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 Alfred Kobsa University of California, Irvine, CA, 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 TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max Planck Institute for Informatics, Saarbruecken, Germany CuuDuongThanCong.com 6595 CuuDuongThanCong.com Alberto Marchetti-Spaccamela Michael Segal (Eds.) Theory and Practice of Algorithms in (Computer) Systems First International ICST Conference, TAPAS 2011 Rome, Italy, April 18-20, 2011 Proceedings 13 CuuDuongThanCong.com Volume Editors Alberto Marchetti-Spaccamela Sapienza University of Rome Department of Computer Science and Systemics "Antonio Ruberti" Via Ariosto 25, 00185 Rome, Italy E-mail: alberto@dis.uniroma1.it Michael Segal Ben-Gurion University of the Negev Communication Systems Engineering Department POB 653, Beer-Sheva 84105, Israel E-mail: segal@cse.bgu.ac.il ISSN 0302-9743 e-ISSN 1611-3349 ISBN 978-3-642-19753-6 e-ISBN 978-3-642-19754-3 DOI 10.1007/978-3-642-19754-3 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011922539 CR Subject Classification (1998): F.2, D.2, G.1-2, G.4, E.1, I.1.2, I.6 LNCS Sublibrary: SL – Theoretical Computer Science and General Issues © Springer-Verlag Berlin Heidelberg 2011 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 The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) CuuDuongThanCong.com Preface This volume contains the 25 papers presented at the First International ICST Conference on Theory and Practice of Algorithms in (Computer) Systems (TAPAS 2011), held in Rome during April 18-20 2011, including three papers by the distinguished invited speakers Shay Kutten, Kirk Pruhs and Paolo Santi In light of the continuously increasing interaction between computing and other areas, there arise a number of interesting and difficult algorithmic issues in diverse topics including coverage, mobility, routing, cooperation, capacity planning, scheduling, and power control The aim of TAPAS is to provide a forum for the presentation of original research in the design, implementation and evaluation of algorithms In total 45 papers adhering to the submission guidelines were submitted Each paper was reviewed by three referees Based on the reviews and the following electronic discussion, the committee selected 22 papers to appear in final proceedings We believe that these papers together with the invited presentations made up a strong and varied program, showing the depth and the breadth of algorithmic research TAPAS 2011 was sponsored by ICST (Institute for Computer Science, Social Informatics and Telecommunications Engineering, Ghent, Belgium) and Sapienza University of Rome Besides the sponsor we wish to thank the people from the EasyChair Conference Systems: their wonderful system saved us a lot of time Finally, we wish to thank the authors who submitted their work, all Program Committee members for their hard work, and all reviewers who helped the Program Committee in evaluating the submitted papers April 2011 CuuDuongThanCong.com Alberto Marchetti-Spaccamela Michael Segal CuuDuongThanCong.com Conference Organization Program Committee Stefano Basagni Andrei Broder Alon Efrat Stefan Funke Michael Juenger Alex Kesselman Alberto Marchetti-Spaccamela Alessandro Mei Michael Segal Hanan Shpungin Jack Snoeyink Leen Stougie Peng-Jun Wan Peter Widmayer Gerhard Woeginger Northeastern University, USA Yahoo Inc., USA University of Arizona, USA University of Stuttgart, Germany University of Cologne, Germany Google Inc., USA Sapienza University of Rome, Italy Co-chair Sapienza University of Rome, Italy Ben-Gurion University of the Negev, Israel Co-chair University of Calgary, Canada University of North Carolina at Chapel Hill, USA VU University, Amsterdam, The Netherlands Illinois Institute of Technology, USA ETH, Switzerland Eindhoven University of Technology, The Netherlands Steering Committee Pankaj Agarwal Imrich Chlamtac Alberto Marchetti-Spaccamela David Peleg Michael Segal Paul Spirakis Roger Wattenhofer Duke University, USA University of Trento, Italy Sapienza University of Rome, Italy Weizmann Institute, Israel Ben-Gurion University of the Negev, Israel University of Patras, Greece ETH, Switzerland External Reviewers Nikhil Bansal Vincenzo Bonifaci Sylvia Boyd Christoph Buchheim Marek Chrobak CuuDuongThanCong.com Daniel Dumitriu Jochen Eisner Martin Gronemann Carsten Gutwenger Cor Hurkens VIII Conference Organization Kirk Pruhs Ashikur Rahman Johan M.M van Rooij Cyriel Rutten Waqar Saleem Daniel Schmidt Andreas Schmutzer Sabine Storandt Zhu Wang Xiaohua Xu Andreas Karrenbauer Leo Kroon Frauke Liers Domagoj Matijevic Sven Mallach Nikola Milosavljevic Matthias Mnich Thomas Moscibroda Rudi Pendavingh Ugo Pietropaoli Stefan Porschen Conference