Hromkovic Giuseppe Italiano Christos Kaklamanis Helen Karatza Ludek Kucera Shay Kutten Catherine McGeoch Simone Martins Bernard Moret Ian Munro Sotiris Nikoletseas Andrea Pietracaprina Tomasz Radzik Rajeev Raman Mauricio Resende Maria Serna Paul Spirakis Eric Taillard Dorothea Wagner Stefan Voss Christos Zaroliagis Politecnico di Milano, Italy Universit´e d’Evry, France University of New Mexico, USA MIT, USA SITE, University of Ottawa, Canada University of Aarhus, Denmark Graz University of Technology, Austria Universita’ degli Studi Roma Tre, Italy Fudan University, Shanghai, China CNRS, Universit´e Paris-Sud, France Rensselaer Polytechnic Institute, USA ETH Zurich, Switzerland Universita’ di Roma Tor Vergata, Italy University of Patras and CTI, Greece Aristotle University of Thessaloniki, Greece Charles University, Czech Republic Technion - Israel Institute of Technology, Israel Amherst College, USA Universidade Federal Fluminense, Brazil University of New Mexico, USA University of Waterloo, Canada University of Patras and CTI, Greece (Chair) University of Padova, Italy King’s College London, UK University of Leicester, UK AT&T Labs Research, USA T.U of Catalonia, Spain University of Patras and CTI, Greece EIVD, Switzerland University of Karlsruhe, Germany University of Hamburg, Germany University of Patras and CTI, Greece VIII Organization Steering Committee Chairs Klaus Jansen Jose Rolim University of Kiel, Germany University of Geneva, Switzerland Organizing Committee Ioannis Chatzigiannakis Rozina Efstathiadou Lena Gourdoupi Athanasios Kinalis CTI, Greece, (Co-chair) CTI, Greece, (Co-chair) CTI, Greece University of Patras and CTI, Greece Referees Nazim Agoulmine Roberto Aringhieri Yossi Azar Ricardo Baeza-Yates Michael Baur Amos Beimel Pietro Belotti Alberto Bertoldo Mauro Bianco Maria Blesa Roderick Bloem Christian Blum Maria Cristina Boeres Thomas Buchholz Costas Busch Sergiy Butenko Roberto Wolfler Calvo Antonio Capone Ioannis Caragiannis Massimiliano Caramia Matteo Cesana Ioannis Chatzigiannakis Yinong Chen Francis Chin Pier Francesco Cortese Yves Crama Cid de Souza Josep Diaz Tassos Dimitriou Rolf Fagerberg Carlo Fantozzi Antonio Fern´ andez Irene Finocchi Dimitris Fotakis Joaquim Gabarr´ o Marco Gaertler Giulia Galbiati Clemente Galdi Giovanni Gallo Efstratios Gallopoulos Fabrizio Grandoni Peter Greistorfer Nir Halman Refael Hassin Martin Holzer Ja Hoogeveen Stanislaw Jarecki Jiang Jun Sam Kamin Howard Karloff Dukwon Kim Athanasios Kinalis Sigrid Knust Elisavet Konstantinou Charalambos Konstantopoulos Spyros Kontogiannis Dimitrios Koukopoulos Joachim Kupke Giovanni Lagorio Giuseppe Lancia Carlile Lavor Helena Leityo Zvi Lotker Abilio Lucena Francesco Maffioli Malik Magdon-Ismail Christos Makris Federico Malucelli Carlos Alberto Martinhon Constandinos Mavromoustakis Steffen Mecke John Mitchell Ivone Moh Gabriel Moruz Pablo Moscato Matthias Mueller-Hannemann Maurizio Naldi Filippo Neri Sara Nicoloso Gaia Nicosia Mustapha Nourelfath Carlos A.S Oliveira Mohamed Ould-Khaoua Organization Andrea Pacifici Evi Papaioannou Panos M Pardalos Paolo Penna Pino Persiano Enoch Peserico Jordi Petit Ugo Pietropaoli Mustafa Pinar Evaggelia Pitoura Maurizio Pizzonia Alexandre Plastino Daniele Pretolani Daniela Pucci de Farias Naila Rahman Massimo Rimondini Isabel Rosseti Harilaos Sandalidis Haroldo Santos Thomas Schank Elad Schiller Frank Schulz Sebastian Seibert Spyros Sioutas Spiros Sirmakessis Riste Skrekovski IX Stˆenio Soares Yannis Stamatiou Maurizio Strangio Tami Tamir Leandros Tassiulas Dimitrios M Thilikos Marco Trubian Manolis Tsagarakis George Tsaggouris Gabriel Wainer Renato Werneck Igor Zwir Sponsoring Institutions – Ministry of National Education and Religious Affairs of Greece – Research Academic Computer Technology Institute (RACTI), Greece – EU-FET R&D project “Foundational Aspects of Global Computing Systems” (FLAGS) – EU-FET R&D project “Dynamically Evolving, Large-Scale Information Systems” (DELIS) Table of Contents Invited Talks T α Παιδ´ια Πα´ιζει The Interaction Between Algorithms and Game Theory Christos H Papadimitriou Using an Adaptive Memory Strategy to Improve a Multistart Heuristic for Sequencing by Hybridization Eraldo R Fernandes, Celso C Ribeiro High-Performance Algorithm Engineering for Large-Scale Graph Problems and Computational Biology David A Bader 16 Contributed Regular Papers The “Real” Approximation Factor of the MST Heuristic for the Minimum Energy Broadcasting Michele Flammini, Alfredo Navarra, Stephane Perennes 22 Implementing Minimum Cycle Basis Algorithms Kurt Mehlhorn, Dimitrios Michail 32 Rounding to an Integral Program Refael Hassin, Danny Segev 44 Rectangle Covers Revisited Computationally Laura Heinrich-Litan, Marco E Lă ubbecke 55 Don’t Compare Averages Holger Bast, Ingmar Weber 67 Experimental Results for Stackelberg Scheduling Strategies A.C Kaporis, L.M Kirousis, E.I Politopoulou, P.G Spirakis 77 An Improved Branch-and-Bound Algorithm for the Test Cover Problem Torsten Fahle, Karsten Tiemann 89 Degree-Based Treewidth Lower Bounds Arie M.C.A Koster, Thomas Wolle, Hans L Bodlaender 101 XII Table of Contents Inferring AS Relationships: Dead End or Lively Beginning? Xenofontas Dimitropoulos, Dmitri Krioukov, Bradley Huffaker, kc c laffy, George Riley 113 Acceleration of