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Complexity of Connected Componentsin Evolving Graphs and the Computationof Multicast Trees in Dynamic Networks ⋆Sandeep Bhadra 1 and Afonso Ferreira 21 Dept. of Electrical Engineering, Indian Institute of Technology,Madras, Chennai, Indiasandy@ee.iitm.ernet.in2 CNRS, I3S & INRIA-Sophia Antipolis, Projet Mascotte,2004 Rt. des Lucioles, BP93, F-06902 Sophia Antipolis, France.Afonso.Ferreira@inria.frAbstract. New technologies and the deployment of mobile and nomadicservices are driving the emergence of complex communicationsnetworks, that have a highly dynamic behavior. This naturally engendersnew route-discovery problems under changing conditions over thesenetworks. Unfortunately, the temporal variations in the topology of dynamicnetworks are hard to be effectively captured in a classical graphmodel. In this paper, we use evolving graphs, which helps capture thedynamic characteristics of such networks, in order to compute multicasttrees with minimum overall transmission time for a class of wirelessmobile dynamic networks. We first show that computing different typesof strongly connected components in evolving digraphs is NP-Complete,and then propose an algorithm to build all rooted directed minimumspanning trees in strongly connected dynamic networks.1 IntroductionInfrastructure-less mobile communication environments, such as mobile ad-hocnetworks and low earth orbiting (LEO) satellite systems, present a paradigmshift from back-boned networks, such as cellular telephony, in that data is transferedfrom node to node via peer-to-peer interactions and not over an underlyingbackbone of routers. Naturally, this engenders new problems regarding optimalrouting of data under various conditions over these dynamic networks [15].In this setting, the generalized case of mobile network routing using shortestpaths or least cost methods are complicated by the arbitrary movement ofthe mobile agents thereby leading to random variations in link costs and connectivity[15]. This variable nature of the topology can be aprehended only bynetwork updates of the link state between moving nodes, thus creating substantialcommunication overhead along the link. This naturally motivates studying⋆ This work was partially supported by the European RTN project ARACNE, theEuropean FET project CRESCCO, and the AS CNRS Dynamo. It was done whilethe first author was visiting the project MASCOTTE, INRIA/CNRS/UNSA.S. Pierre, M. Barbeau, and E. Kranakis (Eds.): ADHOC-NOW 2003, LNCS 2865, pp. 259–270, 2003.c○ Springer-Verlag Berlin Heidelberg 2003
260 S. Bhadra and A. Ferreirathe modeling of such dynamics, and designing algorithms that take it into account[16].Literature related to route discovery issues in dynamic networks started morethan four decades ago, with papers dealing with operations of transport networks(e.g., [5,9,11,10]). Recent work on time-dependent networks deals with flow algorithmsin static networks, with edge traversal times that may depend on thenumber of flow units traversing it at a given moment. If traversal times are discrete,then the approach proposed in [9], namely of expanding the original graphinto T layers representing the time steps (also called space-time approach), maywork for computing several path-related problems (see [13,14] and referencestherein). Unfortunately, this approach leads to non-tractable algorithms, sinceT may be of exponential size.Predictable dynamics. Note, however, that for the case of LEO satellite systems,Unmanned Aerial Vehicles (UAV), and other mobile networks with predestinedtrajectories of the mobile agents, the network dynamics are somewhatdeterministic. Therefore, since the trajectories of the network agents are knownin advance, it is possible to exploit this determinism in optimizing routing strategies[6,17,8].Another setting where the evolution of the network is known was studiedin [7]. The authors used the notion of competitive analysis ([1]) on a dynamicsetting in order to analyze the quality of a protocol and its on-line choices made,forced by the evolution of the network. At the end of the process, the historyof the network is formalized as a sequence of graph topologies on which theapplication can be solved off-line. The merit of the protocol is then the ratio ofthe solution cost found on-line over the optimal off-line cost.Such networks, where the topology dynamics is known or can be predicted beforehand,are henceforth referred to as fixed schedule dynamic networks (FSDN’s)(see Figure 1).(0) (1)(2) (3)Fig. 1. An FSDN represented as an indexed set of networks. The indices correspondto successive time-steps.Evolving graphs. Recently, evolving graphs [2] have been proposed as a formalabstraction for dynamic networks, and can be suited easily to the case of
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Lecture Notes in Computer Science 2
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Samuel Pierre Michel BarbeauEvangel
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PrefaceAd Hoc Networks are wireless
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Program CommitteeG. Alonso, ETHZ, S
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XTable of ContentsResisting Malicio
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2 H. Dubois-Ferrière, M. Grossglau
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10 H. Dubois-Ferrière, M. Grossgla
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SAFAR: An Adaptive Bandwidth-Effici
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14 J. Doshi and P. Kilambiour proto
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22 J. Doshi and P. Kilambiquery its
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26 P. Narayan and V.R. SyrotiukThe
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28 P. Narayan and V.R. Syrotiuk2.2
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30 P. Narayan and V.R. Syrotiukrand
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32 P. Narayan and V.R. Syrotiukenti
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34 P. Narayan and V.R. SyrotiukDSRA
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38 L. Qin and T. Kunzthese two prot
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40 L. Qin and T. Kunzsidered broken
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42 L. Qin and T. KunzP = Prdlog( )
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46 L. Qin and T. KunzFig. 5. Averag
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48 L. Qin and T. Kunz6. J. Broch et
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50 M. Diha and S. Pierrethe propose
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52 M. Diha and S. Pierre3. All proc
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54 M. Diha and S. Pierre3.3 Perform
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56 M. Diha and S. PierreCmip/Cprop0
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58 M. Diha and S. PierreThe tunneli
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Proactive QoS Routing in Ad Hoc Net
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62 Y. Ge, T. Kunz, and L. LamontFig
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Delivering Messagesin Disconnected
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Extending Seamless IP Multicast Edg
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Probabilistic Protocols for Node Di
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106 G. Alonso et al.A node discover
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108 G. Alonso et al.frequency alloc
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110 G. Alonso et al.- D is a diagon
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112 G. Alonso et al.and therefore,
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114 G. Alonso et al.complicated. Fo
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Towards Adaptive WLAN Frequency Man
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A New Framework for Building Secure
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166 H.-P. Bischof, A. Kaminsky, and
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176 G. Calinescuof the UDG 2-hops a
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182 G. CalinescuRyvxFig. 3. The unn
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188 S.O. Krumke et al.desirable in
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190 S.O. Krumke et al.2.2 Bicriteri
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192 S.O. Krumke et al.1. From the g
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194 S.O. Krumke et al.The following
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196 S.O. Krumke et al.1. Let α =6l
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200 S. PatilIDEA uses the concept o
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202 S. PatilAfter flooding the quer
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206 S. PatilAvg. Aggregate Energy C
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Page 305 and 306: 3BerlinHeidelbergNew YorkHong KongL
Page 307 and 308: Series EditorsGerhard Goos, Karlsru
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Page 312 and 313: Table of ContentsSpace-Time Routing
Page 314: Author IndexAlonso, G. 104Bachir, A
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