Coordinator Elena Fezzardi CuuDuongThanCong.com ICST Table of Contents Distributed Decision Problems: The Locality Angle (Invited Talk) Shay Kutten Managing Power Heterogeneity (Invited Talk) Kirk Pruhs The Mathematics of Mobility (Invited Talk) Paolo Santi Speed Scaling to Manage Temperature Leon Atkins, Guillaume Aupy, Daniel Cole, and Kirk Pruhs Alternative Route Graphs in Road Networks Roland Bader, Jonathan Dees, Robert Geisberger, and Peter Sanders 21 Robust Line Planning in Case of Multiple Pools and Disruptions Apostolos Bessas, Spyros Kontogiannis, and Christos Zaroliagis 33 Exact Algorithms for Intervalizing Colored Graphs Hans L Bodlaender and Johan M.M van Rooij 45 L(2,1)-Labeling of Unigraphs (Extended Abstract) Tiziana Calamoneri and Rossella Petreschi 57 Energy-Efficient Due Date Scheduling Ho-Leung Chan, Tak-Wah Lam, and Rongbin Li 69 Go with the Flow: The Direction-Based Fr´echet Distance of Polygonal Curves Mark de Berg and Atlas F Cook IV A Comparison of Three Algorithms for Approximating the Distance Distribution in Real-World Graphs Pierluigi Crescenzi, Roberto Grossi, Leonardo Lanzi, and Andrea Marino Exploiting Bounded Signal Flow for Graph Orientation Based on Cause–Effect Pairs Britta Dorn, Falk Hă uner, Dominikus Kră uger, Rolf Niedermeier, and Johannes Uhlmann On Greedy and Submodular Matrices Ulrich Faigle, Walter Kern, and Britta Peis CuuDuongThanCong.com 81 92 104 116 Speed Scaling for Energy and Performance with Instantaneous Parallelism 251 also in processor allocation The former mistake leads to bad performance since jobs that complete early may in fact be slowed down to save energy, and this has contributed to the lower bound of IP-clairvoyant algorithms shown in this paper The situation may deteriorate further in the non-clairvoyant setting as more energy will be wasted or slower execution rate will result if a wrong number of processors is also allocated to a job References [1] Albers, S.: Energy-efficient algorithms Communications of the ACM 53(5), 86–96 (2010) [2] Albers, S., Fujiwara, H.: Energy-efficient algorithms for flow time minimization In: Durand, B., Thomas, W (eds.) STACS 2006 LNCS, vol 3884, pp 621–633 Springer, Heidelberg (2006) [3] Bansal, N., Chan, H.-L., Pruhs, K.: Speed scaling with an arbitrary power function In: SODA, pp 693–701 (2009) [4] Bansal, N., Pruhs, K., Stein, C.: Speed scaling for weighted flow time In: SODA, pp 805–813 (2007) [5] Brooks, D.M., Bose, P., Schuster, S.E., Jacobson, H., Kudva, P.N., Buyuktosunoglu, A., Wellman, J.-D., Zyuban, V., Gupta, M., Cook, P.W.: Power-aware microarchitecture: Design and modeling challenges for next-generation microprocessors IEEE Micro 20(6), 26–44 (2000) [6] Chan, H.-L., Edmonds, J., Lam, T.-W., Lee, L.-K., Marchetti-Spaccamela, A., Pruhs, K.: Nonclairvoyant speed scaling for flow and energy In: STACS 2009, pp 409–420 (2009) [7] Chan, H.-L., Edmonds, J., Pruhs, K.: Speed scaling of processes with arbitrary speedup curves on a multiprocessor In: SPAA, pp 1–10 (2009) [8] Edmonds, J.: Scheduling in the dark In: STOC, pp 179–188 (1999) [9] Grunwald, D., Morrey III, C.B., Levis, P., Neufeld, M., Farkas, K.I.: Policies for dynamic clock scheduling In: OSDI, pp (2000) [10] Irani, S., Pruhs, K.: Algorithmic problems in power management SIGACT News 36(2), 63–76 (2005) [11] Lam, T.W., Lee, L.-K., To, I.K.-K., Wong, P.W.H.: Speed scaling functions for flow time scheduling based on active job count In: Halperin, D., Mehlhorn, K (eds.) ESA 2008 LNCS, vol 5193, pp 647–659 Springer, Heidelberg (2008) [12] Pruhs, K.R., van Stee, R., Uthaisombut, P.: Speed scaling of tasks with precedence constraints In: Erlebach, T., Persinao, G (eds.) WAOA 2005 LNCS, vol 3879, pp 307–319 Springer, Heidelberg (2006) [13] Robert, J., Schabanel, N.: Non-clairvoyant batch sets scheduling: Fairness is fair enough In: Arge, L., Hoffmann, M., Welzl, E (eds.) ESA 2007 LNCS, vol 4698, pp 741–753 Springer, Heidelberg (2007) [14] Sun, H., Cao, Y., Hsu, W.-J.: Non-clairvoyant speed scaling for batched parallel jobs on multiprocessors In: CF, pp 99–108 (2009) [15] Sun, H., He, Y., Hsu, W.