Shortest Path and Constrained Shortest Path Computation Ekkehard Kă ohler, Rolf H Mă ohring, Heiko Schilling 126 A General Buffer Scheme for the Windows Scheduling Problem Amotz Bar-Noy, Jacob Christensen, Richard E Ladner, Tami Tamir 139 Implementation of Approximation Algorithms for the Multicast Congestion Problem Qiang Lu, Hu Zhang 152 Frequency Assignment and Multicoloring Powers of Square and Triangular Meshes Mustapha Kchikech, Olivier Togni 165 From Static Code Distribution to More Shrinkage for the Multiterminal Cut Bram De Wachter, Alexandre Genon, Thierry Massart 177 Partitioning Graphs to Speed Up Dijkstra’s Algorithm Rolf H Mă ohring, Heiko Schilling, Birk Schă utz, Dorothea Wagner, Thomas Willhalm 189 Efficient Convergence to Pure Nash Equilibria in Weighted Network Congestion Games Panagiota N Panagopoulou, Paul G Spirakis 203 New Upper Bound Heuristics for Treewidth Emgad H Bachoore , Hans L Bodlaender 216 Accelerating Vickrey Payment Computation in Combinatorial Auctions for an Airline Alliance Yvonne Bleischwitz, Georg Kliewer 228 Algorithm Engineering for Optimal Graph Bipartization Falk Hă uner 240 Empirical Analysis of the Connectivity Threshold of Mobile Agents on the Grid Xavier P´erez 253 Finding, Counting and Listing All Triangles in Large Graphs 609 time in O m3/2 Both algorithms can be further improved with certain methods relying on hashing, see [SW05] for details Experiments The algorithms are tested in two ways On the one hand we list the execution time of the algorithms Additionally, we give the number of triangle operations, which in essence captures the asymptotic running time of the algorithm without preprocessing The algorithms are implemented in C++ The experiments were carried out on a 64-bit machine with a AMD Opteron Processors clocked at 2.20G-Hz Figure shows the results on generated Gn,m graphs where m edges are inserted randomly between n nodes These Gn,m graphs tend to have no high degree nodes and to have a very low deviation from the average degree However, this seems to be not true for many real networks √ [FFF99] Therefore, Figure shows results on modified Gn,m graphs with O( n) high degree nodes Conclusion The two known standard Algorithms node-iterator and edge-iterator are asymptoticly equivalent However, the Algorithm edge-iterator can be implemented with a much lower constant overhead It works very well for graphs where the degrees not differ much from the average degree If the degree distribution is skewed refined algorithms are required The Algorithm forward shows to be the best compromise It is asymptotically efficient and can be implemented to have a low constant factor with respect to execution time References AYZ97 Noga Alon, Raphael Yuster, and Uri Zwick Finding and counting given length cycles Algorithmica, 17(3):209–223, 1997 BM01 Vladimir Batagelj and Andrej Mrvar A subquadratic triad census algorithm for large sparse networks with small maximum degree Social Networks, 23:237–243, 2001 FFF99 Michalis Faloutsos, Petros Faloutsos, and Christos Faloutsos On power-law relationships of the Internet topology In Proceedings of SIGCOMM’99, 1999 SW05 Thomas Schank and Dorothea Wagner Finding, counting and listing all triangles in large graphs Technical report, Universită at Karlsruhe, Fakultă at fă ur Informatik, 2005 Selecting the Roots of a Small System of Polynomial Equations by Tolerance Based Matching H Bekker∗ , E.P Braad∗ , and B Goldengorin∗∗ ∗ Department of Mathematics and Computing Science, ∗∗ Faculty of Economic Sciences, University of Groningen, P.O.Box 800, 9700 AV Groningen, The Netherlands bekker@cs.rug.nl, e.p.braad@wing.rug.nl, b.goldengorin@eco.rug.nl Abstract The roots of a system of two bivariate polynomial equations are calculated using a two-step method First all x-roots and y-roots are determined independently Then tolerance based weighted matching is used to form (x, y)-pairs that together form a minimum-error solution to the system Keywords: combinatorial optimization, tolerance based bipartite matching, solving polynomial equations Introduction Consider a system of two polynomial equations f (x, y) = g(x, y) = (1) with symbolic constants and of low degree By assigning numerical values to the constants we obtain a problem instance Assuming that it is known that (1) has a finite number of solutions the conventional method to calculate the solutions is as follows From (1) a univariate polynomial, say p(x), is derived by eliminating y For every problem instance the symbolic constants in p(x) = 0, f (x, y) = and g(x, y) = are replaced by numerical values and p(x) = is solved numerically giving the roots x1 , , xn Subsequently, for every root xi the corresponding root yi has to be determined To that end, xi is backsubstituted in f (x, y) = and g(x, y) = 0, giving the univariate polynomial equations f (xi , y) = and g(xi , y) = Solving f (xi , y) = for y gives the solutions yf1 , , yfl , and solving g(xi , y) = for y gives the solutions yg1 , , ygm The value yi occurring both in yf1 , , yfm and yg1 , , ygm is the desired value, i.e., the pair xi , yi is a root of (1) During this process, a number of complications may occur: The equation f (xi , y) = may be degenerate, i.e may be = 0, or even worse, may be near degenerate within the noise margin The