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Range Assignment for High Connectivity in Wireless Ad Hoc Networks 2370.5 0.5 110.50.50.50.51(a) (b) (c)Fig. 2. Asymmetric topology may have higher connectivity than symmetric topology.(a). The nodes lie in a regular hexagon of side equal to one, and their transmissionranges are given beside the nodes. (b) The asymmetric topology is connected. (c). Thesymmetric topology is disconnected.(resp., k-edge) connected. For k = 1, edge and node connectivity are identicalto each other, and thus are simply referred to as connectivity. For k = 2, 2-nodeconnectivity is simply referred to as biconnectivity, and 2-edge connectivity issimply referred to as edge-biconnectivity. With the same transmission ranges, theasymmetric connectivity is always not lower than the symmetric connectivity. Ifthe transmission ranges are not identical, the asymmetric connectivity may behigher than the symmetric connectivity. Figure 2 shows an example in which theasymmetric topology is connected but the symmetric topology is disconnected.The network consists of nine nodes lying on a regular hexagon of side equal toone, with six nodes at the vertices of the hexagon and the other three nodes atthe midpoints of three alternate sides of the hexagon. Three alternate nodes atthe vertices have transmission range of one, and all others have the transmissionrange of one half. The asymmetric topology is connected, but the symmetrictopology is not. On the other hand, if all nodes have the same transmissionrange, the asymmetric topology and the symmetric topology always have thesame connectivity.The requirement on the network connectivity (either asymmetric or asymmetric)imposes a constraint on the transmission ranges of all nodes. A crucialissue is how to find a range assignment of the smallest total power to meet a specifiedconnectivity requirement. The Min-Power Symmetric (resp., Asymmetric)k-Node Connectivity problem seeks a range assignment of minimum total powersubject to the constraint the induced symmetric (resp. asymmetric) topology isk-connected. Similarly, the Min-Power Symmetric (resp., Asymmetric) k-EdgeConnectivity problem seeks a range assignment of minimum total power subjectto the constraint the induced symmetric (resp., asymmetric) topology is k-edgeconnected. Clearly, the smallest total power for asymmetric k-node (resp., edge)connectivity is no more than the smallest total power for symmetric k-node(resp., edge) connectivity.The study of the Min-Power Asymmetric Connectivity problem was startedby Chen and Huang [5], who gave a 2-approximation algorithm based on mini-
238 G. Calinescu and P.-J. Wanmum spanning tree. Further contributions were made in [19] and [8]. The relatedbroadcast problem was studied in [27], [25], and [6]. The recent survey [9] presentsthe state of the art for these “asymmetric” problems. The Min-Power SymmetricConnectivity problem was proposed in [2] and [4]. Both papers claim thatMin-Power Symmetric Connectivity is NP-Hard, and [4] presents a (1 + ln 2)-approximation algorithm. In the journal submission of [4], this approximationratio is improved to 5/3.The Min-Power Symmetric Biconnectivity problem has been first studied byRamanathan and Rosales-Hain [23], which proposed one reasonable heuristic butwithout a proven approximation ratio. Lloyd et. al [21] studied both Min-PowerSymmetric Biconnectivity and Min-Power Symmetric Edge-Biconnectivity.Among other results, they show that the range assignment based the approximationalgorithm of Khuller and Raghavachari [17], which we refer to as AlgorithmKR, has an approximation ratio of at most 2(2 − 2/n)(2+1/n) forMin-Power Symmetric Biconnectivity, and the range assignment based on theapproximation algorithm of Khuller and Vishkin [18], which we refer to as AlgorithmKV, has an approximation ratio of at most 8(1 − 1/n) for Min-PowerSymmetric Edge-Biconnectivity.In this paper, we present a reduction that establishes the NP-Hardness ofboth Min-Power Symmetric Two-Node Connectivity and Min-Power SymmetricTwo-Edge-Connectivity. The NP-Hardness holds for plane instances, not onlyfor arbitrary graph weights. We show that the range assignment based on theAlgorithm KR has an approximation ratio of at most 4 for both Min-PowerSymmetric Biconnectivity and Min-Power Asymmetric Biconnectivity. Specifically,we prove that the total power of this range assignment is less than fourtimes the smallest power for asymmetric biconnectivity. We also show that therange assignment based on Algorithm KV has an approximation ratio of atmost 2k for both Min-Power Symmetric k-Edge Connectivity and Min-PowerAsymmetric k-Edge Connectivity. Specifically, we prove that the total power ofthis range assignment is less than 2k times the smallest power for asymmetric k-edge connectivity. As both algorithms are graph algorithms, the approximationratios hold also if the nodes are in the three dimensional space, if the possibleranges come from a discrete set of values, if obstacles completely block the communicationin between certain pairs of nodes, and if there is a maximum valueon the ranges.Although the range assignments based Algorithm KR and Algorithm KVhave constant approximation ratios, they have very complicated implementationsand are not practical for wireless ad hoc networks. This motivates us to seek atrade-off between the approximation ratio and the implementation complexity.We propose a very simple range assignment which achieves both symmetric andasymmetric biconnectivity. The total power of this range assignment is less than8 for κ =2,or3.2 · 2 κ for κ > 2 times the smallest power for asymmetricconnectivity for plane instances.The remaining of this paper is organized as follows. Due to space limitations,we omit the reduction proving the NP-hardness of Min-Power Symmetric
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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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16 J. Doshi and P. Kilambipacket th
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18 J. Doshi and P. Kilambibandwidth
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20 J. Doshi and P. Kilambi5 Simulat
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22 J. Doshi and P. Kilambiquery its
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24 J. Doshi and P. Kilambi4. David
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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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36 P. Narayan and V.R. Syrotiuk7. A
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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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64 Y. Ge, T. Kunz, and L. Lamontits
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66 Y. Ge, T. Kunz, and L. Lamonttha
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68 Y. Ge, T. Kunz, and L. Lamontwhi
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70 Y. Ge, T. Kunz, and L. Lamontver
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Delivering Messagesin Disconnected
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Extending Seamless IP Multicast Edg
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86 P.M. Ruiz et al.Section 4 presen
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88 P.M. Ruiz et al.Access NetworkCo
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90 P.M. Ruiz et al.Access NetworkR
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92 P.M. Ruiz et al.MIGAccess Networ
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94 P.M. Ruiz et al.10.90.81-hop-750
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A Uniform Continuum Model for Scali
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98 E.W. Grundke and A.N. Zincir-Hey
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100 E.W. Grundke and A.N. Zincir-He
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102 E.W. Grundke and A.N. Zincir-He
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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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118 F. Gamba, J.-F. Wagen, and D. R
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Analyzing Split Channel Medium Acce
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Preventing Replay Attacks for Secur
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142 J. Zhen and S. SrinivasTraffic
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144 J. Zhen and S. SrinivasFig. 2.
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146 J. Zhen and S. Srinivasbeginnin
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152 M. Just, E. Kranakis, and T. Wa
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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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178 G. CalinescuSection 3 describe
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180 G. CalinescuWhen receiving a pa
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182 G. CalinescuRyvxFig. 3. The unn
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184 G. Calinescu< nodeID, pieceID,
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288 L. Hughes, K. Shumon, and Y. Zh
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3BerlinHeidelbergNew YorkHong KongL
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Series EditorsGerhard Goos, Karlsru
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Organizing CommitteeConference Co-c
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Table of ContentsSpace-Time Routing
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Author IndexAlonso, G. 104Bachir, A
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