-J.: Energy-Efficient Multiprocessor Scheduling for Flow Time and Makespan CoRR abs/1010.4110 (2010) [16] Yao, F., Demers, A., Shenker, S.: A scheduling model for reduced CPU energy In: FOCS, pp 374–382 (1995) CuuDuongThanCong.com Algorithms for Scheduling with Power Control in Wireless Networks Tigran Tonoyan TCS Sensor Lab Centre Universitaire d’Informatique Route de Drize 7, 1227 Carouge, Geneva, Switzerland tigran.tonoyan@unige.ch http://tcs.unige.ch Abstract In this work we study the following problem of scheduling with power control in wireless networks: given a set of communication requests, one needs to assign the powers of the network nodes, and schedule the transmissions so that they can be done in a minimum time, taking into account the signal interference of parallelly transmitting nodes The signal interference is modeled by SINR constraints We correct and complement one of recent papers on this theme, by giving approximation algorithms for scheduling with power control for the case, when the nodes of the network are located in a doubling metric space Keywords: wireless network, scheduling, algorithm Introduction One of the basic issues in wireless networks is that concurrent transmissions may cause interference We are interested in the problem of scheduling with power control, i.e we choose the power levels of the nodes and then schedule the set of communication requests with respect to the chosen power settings The scheduling problem has been studied in several communication models It has been shown that the results obtained in different models differ essentially One of the factors on which the scheduling problem depends crucially is the model of interference Wireless networks have often been modeled as graphs The nodes of this communication graph represent the physical devices, two nodes being connected by an edge if and only if the respective devices are within mutual transmission range In this graph-theoretic model a node is assumed to receive a message correctly if and only if no other node in close physical proximity transmits at the same time Clearly, the graph-theoretic model fails to capture the accumulative nature of actual radio signals If the power levels of the nodes are chosen properly, then a node may successfully receive a message in spite of being in the transmission range of other simultaneous transmitters In contrast, in last several years there has been a significant research done considering the problem of scheduling in models of wireless networks which are Research partially founded by FRONTS 215270 A Marchetti-Spaccamela and M Segal (Eds.): TAPAS 2011, LNCS 6595, pp 252–263, 2011 c Springer-Verlag Berlin Heidelberg 2011 CuuDuongThanCong.com Algorithms for Scheduling with Power Control in Wireless Networks 253 more realistic (and more efficient, see [16]) than graph-theoretic models The standard model is the signal-to-interference-plus-noise (SINR) model The SINR model reflects physical reality more accurately and is therefore often simply called the physical model More formally, given is an arbitrary set of links, each a sender-receiver pair of points on a metric space We seek an assignment of powers to the senders and a partition of the linkset into a minimum number of subsets or slots, so that the links in each slot satisfy the SINR-constraints We refer to this as the problem of scheduling with power control, or simply as PC-scheduling problem in directed model In the bidirectional model both nodes in a link may be transmitting, which implies stronger constraints We are trying to design algorithms that result in efficient schedules We are particularly interested in schedules using so-called oblivious power assignments, which depend only on the length of the given link Oblivious assignments appear unavoidable in the distributed setting of the problem, as the nodes in that case “do not know” the topology of the whole network So it is desirable to find short schedules using these power assignments, or find out how much worse can perform such power assignments in comparison to the optimal power assignment Related Work and Our Results The body of algorithmic work on the scheduling problem is mostly on graph-based models The inefficiency of graph-based protocols is well documented and has been shown theoretically as well as experimentally (see [7] and [16] for example) The algorithmic study of the problem from the perspective of SINR model started recently, with papers as [17], [14] and [4] Here the performance ratio of the algorithms is evaluated, and it depends on some structural properties of the network which can grow linearly with the number of nodes/links In [1] an O(log Λ)-approximation algorithm is given for the Single-Slot scheduling problem, which is to find the maximum SINR feasible subset of links Here Λ is the ratio between the longest and the shortest link lengths In [6] a randomized algorithm is given for the scheduling problem using the linear power assignment that uses O(OP T log Λ + log2 n) slots, where OPT is the number of slots in the optimum schedule and n is the number of all links All these results are for the directed model of scheduling In [5] a construction is given, that shows that schedules based on any oblivious power assignment can be a factor of n from the optimum However, in [8] it is shown that in terms of Λ, the gap is actually Ω(log log Λ), using similar constructions In [5] the