case of exact S.E Nikoletseas (Ed.): WEA 2005, LNCS 3503, pp 610–613, 2005 c Springer-Verlag Berlin Heidelberg 2005 Selecting the Roots of a Small System of Polynomial Equations 611 degeneracy is easily detected but it is not trivial to detect near degeneracy In both cases every solution of the other equation, that is, of g(xi , y) = is a correct root Analogously, g(xi , y) = may be degenerate, giving the same problems As we know that (1) has a finite number of solutions the situation that f (xi , y) = and g(xi , y) = are both degenerate will not occur It is sometimes hard to select from yf1 , , yfl and yg1 , , ygm the collective value yi because, by numerical errors, the actual value of yi will be different in the two sets p(x) = may have multiple roots, that is, the roots x1 , , xn may contain (near) identical values Let us assume that there is a double root, given by the the identical values xi and xi Then there will be two matching roots yj and yj , not necessarily with the same value When yj is matched to xi , in a later stage yj should be matched to xi and not to xi When solving a problem of computational geometry we ran into these problems, first using our own multivariate polynomial solver and later using methods from packages As a result a small but significant part of the roots, notably multiple roots, were missed or were completely wrong The CORS Method To avoid the aforementioned complications we propose and test a two-step method, called the CORS method (Combinatorial Optimization Root Selection) First from (1) two univariate polynomials p(x) and q(y) are derived by eliminating y and x respectively Whether this is done by calculating resultants or a Groebner basis is irrelevant Now for every problem instance the symbolic constants in p(x), q(y), f (x, y) and g(x, y) are replaced by numerical values and the roots in C of p(x) and q(y) are calculated numerically Both p(x) and q(y) have n roots represented by x1 , , xn and y1 , , yn , respectively These roots are used to calculate n2 weights, where wi,j is defined as w(i, j) = (f (xi , yj ))2 + (g(xi , yj ))2 (2) Subsequently a complete weighted bipartite graph G(V, E) is constructed with V = X ∪ Y and |X| = |Y | = n The nodes in X consist of the values x1 , , xn , and the nodes in Y consist of the values y1 , , yn The arc between nodes xi and yj is assigned the weight w(i, j) On G the minimum-weight perfect matching π0 is calculated The n arcs in π0 represent the optimal solutions of (1) Here, optimal means that the sum of the errors is minimal In the following this method of roots selection is called CORS1 Instead of minimizing the sum of the errors it is more natural to minimize the maximum error This is done as follows All n2 arcs and their weights are stored in a linear list L Subsequently, L is sorted in increasing order of weights Now the weight in the first entry in L is set to 1, and the weight of item i is equal to the sum of i−1 the weights in previous items plus one, i.e weight[i] = ( j=1 weight[j]) + Thus the weights are 1, 2, 4, 8, 16, A new graph G is constructed, identical to G but 612 H Bekker, E.P Braad, and B Goldengorin with the new weights Of G the minimum-weight perfect matching π0 is calculated The n arcs in π0 represent the optimal solutions of (1) Here, optimal means that the maximum error of π0 is minimal We call this method CORS2 We here outline the three steps of a proof that this procedure minimizes the maximum weight when no identical weights in G occur, without this assumption the proof is similar but more complex 1: π0 is unique because the total weight of π0 can be constructed only in one way from the weights in G 2: In π0 there is only one element em with the maximum weight 3: There is no matching of G without em , with a lower weight q.e.d The weights in G become very large causing overflow on standard integer arithmetic Therefore the infinite precision integer type should be used The weights in G have the nice property that none of the weights can be constructed from other weights This makes G very suitable for tolerance based matching Tolerance Based Matching A Feasible Assignment (matching, permutation) (FA) π on the bipartite graph G is a mapping π of X onto Y with w(π) = (i,j)∈π w(i, j) < ∞ and the set of all FAs is Π The Linear Assignment Problem (LAP) is the problem of finding a FA π0 ∈ arg min{w(π) : π ∈ Π}, and all algorithms are based on shortest paths and the Kă onig-Egervarys theorem with O(n3 ) time complexity when applied to dense instances [1] We sketch the idea of algorithms which are based only on tolerances for the Relaxed LAP (RLAP) without using the Kă onig-Egervarys theorem A Relaxed FA (RFA) θ is defined on the same graph G as a mapping θ of X into Y with w(θ) = (i,j)∈θ w(i, j) < ∞ The RLAP is the problem of finding min{w(θ) : θ ∈ Θ} = i∈X min{w(i, j) : j ∈ Y } = w(θ0 ) ≤ w(π0 ) on the set of RFA Θ ⊃ Π A FA π on G is a set of n arcs (i, j) such that the out-degree od(i) = for all i ∈ X and the in-degree id(j) = for all j ∈ Y , and a RFA θ is a set of n arcs (i, j) with od(i) = for all i ∈ X and j∈Y id(j) = n Note that θ is a FA if the id(j) = for all j ∈ Y For each fixed row i of the matrix W = ||w(i, j)|| let w[i, j1 (i)] ≤ w[i, j2 (i)] ≤ ≤ w[i, jn (i)] be the ordered set of entries in a non-decreasing order We define the reduced matrix W r = ||wr (i, j)|| with wr (i, j) = w(i, j) − w[i, j1 (i)] for all i ∈ X and j ∈ Y The tolerance problem for the RLAP is the problem of finding for each arc (i, j) ∈ X × Y the maximum decrease l(i, j) and the maximum increase u(i, j) of the arc weight w(i, j) preserving the optimality of θ0 under the assumption that the weights of all other arcs remain unchanged Now for an arc [i, j1 (i)] ∈ θ0 the upper tolerance u[i, j1 (i)] = w[i, j2 (i)], and the lower tolerance l[i, j1 (i)] = ∞ Similarly, for an arc (i, j) ∈ / θ0 the lower tolerance l(i, j) = wr (i, j) and the upper tolerance u(i, j) = ∞ Let us show that the bottleneck tolerance b(θ0 ) = max{u(θ0 ), l(θ0 )} is a tightness measure between known value of w(θ0 ) and the unknown value of w(π0 ) For a fixed θ0 we partition the set Y into three subsets of vertices: the unassigned set Y0 = {j ∈ Y : id(j) = 0}, assigned set Y1 = {j ∈ Y : id(j) = 1}, Selecting the Roots of a Small System of Polynomial Equations 613 and overassigned set Y2 = {j ∈ Y : id(j) > 1} For each fixed j ∈ Y2 we order the corresponding upper tolerances in non-decreasing order u[i1 (j), j] ≤ u[i2 (j), j] ≤ pj −1 ≤ u[ipj (j), j] and compute u(θ0 ) = j∈Y2 u(j) with u(j) = t=1 u[it (j), j] Similarly, for each fixed j ∈ V0 , l[i(j), j] = min{l(i, j) : i ∈ X}, l[Y0 (j)] = k max{l[i(t), t] : t ∈ Y0 (j)} and l(θ0 ) = j=1 l[Y0 (j)] with Y0 (j) = {t ∈ Y0 : i(t) = i(j)} Here, Y0 (1), , Y0 (k) is a partition of Y0 Further we treat each π, and each θ as the sets of corresponding arcs such that |π| = |θ| = n Note that if either Y0 = ∅ or Y2 = ∅ then |Y1 | = n and θ0 is a FA Hence, for each θ0 ∈ / Π we may use the number of unassigned columns |Y0 | = j∈Y2 |id(j) − 1| in the reduced matrix W r as a measure of structural infeasibility of θ0 to the LAP, for which the bottleneck tolerance b(θ0 ) ≤ w(π0 ) − w(θ0 ) Our algorithm for solving the LAP recursively fixes the arc (i, j) ∈ θ0 with the largest tolerance and replaces all other arcs from Y2 by the arcs representing the tolerances ordered in a non-increasing order, regardless of either upper or lower tolerance will be the next tolerance induced by that order Therefore, the first obtained θ ∈ Π is θ = π0 , and hence the time complexity of LAP for CORS2 is O(n2 ) Implementation, Tests and Results Implementation We tested CORS on our computational geometry problem Of this class of problems it is known that every instance has eight solutions The univariate polynomials p(u) and q(w) are derived with MAPLE The numerical calculations are implemented in C++ in double precision Laguerre’s method [2] is used to compute the roots of the polynomials p(u) and q(w) The LEDA [3] implementation of the minimum weight bipartite matching algorithm is used Tests We tested the CORS1 and CORS2 method Every problem instance is solved in two ways: with the CORS method and with SYNAPS, a C++ package for solving polynomial equations [4] We solved 104 problem instances with CORS and SYNAPS, and ≈ 400 with MAPLE The latter problem instances were solved correctly by CORS and were missed by SYNAPS, i.e we use MAPLE to decide whether CORS or SYNAPS gave the correct result Results In general the results of CORS1 and CORS2 are identical In the tests approximately 2.4% of the solutions is missed by SYNAPS and are found by CORS No solutions were missed by CORS The average error of the solutions found by SYNAPS is 1.3 10−10 and of CORS 6.5 10−11 Running 105 problem instances with CORS takes 14 sec and with SYNAPS 475 sec References Burkard, R E Selected topics on assignment problems Discrete Applied Mathematics 123, 257–302, 2002 Press, W H., Flannery, B P., Teukolsky, S A., and Vetterling, W T Numerical Recipes in C++ Cambr Univ Press, New York K Melhorn, Nă aher, S LEDA A Platform for Combinatorial and Geometric Computing Cambridge University press,Cambridge 1999 Synaps Available at: http://www.inria.fr/galaad/logiciels/synaps/inex.html Developing Novel Statistical Bandwidths for Communication Networks with Incomplete Information Janos Levendovszky and Csego Orosz Budapest University of Technology and Economics, Department of Telecommunications, Magyar tudosok korutja 2, H-1117 Budapest, Hungary {levendov, oroszcs}@hit.bme.hu Abstract In this paper, the concept of statistical bandwidth of multiaccess systems are studied and extended to the case of unknown statistical descriptors The results can improve the statistical characterization of the tail distribution of aggregated load presented to a multi-access system which is traditionally based on the logarithmic moment generation function (LMGF)[1] In the paper, an extended moment generating function is introduced for calculating the statistical bandwidth and as a result a novel admission algorithm is presented To further maximize the admitted load into the multi-access system the free parameter of the extended statistical bandwidth is optimized based on the geometrical optimization of polygonal surfaces In this way, the system utilization can be near-optimal Introduction Let us assume that the following quantities are given: (i) a number of traffic classes (e.g video, ftp, voice etc.) denoted by i = 1, , M ; (ii) the random (i) traffic emitted by source j from class i is denoted by Xj (sources form the same class are assumed to be homogenous); (iii) traffic class i is characterized by traffic descriptors ri = (ri1 , , riV ) (e.g in the case of On/Off sources ri = (mi , hi ), where mi refers to the average rate, whereas hi stands for the peak rate, respectively); (iv) ri , i = 1, , M are supposed to be random variables due to imperfect measurements and the corresponding p.d.f.