bidirectional version of PC-scheduling problem is considered, and a O(log4.5+α n)approximation algorithm is given, using the mean power assignment in general metrics, where α > is the so called path loss exponent Properties of wireless networks in SINR setting has been investigated also from the point of view of network connectivity, as in [15] and [3] In [2] so called SINRdiagrams are considered, which are the reception zones of the sender nodes, and particularly the convexity and fatness of these zones is shown when the powers are uniform, and α = CuuDuongThanCong.com 254 T Tonoyan In this work we discuss the results from [8] They consider the problem of PC-scheduling in the SINR model Among others, they state results regarding scheduling links with arbitrary length: there is an algorithm approximating PC-scheduling within a factor of O(log n log log Λ) using the mean power assignment in the directed model, and there is an algorithm approximating PC-scheduling within a factor O(log n) using the mean power assignment in the bidirectional model Here we give a counter-example for a key lemma from [8], which shows that the statements and are still unproven Next we prove the non-constructive versions of and 2.: the mean power assignment is a O(log n)-approximation for the problem of PC-scheduling in the bidirectional model, and O(log n log log Λ)-approximation in the directed model, when the network is placed in a fading metric Next we present a O(log n)-approximation algorithm for the bidirectional model, which uses the mean power assignment, and O(log n log log Λ)-approximation algorithm for the directed model1 These algorithms both can be used as O(log n)-approximation algorithms for scheduling problem with mean power assignment Preliminaries Here we mainly follow the definitions used in [8] Given is a set L = {1, 2, , n} of links, where each link v represents a communication request between a sender node sv and a receiver node rv The nodes are located in a metric space with distance function d The asymmetric distance dvw from a link v to a link w is defined as follows: when the directed model of communication is adopted, then dvw = d(sv , rw ), and when the bidirectional model of communication is adopted, then dvw = min{d(sv , rw ), d(sv , sw ), d(rv , rw ), d(rv , sw )} Note that in the latter case dvw = dwv (i.e the distance is actually symmetrical), but in the former case for some pairs v,w it can be dvw = dwv The length of a link v is lv = d(sv , rv ) Each node v is assigned a transmitting power Pv > In the bidirectional model of communication both sender and receiver nodes of a link are assigned the same power, as in this case during a data transmission the receiver also sends some information to the sender We adopt the path loss radio propagation model for the reception of signals, where the signal received from a node x of the link v at some node y is Pv /d(x, y)α , where α > denotes the path loss exponent We adopt the physical interference model, where a communication v is done successfully if and only if the following condition holds: Pv /lvα ≥ β, (1) α w∈S\{v} Pw /dwv + N In the same setting, a better, O(log n log log Λ)-approximation is achieved by M.M Halld´ orsson, see [9] CuuDuongThanCong.com Algorithms for Scheduling with Power Control in Wireless Networks 255 where N is the ambient noise, S is the set of concurrently scheduled links in the same slot, and β ≥ denotes the minimum SINR(signal-to-interference-plusnoise-ratio) required for the transmission to be successfully done We say that S is SINR-feasible if (1) holds for each link in S As in [8], we assume N = (i.e there is no ambient noise), β = 1, and strict inequality in (1) We will show that thanks to Theorem those assumptions not have essential effect on the results In the problem of scheduling with power control given the set L of links, one needs to assign the powers of the nodes, and split L into SINR-feasible subsets (slots) with respect to the chosen power assignment, such that the number of slots is the minimum The collection of such subsets is called schedule, and the number of slots in a schedule is called the length of the schedule We will refer to this problem as PC-scheduling problem In the problem of scheduling with given powers given the set L and the power assignments, one needs to schedule L into minimum number of slots with respect to the given power assignment In this work we are interested in the problem of PC-scheduling Note that each of these problems can be stated for both directed and bidirectional model If for some statement we don’t explicitly mention the model, then it is stated for both models The affectance of a link v caused by a set of links S is the sum of the interferences of the links in