-s are denoted by pi (x), i = 1, , M ; (v) the traffic state of the network is described by a traffic state vector n = (n1 , , nM ), the ith component of which indicates the number of users being present from class i; (vi) the network (or access point) capacity is denoted by C; (vii) QoS is measured by the cell loss probability (zero buffer approximation) meaning that ⎛ P⎝ M ni i=1 j=1 ⎞ Xj > C ⎠ < e−γ , (i) S.E Nikoletseas (Ed.): WEA 2005, LNCS 3503, pp 614–617, 2005 c Springer-Verlag Berlin Heidelberg 2005 (1) Developing Novel Statistical Bandwidths for Communication Networks 615 where γ is the QoS parameter Our concern is to evaluate equation (1) when traffic descriptors are only given by their p.d.f and possibly reduce this expression M into i=1 ni βi < Ψ (C, γ), where βi is referred to as statistical bandwidth of class i (due to the additive rule) and Ψ is an appropriate function Traditionally, statistical bandwidth has been calculated on the basis of Chernoff inequality assuming known traffic descriptors ⎞ ⎛ P⎝ M ni i=1 j=1 where µi (s) := log E esX Xj > C ⎠ < e (i) (i) M i=1 and s∗ : mins ni µi (s∗ )−s∗ C M i=1 , (2) ni µi (s) − sC In this way inequality (1) can easily be evaluated by checking the equivalent inequality M ∗ ∗ i=1 ni µi (s ) < s C − γ This form defines the statistical bandwidth as µi (s), since it follows the desired additive rule One must note that this expression defines a dichotomy over the traffic state space N expanded by the traffic vectors Unfortunately, µi (s) can only be calculated if the source is fully characterized by its traffic descriptors, i.e in the case of On/Off sources when ri = (mi , hi ), mi i P (Xi = 0) = − m hi and P (Xi = hi ) = hi , the statistical bandwidth becomes mi shi i In the case of unknown ri , i = 1, , M are not µi (s) = log − m hi + hi e known then new methods must be developed for CAC Extension of Statistical Bandwidth In this section, we embark on extending the statistical bandwidth when ri , i = 1, , M are not given by their exact values but assumed to be random variables and only the family of p.d.f.-s, pi (z), i = 1, , M is given This extension is based on the following theorem: Theorem The function βi (s) := log z1 , ,zV pi (z1 , , zV )eµi (z1 , ,zV ,s) dz1 , , dzV (3) is additive in the sense that calls can be accepted by checking if M ni βi (s) < sC − γ i=1 The proof drops out form the conditional form of the Chernoff inequality, due to the limited we not detail the steps As a result we call βi (s), i = 1, , M as Generalized Statistical Bandwidth (GSB) One must note that βi (s) does not need the exact values of the traffic descriptors, but only their p.d.f is necessary As was mentioned earlier, the 616 J Levendovszky and Cs Orosz modified Chernoff-bound is valid for each positive s Therefore, one can select the tightest upper bound given as follows: ⎞ ⎛ P⎝ M ni i=1 j=1 Xj > C ⎠ < e (i) M i=1 ni βi (s∗ )−s∗ C , where s∗ : s M ni βi (s∗ ) − s∗ C i=1 (4) Consequently, the CAC algorithm in the case of unknown traffic descriptors can be performed as follows: Given: n, C and γ M (i) Calculate βi (s) := log z(i) eµi (s) pi z(i) dz(i) ; (ii) Calculate s∗ : mins i=1 M ni βi (s∗ ) − s∗ C; (iii) Check if i=1 ni β( s∗ ) < s∗ C − γ; (iv) If YES then accept traffic vector n, otherwise refuse it One can see that this algorithm is rather tiresome in the sense that parameter s has to be re-optimized for each entering traffic vector n Therefore, it is not suitable for real-time CAC algorithm To get rid of this computational burden, it M s) < s˜C − γ defines a is easy to see that with a constant s˜ the formula i=1 ni βi (˜ set-separation where the separation surface is a linear hyperplane Therefore, for a given s˜ the admitted traffic volume can be calculated as counting the number of traffic vectors n for which the formula holds This volume can be expressed M Hence, one can use a fixed s˜ which maximizes the as V ol(˜ s) := (˜sC−γ) M M! i=1 βi (˜ s) volume defined above, setting sopt : sopt : maxs˜ (˜ sC−γ)M M! M i=1 βi (˜ s) Since this sopt does not depend on the incoming traffic vector n, it can be calculated once for all, demanding only off-line complexity Numerical Results The aim of this section is to evaluate the traffic volumes accepted by SB-CAC (ex.SB-CAC: SB-CAC with expected values ; av.SB-CAC: SB-CAC with random selection and averaging) and GSB-CAC methods, respectively The comparison of these volumes will characterize the loss of traffic due to the fact of uncertain link descriptors given only by their p.d.f.-s The new statistical bandwidths, βi (s), i = 1, , M are calculated when the mean rates are considered to be random variables subject to Gaussian and uniform p.d.f The second column of Table shows the sopt parameters optimized off-line and calculated by maximizing the traffic volume All calculations were made with the following initial parameters: (i) the number of traffic classes: M = 2; (ii) the network capacity is chosen to be: C = 10000 kbps; (iii) the QoS parameter is chosen to be: γ = 10, (iv) the deviation of the Gaussian p.d.f is chosen to be: σ = 10; (v) the traffic rates: [mi , hi ]T C1 = [32, 64]kbps ; [mi , hi ]T C2 = [96, 128]kbps Using the sopt parameters for calculating the volume of the accepted users, the results can easily be demonstrated by counting the accepted traffic vectors n Table shows the calculated accepted volume of the user vectors As one can see, there is a loss of accepted traffic volume due to the uncertain information Developing Novel Statistical Bandwidths for Communication Networks 617 Table The calculated sopt parameters, and the accepted volume in the cases of different p.d.f.s and CAC methods volume sopt ex.SB-CAC av.SB-CAC GSB-CAC Gaussian pdf 0.0079 10149 5626 13496 Uniform pdf 0.0088 10240 5030 13252 (unknown traffic descriptors) The larger the variance of the p.d.f of the traffic descriptor (i.e the larger the amount of uncertainty), the larger these losses become The loss in traffic volume is the trade-off for only accepting those traffic configurations which not put the QoS in jeopardy These numerical results demonstrate the advantage of using the GSB concept in admission control Conclusion In this paper the concept of statistical bandwidth was extended to the case of unknown traffic descriptors A novel statistical bandwidth was derived by using the LMGF of the traffic descriptors, furthermore the tail estimator was optimized to admit the maximal load volume without violating the QoS requirements As the numerical results have demonstrated the new method yield minimal loss of load volume despite the incompleteness in characterization of the traffic descriptors Acknowledgement The research carried out here was supported by the the Laboratory of Analogic Computing, Hungarian Academy of Sciences and by the project GOA/98/06 of Research Fund, Katholieke Universiteit Leuven References F P Kelly: ”Notes on effective bandwidths”, Stochastic Networks: Theory and Applications, Editors F.P Kelly, S Zachary and I.B Ziedins), Royal Statistical Society Lecture Notes, Series, , Oxford University Press, 1996 141-168 R.J Gibbens, F.P Kelly , P.B Key.: ”A decesion theoretic approach to call admission control in ATM networks”, IEEE Journal on Selected Areas in Communication, Vol.13, No 6., Agust 1995 J Hui: ”Switching and traffic theory for integrated broadband networks”, Kluwer Academic Publisher, 1990 Levendovszky J., Vegso Cs., van der Meulen, E.C.: ”Nonparametric decision algorithms for CAC in ATM networks”, Performance Evaluation - Elsevier, Vol.41, pp 133-147, 2000 Dynamic Quality of Service Support in Virtual Private Networks Yuxiao Jia1, Dimitrios Makrakis2, Nicolas D Georganas2, and Dan Ionescu2 Nortel Networks, Ottawa, Ont., Canada School of Information Technology & Engineering, University of Ottawa, 161 Louis Pasteur St., P.O Box 450 Stn A, Ottawa, Ont., K1N 6N5, Canada Tel: 1-613-5625800 ext {6202, 6225, 6209} jenniej@nortelnetworks.com, {dimitris, georgana, ionescu}@site.uottawa.ca Abstract This paper presents a framework for the provision of dynamic Quality of Service (QoS) support in Virtual Private Networks (VPNs) running over an MPLS-enabled public network infrastructure A Dynamic Bandwidth Allocation scheme that includes traffic estimators and the development of a resource reservation algorithm, capable of modifying the resource allocation in real-time, is used Three traffic estimation algorithms are implemented and tested This system can automatically adjust to the bandwidth size of a VPN tunnel The technique is beneficial to Internet Service Providers (ISPs) and corporate users alike The superior resource management achieved through the examined approach can produce lower costs for the users and higher profits to the ISP Implementation and experimental evaluation of the technique, using our MPLS and Diffserv enabled Linux test-bed, confirmed its ability to provide better resource utilization Introduction VPN services have been offered in various forms over an extended period of time and typically have been implemented at the data link layer using Frame Relay and Asynchronous Transfer Mode (ATM) networking technologies VPN services based on IP/MPLS are quickly gaining public interest and market acceptance Most work on VPNs has mainly dealt with the security issue However, with the advancement of technology and the introduction of sophisticated applications, users don’t only demand security, but expect provision of QoS guarantees, in several cases compared to those provided by leased line services Thus, incorporation of QoS support in the VPN technology, in a resource efficient manner, becomes increasingly important In this paper we describe and evaluate a dynamic QoS supporting technique, suitable for VPNs It was implemented in an MPLS and DiffServ enabled experimental network, incorporating Traffic Engineering and RSVP-TE for the establishment of connections Three traffic estimators and the CBQ, modified in order to be able to support resource allocation in a dynamic fashion, were implemented S.E Nikoletseas (Ed.): WEA 2005, LNCS 3503, pp 618 – 621, 2005 © Springer-Verlag Berlin Heidelberg 2005 Dynamic Quality of Service Support in Virtual Private