S on v relative to the signal between the nodes of v: aS (v) = w∈S\{v} Pw /dα wv = Pv /lvα w∈S\{v} Pw lvα · Pv d α wv Note that the affectance is additive, i.e if there are two disjoint sets S1 and S2 , then aS1 ∪S2 (v) = aS1 (v) + aS2 (v) A p-signal set or schedule is one where the affectance of any link is less than 1/p Note that a set is SINR-feasible if and only if it is a 1-signal set We will call 1-signal schedule a SINR-feasible schedule We describe the doubling metric spaces Consider a metric space X with metric d The ball of radius r centered at a point x ∈ X is the set B(x, r) = {y ∈ X|d(x, y) < r} A set Y ⊂ X is an r-packing if d(x, y) > 2r for any pair x, y ∈ Y of different points The packing number Π(X, r) is the size of the largest r-packing The doubling dimension of X is the value t, such that supx∈X,R>0 Π(B(x, R), eR) = C/et as e → 0, where C is an absolute constant The doubling metric spaces are precisely the spaces with finite doubling dimension It is known that the k-dimensional Euclidean space is a doubling metric with doubling dimension k (see [11]) Usually we will consider the nodes of the network on a doubling space, and the path loss exponent α being greater than the doubling dimension of the space The pair of a doubling space and the path loss exponent greater than the dimension is called a fading metric In [8] for approximating the problem PC-scheduling the mean power assignment is considered, which is given by assigning to a node of the link v a power CuuDuongThanCong.com 256 T Tonoyan α/2 Pv = clv , where c > In this case the affectance of a link v by √0 is a constant α a link w is aw (v) = lv lw /dwv We call two links lv and lw q-independent with power scheme {Pv }, if the affectance (with the specified powers) of each of those links by the other one is less than q α It is easy to check, that two links lv and lw are q-independent with the mean powers if and only if the following condition holds: dvw > q lw lv and dwv > q lw lv As for the bidirectional case the distances dwv and dvw are the same, the links lv and lw are q-independent with the mean powers if and only if dvw > q lw lv We call two links lv and lw q-independent, if the following inequality holds: dvw dwv > q lw lv Note that for the bidirectional model two links are q-independent if and only if they are q-independent with the mean power assignment A set S of links is a q-independent set if each pair of links in S is q-independent The following fact immediately follows from the definition of q-independence Lemma A set of links that belong to the same q α -signal slot in some schedule, is q-independent We say that a set of links is nearly equilength, if the lengths of any pair of links in the set differ not more than two times The following theorem from [8] shows that each q-independent set S of nearly equilength links in a fading metric is a Ω(q α )-signal slot when the uniform powers are used, i.e all nodes have the same power P , for some P > Theorem [8] Let L be a q-independent set of nearly equilength links in a fading metric Then L is a Ω(q α )-signal set when the powers are uniform We say that a set S of links is well-separated, if for each two links from S the ratio between the longer link length and the shorter link length is not more than or not less than n2 Two links v and w are said to be τ -close under the mean power assignments if max{av (w), aw (v)} ≥ τ , i.e at least one affects the other one more than by τ We call a set of links S ⊆ L p-bounded for p > 0, if for each link lv ∈ L, there are at most p links lw in S, such that n2 lv ≤ lw and lw is -close to lv 2n Let Λ denote the ratio between the maximum and the minimum length of links The following theorem is proven (in a slightly different statement) in [8] Theorem In the case of directed scheduling each 3-independent set of links is p-bounded with p = O(log log Λ) In the case of bidirectional scheduling each 2-independent set of links is 1-bounded CuuDuongThanCong.com Algorithms for Scheduling with Power Control in Wireless Networks 257 Note that in [8] the first part of Theorem is stated for well-separated SINRfeasible sets, but with exactly the same proof the result holds for just 3independent sets With the stronger assumption of q-independence with mean powers, in fading metrics a stronger bound holds for the directed model Theorem In the directed model each 3-independent set of links is O(1)-bounded The following result demonstrates the robustness of schedules in the model we use, and is proven in [10] We assume the power assignment of the nodes is given Theorem [10] There is a polynomial-time algorithm that takes a p-signal schedule and refines into a p -signal schedule, for p > p, increasing the number of slots by a factor of at most 2p /p The algorithm described in Theorem works for both communication models The Counterexample In [8] the following claim is stated, which is used as a key feature in the proofs of a number of theorems Claim [8] Let L be a set of links partitioned into length groups L1 , L2 , , Lt such that links in the same group differ by a factor of at most but links in different groups differ by a factor of at least n2 Suppose each group Li has been scheduled with uniform powers using Γi slots Then, there is an algorithm that produces a combined schedule of L with the mean power assignment using O(log log Λ · maxi Γi ) slots in the directed model and O(maxi Γi ) slots in the bidirectional model We bring an example that shows that the claim does not hold The example is for the directed model, but the same works for the bidirectional model Let each Lv consist of only one link v: Lv = {v}, so that we have maxi Γi = We also assume t = n We define d(rv , rw ) = for all pairs v, w, i.e all receiver nodes are at the same point It follows then that each link must be scheduled in a separate slot (using any power assignment), which gives n slots But then we can choose the lengths of the links, so that they are still well-separated, but log log Λ 0, schedules Q into O(p log n) slots with the mean power assignment A similar algorithm was used in [8] for proving the erroneous claim above We modify their algorithm, and prove that it is an approximation algorithm for scheduling q-independent sets The description of the procedure follows We will refer to the algorithm as ScheduleIndependent Input: a q-independent p-bounded set Q, for some p > and q ≥ Let Q = ∪i Qi , where Qi = {t ∈ Q|lt ∈ [2i−1 lmin , 2i lmin )} Assign Bi = ∪j Qi+j·2 log n , for i = 1, 2, , log n Schedule each Bi = ∪j Kj , where Kj = Qi+j·2 log n , the following way 4.1 Using the algorithm from Theorem transform each Kj into an f -signal kj schedule Σj = {Sjs }s=1 with f = 2α/2+1 4.2 s ← 4.3 Assign S ← ∪j Sjs : if for some j, kj < s, then we take Sjs = ∅ 4.4 Sort S in the non-increasing order of link lengths: l1 ≥ l2 ≥ l|S| 4.5 Tsr ← ∅, r = 1, 2, , p + 4.6 For k = 1, 2, , |S| do: find a Tsr not containing links u with lu > n2 lk which are 1/(2n)-close to lk , and assign Tsr ← Tsr ∪ {lk } 4.7 s ← s + 1: if s ≤ max kj , then go to step 4.3, otherwise the schedule for Bi is {Tsr |Tsr = ∅} Output the union of the schedules of all Bi The algorithm splits the input set into a logarithmic number of well-separated subsets Bi , then schedules each Bi separately First Bi is split into maximal equilength subsets Qj Then each Qj is scheduled into a constant number of slots with the mean power assignment, using Theorem To schedule Bi , the algorithm takes the union of the first slots of the schedules for all Qj (which are contained in Bi ), and schedules them into p + slots, using the p-bounded property So we get a schedule with O(p) slots for each Bi , and a schedule with O(p log n) slots for Q The correctness of the algorithm is proven in the following theorem Theorem Let Q = {1, 2, , k} be a q-independent p-bounded subset of L for q ≥ Then ScheduleIndependent schedules Q into O(p log n) slots with the mean power assignment CuuDuongThanCong.com Algorithms for Scheduling with Power Control in Wireless Networks 259 Using the above mentioned algorithm one gets “short” schedules for a given qindependent set of links, so the next step is to split the set L into a small number of q-independent subsets At this point we already can prove bounds for the mean power assignments Note that according to Lemma a SINR-feasible set is a 1-independent set, i.e each schedule splits the set L into 1-independent subsets, with the number of subsets equal to the length of the schedule So we have the following corollary of Theorem Corollary For the directed model of communication the mean power assignment is a O(log n log log Λ)-approximation for the problem PC-scheduling in fading metrics For bidirectional model of communication the mean power assignment is a O(log n)-approximation for the problem PC-scheduling in fading metrics Proof We prove the claim for the directed model, the other case can be proven similarly Suppose we are given the optimal power assignment and the optimal schedule Σ for that power assignment Obviously, Σ is a 1-signal schedule (according to our notation) Using the algorithm from Theorem 4, Σ can be converted to a 3α -signal schedule Σ = (S1 , S2 , , Sk ), by increasing the length only by a constant factor Then according to Lemma each Si is a 3-independent set According to Theorem the set Si is p-bounded with p = O(log log Λ), so by applying Theorem 5, each Si can be scheduled into O(log n log log Λ) slots, so the whole set L can be scheduled using O(log n log log Λ · k) slots with the mean power assignment, which completes the proof Splitting L into a Small Number of q-independent Subsets First we present an algorithm for coloring a certain class of graphs, which we call t-strong graphs Let G be a simple undirected graph We denote by V (G) the vertex-set of G For a vertex v of G we denote by NG (v)(or simply N (v)) the subgraph of G induced by the set of neighbors of v in G For an integer t > we say G is a t-strong graph if for each induced subgraph G of G there is a vertex v in G , such that the graph NG (v) does not have independent sets of size more than t Using the ideas of [13] for coloring Unit Disk Graphs, we prove that there is a t-approximation algorithm for coloring a t-strong graph The following theorem from [12] describes the algorithm which we use It is based on the results of [18] Theorem [12] Let G = (V, E) be a simple undirected graph and let δ(G) denote the largest δ such that G contains a subgraph in which every vertex has a degree at least δ Then there is an algorithm coloring G with δ(G) + colors, with running time O(|V | + |E|) We will refer to the algorithm from Theorem as Hochbaum’s algorithm The proof of the following theorem is similar to the proof of Theorem 4.5 of [13] CuuDuongThanCong.com 260 T Tonoyan Theorem Hochbaum’s algorithm applied to a t-strong graph G gives a tapproximation to the optimal coloring Next we apply Hochbaum’s algorithm to split L into a small number of qindependent sets For q ≥ 1, when the directed model of communication is considered, let Dq (L) be the graph with vertex set L (i.e the vertices are the links from L), where two vertices v and w are adjacent in Dq (L) if and only if v and w are not q-independent with the mean power assignment, i.e either dvw ≤ q lw lv or dwv ≤ q lw lv (2) For the bidirectional model let Bq (L) be the graph with vertex set L and with two vertices v and w adjacent if and only if they are not q-independent, i.e dvw ≤ q lw lv (3) We show that Bq (L) is t-strong, and Dq (L) is t -strong for some constants t, t > 0, so that Hochbaum’s algorithm finds colorings for those graphs, which approximate the respective optimal colorings within constant factors We will need the following lemma Lemma Let {t0 , t1 , t2 , , tk } be a set of points in an m-dimensional doubling metric space and c1 , c2 , c3 and {b0 , b1 , b2 , , bk } be positive reals, such that 1) b0 ≤ c1 bi , for i = 1, 2, , k, 2) d(t0 , ti ) ≤ c2 b0 bi for i = 1, 2, , k and 3) d(ti , tj ) > c3 bi bj for i, j = 1, 2, , k, i = j 4c2 Then k ≤ C( + 1)m + c1 c3 Theorem The graph Bq (L) is O(1)-strong Proof We need the following lemma Consider the vertex v with lv being minimum over all links Then for each vertex w of the √ subgraph N (v) we have lw ≥ lv On the other hand, from (3) we have dvw ≤ q lv lw Consider a subset I = {1, 2, , k} of vertices of N (v), which is an independent set in N (v) Our goal is to show that |I| = O(1) Consider the set of nodes R = {t1 , t2 , , tk }, where ti is the node (sender or receiver) of the link i, closest to the link v (in terms of the distance between two sets of points) R can be split into two subsets, first with nodes for which the closest node of v is the sender of v, and the others for which the receiver of v is closer We assume that R is anyone of that subsets: if we show that |R| = O(1), then the proof follows We denote by t0 the node of v which is closer to R than the other one √ √ Let us denote bi = li for each link i, and b0 = lv According to (3) we have d(t0 , ti ) ≤ qb0 bi CuuDuongThanCong.com (4) Algorithms for Scheduling with Power Control in Wireless Networks 261 d(ti , tj ) > qbi bj , for i, j = 1, 2, , k, i = j, (5) which means that we can apply Lemma with points t0 , t1 , , tk , real numbers b0 , b1 , , bk and c1 = 1, c2 = c3 = q, getting m |R| = k ≤ C (4/q + 1) + 1, thus completing the proof The following theorem is proven using similar technique Theorem For a constant q the graph Dq (L) is O(1)-strong Now let us go back to the problem of PC-scheduling in a fading metric Consider the following algorithm for scheduling L We refer to it as Schedule Construct the graph B2 (L) (respectively D3 (L) for the directed model) Applying the algorithm from Theorem on the resulting graph, split L into 2-independent (3-independent) subsets S1 , S2 , , Sk For i = 1, 2, , k apply the algorithm ScheduleIndependent to the set Si , getting a schedule Σi = {Si1 , Si2 , , Siki } Output the schedule ∪i Σi Theorem 10 In the bidirectional model of communication the algorithm Schedule approximates PC-scheduling within a factor O(log n) in fading metrics For the directed model the algorithm Schedule approximates PC-scheduling within a factor O(log2 n log log Λ) in fading metrics Proof Consider the bidirectional