Networks 619 Traffic Estimators We assume that the measurements comprise samples gathered at regularly spaced instants during a measurement window Tmeas The measurements are used to estimate the bandwidth for the traffic flow over some reservation window Y The parameters are: X: inter-sample interval; Y: reservation window; W: number of samples taken within the measurement window Tmeas; N: number of samples taken within the reservation window Y; Ri: average sampling rate The traffic estimation algorithms we implemented and evaluated are the Maximum Estimator (ME) [1], Gaussian Estimator (GE) [2] and α−stable Estimator (ASE) [5] The following relations between the above defined parameters hold: Y = N*X (N = 1,2, … …; N is positive integer) The expressions of the renegotiated bandwidth Rren for the three estimators are given in equation In the case of ME (equation 1(a)), the renegotiated rate Rren is the maximum of the rate samples collected during the measurement window Tmeas, i.e.: Rren = max{Ri } (a) Rren = m + a v (b) Wˆ j = m + Kσ W (c) (1) The GE is based on the assumption that aggregated traffic can be characterized by the Gaussian distribution The renegotiated rate Rren for GE is given in equation 1(b), where m and v are respectively the mean and variance of the rates sampled during the measurement window, and a is a scale factor that controls the extent to which the negotiated rate accommodates the variability of the samples The ASE estimator assumes that the statistical behavior of aggregate traffic streams is statistically described through α−stable long-range dependent stochastic processes [3] This is a valid assumption, since the authors of [3] proved that such distributions describe more accurately aggregate traffic passing through modern networks as compared to earlier models In [4], traffic estimators for α−stable long-range dependent traffic have been proposed In [5], the concept of probabilistic envelope processes was extended to the α−stable case, providing us with a very practical and simple to implement traffic estimator The bandwidth demand imposed by the traffic is represented by an envelope process The following envelope process Wˆ j is used (see equation 1(c)), where m is the mean of the number of arrivals per unit time, and σ W is the scale parameter It is similar to the variance of the Gaussian distribution How to compute σ W can be found in [3,4,5] K is determined by the overshoot probability ε [5] and index of stability α [3] Experimental Set-Up and Performance Evaluation Figure shows the architecture of our test-bed The source customer network includes an IP router used to mark IP packets (setting up the DSCP value) by using iptables The MPLS backbone network is made up of two edge LERs and one core LSR, which establish one or more Diffserv enabled LSPs All routers run under the Linux RedhatTM system At the core router, CBQ has been implemented for the allocation of bandwidth The CBQ has been modified in order to be able to change the allocation in 620 Y Jia et al real-time The speed of the segment, to which the core LSR’s output is connected, is 10 Mbps, becoming the source of congestion The other segments have 100 Mbps speed In our experiments, we used our α−stable Long Range Dependent (LRD) traffic generator, set to produce traffic streams with index of stability: i) α=1.95, ii) α=1.60, and a Short Range Dependent (SRD) UDP Poisson traffic generator The QoS parameters we measure are the packet loss rate, average delay, delay jitter and the ratio of average reserved bandwidth over the average traffic volume The parameters we examine in terms of their impact on the performance are: i) buffer size at the core router B; ii) sample interval X; iii) measurement window Tmeas ; iv) resource reservation window Y For each parameter, we plot: a) the amount of packet losses at the core router; b) the average forwarding delay at the core router; c) the average delay jitter (IPDV) at the core router; d) the ratio of average reservation over the average traffic volume LSR 172.16.30.0 E1 IP router Host B 172.16.20.0 172.16.10.3 E0 172.16.40.0 E1 E0 E1 Ingress Egress 172.16.10.0 E0 172.16.5.0 Host A 172.16.5.1 Fig Test-bed’s architecture Due to the limited space, we only show the performance as function of Tmeas for α−stable traffic with α=1.95 when the GE and ASE are used The results are presented in figure We consider the cases where the traffic process exceeds its envelope by 0.1, 0.01 and 0.001 Please note that differently from the Gaussian case, the α−stable estimator is able to provide better performance when the QoS requirement becomes more stringent Dynamic bandwidth allocation jointly with the ASE provides more reliable performance, especially when the traffic exhibits high variability Conclusions This paper presents a framework for QoS provisioning in VPNs by combining dynamic Bandwidth Allocation and traffic monitoring Three traffic estimation algorithms were assessed by developing the proposed scheme on an experimental testbed with DiffServ MPLS capabilities GE k = ASE e = 0.1 GE k = GE k = ASE e = 0.1 GE k = ASE e = 0.01 GE k = ASE e = 0.001 ASE e = 0.01 GE k = ASE e = 0.001 1.4 Average forwarding delay at the core router [ms] Ratio of reservation over average traffic volume Dynamic Quality of Service Support in Virtual Private Networks 1.3 1.2 1.1 1 40 30 20 10 Measurement window [min] GE k = ASE e = 0.01 GE k = ASE e = 0.001 Packet loss at the core router [%] IPDV jitter at the core router [ms] ASE e = 0.1 3 Measurement window [min] (c) (b) GE k = Measurement window [min] (a) 621 GE k = ASE e = 0.1 GE k = ASE e = 0.01 GE k = ASE e = 0.001 0.8 0.6 0.4 0.2 Measurement window [min] (d) Fig (a) Ratio of reservation over average traffic volume; (b) average forwarding delay [ms] measured at the core router ; (c) IPDV jitter [ms] measured at the core router; (d) packet loss [%] measured at the core router, vs Tmeas , for GE and ASE when α−stable traffic with α=1.95 is applied References Cisco White Paper, “Cisco MPLS Auto-Bandwidth Allocator for MPLS Traffic Engineering: A Unique New Feature of Cisco IOS Software”, http://www.cisco.com/warp/public/ cc/pd/iosw/prodlit/mpatb_wp.htm N G Duffield, P Goyal, A Greenberg, P Mishra, “A Flexible Model for Resource Management in Virtual Private Networks”, ACM SIGCOMM’ 99, Oct 1999, Cambridge, MA, USA J R Gallardo, D Makrakis, L Orozco-Barbosa, “Use of Alpha-Stable Self-Similar Stochastic Process for Modeling Traffic in Broadband Networks”, Performance Evaluation, 40 (1-3), pp 71-98, 2000 J R Gallardo, D Makrakis, M Angulo, “Dynamic Resource Management Considering the Real Behavior of Aggregate Traffic”, IEEE Trans On Multimedia, Vol 3, No 2, pp 177185, June 2001 M L Guerrero, L Orozco-Barbosa, D Makrakis, “Probabilistic Envelope Processes for alpha-Stable Self-Similar Traffic Models and their Application to Resource Provisioning”, to appear in the Performance Evaluation Journal Author Index Albuquerque, Paul 341 Andreou, M.I 302 Asgeirsson, Eyjolfur 545 Ayala-Rincon, Mauricio 464 Bachoore, Emgad H 216 Bader, David A 16 Bar-Noy, Amotz 139 Bast, Holger 67 Becker, Bernd 452 Behle, Markus 452 Bekker, H 610 Bleischwitz, Yvonne 228 Bodlaender, Hans L 101, 216 Botelho, Fabiano C 488 Boughaci, Dalila 501 Boukerche, Azzedine 403, 464 Braad, E.P 610 Cantone, Domenico 265, 428 Christensen, Jacob 139 claffy, kc 113 Cristofaro, Salvatore 428 de Andrade, Marcos R.Q 558 de Andrade, Paulo M.F 558 de Melo, Alba Cristina Magalhaes Alves 403, 464 de Souza, Cid C 328 De Wachter, Bram 177 Dimitropoulos, Xenofontas 113 Drias, Habiba 501 Eisenbrand, Friedrich Elmasry, Amr 597 452 Fahle, Torsten 89 Faro, Simone 428 Fernandes, Eraldo R Ferro, Alfredo 265 Festa, Paola 367 Flammini, Michele 22 Gambin, Anna 534 Genon, Alexandre 177 Georgakopoulos, George F 570 Georganas, Nicolas D 618 Geraci, Filippo 580 Giugno, Rosalba 265 Goldberg, M 513 Goldengorin, B 610 Gomulkiewicz, Marcin 415 Grossi, Roberto 580 Hammad, Abdelrahman 597 Hassin, Refael 44 Heinrich-Litan, Laura 55 Hollinger, D 513 Huaker, Bradley 113 Hă uner, Falk 240 Hyyră o, Heikki 380 Ionescu, Dan Jia, Yuxiao 618 618 Kamphans, Tom 593 Kaporis, A.C 77 Kchikech, Mustapha 165 Kim, Dong Kyue 315 Kim, Ji Eun 315 Kimbrel, Tracy 391 Kirousis, L.M 77 Kliewer, Georg 228 Koch, Jeferson 403 Kochol, Martin 602 Kohayakawa, Yoshiharu 488 Kă ohler, Ekkehard 126 Koster, Arie M.C.A 101 Krioukov, Dmitri 113 Krivoˇ na ´kov´ a, Nad’a 602 Kutylowski, Miroslaw 415 Ladner, Richard E 139 Langetepe, Elmar 593 Leone, Pierre 341 Levendovszky, Janos 614 Liebchen, Christian 354 Lo Presti, Giuseppe 265 624 Author Index Lu, Qiang 152 Lă ubbecke, Marco E 55 Magdon-Ismail, M 513 Makrakis, Dimitrios 618 Martins, Simone L 558 Massart, Thierry 177 Mazza, Christian 341 Mehlhorn, Kurt 32 Michail, Dimitrios 32 Mă ohring, Rolf H 126, 189 Moura, Arnaldo V 328 Na, Joong Chae 315 Navarra, Alfredo 22 Orogz, Csego 614 Orponen, Pekka 524 Ossamy, Rodrigue 290 Panagopoulou, Panagiota N 203 Papadimitriou, Christos H Papadopoulou, V.G 302 Pardalos, Panos M 367 Park, Kunsoo 315 Pereira, Romulo A 328 Perennes, Stephane 22 P´erez, Xavier 253 Pinzon, Yoan 380 Pitsoulis, Leonidas S 367 Plastino, Alexandre 558 Politopoulou, E.I 77 Pulvirenti, Alfredo 265 Resende, Mauricio G.C Ribeiro, Celso C Riley, George 113 Rolim, Jose 341 367 Santana, Thomas M 464 Sawitzki, Daniel 277 Schaeffer, Satu Elisa 524 Schank, Thomas 606 Schilling, Heiko 126, 189 Schă utz, Birk 189 Schwartz, Justus 476 Segev, Danny 44 Shinohara, Ayumi 380 Smejov´ a, Silvia 602 Spirakis, Paul G 77, 203, 302 Steger, Angelika 476 Stein, Cliff 545 Steinder, Malgorzata 391 Sviridenko, Maxim 391 Tamir, Tami 139 Tantawi, Asser 391 Theodorides, B 302 Tiemann, Karsten 89 Togni, Olivier 165 Vierhaus, Heinrich Theodor Wagner, Dorothea 189, 606 Weber, Ingmar 67 Weißl, Andreas 476 Weinard, Maik 440 Willhalm, Thomas 189 Wimmer, Ralf 452 Wla´z, Pawel 415 W´ ojtowicz, Damian 534 Wolle, Thomas 101 Xeros, A 302 Zhang, Hu 152 Ziviani, Nivio 488 415 ... the help of trusted external referees In addition to the 54 papers included in this volume, papers were accepted as poster presentations: these papers were published in a separate poster proceedings... markets (besides being one itself), the Internet challenged economists, and especially game theorists, in new ways At the other bank, computer scientists were faced for the first time with a mysterious... Thomas Schank Elad Schiller Frank Schulz Sebastian Seibert Spyros Sioutas Spiros Sirmakessis Riste Skrekovski IX Stˆenio Soares Yannis Stamatiou Maurizio Strangio Tami Tamir Leandros Tassiulas Dimitrios