model According to Theorem 4, for a constant q ≥ an optimal q α -signal schedule is a constant factor approximation for an optimal SINR-feasible schedule But from Lemma we know that each q α signal schedule induces a coloring of the graph Bq (L), so the chromatic number of Bq (L) is not more than the length of the optimal q α -signal schedule So if we denote the length of an optimal SINR-feasible schedule by OP T , then on the second step of the algorithm we have k = O(OP T ) According to Theorem 2, on the third step of the algorithm for all i = 1, 2, , k we have ki = O(log n), so the k length of the resulting schedule on the fourth step is i=1 ki = O(log nOP T ) for the bidirectional model Now consider the directed model It is easy to see, that for q ≥ each q α -signal schedule, which uses the mean power assignment, induces a coloring of the graph Dq (L), so the chromatic number of Dq (L) is not more than the optimal q α -signal schedule with the mean power assignment On the other hand, from Corollary we know that the mean power assignment approximates the problem of PC-scheduling within a factor of O(log n log log Λ), so if the optimal SINR-feasible schedule length (with the optimal power assignment) is OP T , then on the second step we have k = O(log n log log ΛOP T ) According to Theorem 3, on the third step of the algorithm for all i = 1, 2, , k we have ki = O(log n), so the length of the resulting schedule on the fourth step k is i=1 ki = O(log2 n log log ΛOP T ) for the directed model CuuDuongThanCong.com 262 T Tonoyan Introducing the Noise Factor All the results we derived are for the case when there is no ambient noise factor in SINR formula To see how much is the impact of introducing the noise factor into the formula on the schedule length, first let us notice that if there is a noise N , then for each power assignment, which is a solution for the problem PC-scheduling, the following must hold: Pv /lvα ≥ N , for each link v (6) This is the minimum power needed to deliver a message to the receiver of v even if there are no other transmissions Then, if there is a set S, which is SINRfeasible with powers {Pv } and without a noise, then for each v ∈ S we have α α Pv /lvα > w∈S\v Pw /dα wv Then 2Pv /lv > w∈S\v Pw /dwv + N , which means if we introduce the noise factor, then for S the SINR condition holds with β = 1/2, and using Theorem we can split S into subsets for which the SINR condition holds with β = and the noise factor N included Thus we have: Proposition If (6) holds, then each zero-noise schedule of length T can be transformed into a non-zero-noise schedule of length no more than 4T This comes to show that all above results hold also for the case with a non-zero noise, as we didn’t any assumptions on the coefficients of the mean power assignment we used Conclusion In this work we pointed out a flaw in proofs in paper [8], and tried to prove their claims which were dependent on the erroneous statement Thus we showed that in fading metrics the mean power assignment approximates the problem of PC-scheduling for bidirectional and directed models with factors O(log n) and O(log n log log Λ) respectively Moreover, we presented approximation algorithms for both models with approximation guarantee O(log n) and O(log2 n log log Λ) respectively Note that both algorithms can be used as O(log n)-approximation algorithms for the problem of scheduling with mean power assignment As the scheduling problem is interesting in general metrics, it is an open problem to find good approximation for PC-scheduling problem for networks placed in general metric spaces It is also desirable to further investigate the capabilities of oblivious power assignments Acknowledgment Author thanks Prof M.M Halld´ orsson for helpful discussions CuuDuongThanCong.com Algorithms for Scheduling with Power Control in Wireless Networks 263 References Andrews, M., Dinitz, M.: Maximizing capacity in arbitrary wireless networks in the SINR model: Complexity and game theory In: 29th Annual IEEE Conference on Computer Communications, INFOCOM (2009) Avin, C., Emek, Y., Kantor, 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Practice of Algorithms in (Computer) Systems (TAPAS 2011) , held in Rome during April 18-20 2011, including three papers by the distinguished invited speakers Shay Kutten, Kirk Pruhs and Paolo Santi In. .. Santi In light of the continuously increasing interaction between computing and other areas, there arise a number of interesting and difficult algorithmic issues in diverse topics including coverage,... study of reaching consensus in the face of the uncertainty concerning process failures The impossibility of solving this problem in asynchronous systems was established in the seminal paper of [14]