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Computing 2-Hop Neighborhoods in Ad Hoc Wireless Networks 179Fig. 2. An illustration of the virtual backbone of Alzoubi, Wan, and Frieder. The solidround nodes are the MIS node, which form a dominating set. Two virtually-adjacentMIS nodes are connected by paths of length at most three through connector nodes -the empty tiny circles in the figure. Nodes not in the virtual backbone are small solidsquares in the figureto an MIS node adjacent to the destination of the path. This fact was noticed byAlzoubi [1], and by Wang and Li [22], which also planarize the virtual backbonewhile keeping all its attractive properties.3 Geographic Position AvailableIn this section we describe the distributed algorithm which allows every nodeto construct the list of its 2-hop neighbors, assuming every node knows its geographicalposition. With this information, every node can also easily computethe links between its 1-hop and 2-hop neighbors. Our algorithm is described inthe simplest version, and we do not try to optimize the constant hidden in theO notation.We start from the moment the virtual backbone is already constructed, andevery node knows the ID and the position of its neighbors. The idea of thealgorithm is for every node to efficiently announce its ID and position to asubset of nodes which includes its 2-hop neighbors.The responsibility for announcing the ID and position of a node v is taken bythe MIS nodes adjacent to v. Each such MIS node assembles a packet containing:< ID, position, counter >, with the ID and position of v, and a counter variablebeing set to 2. The MIS node then broadcasts the packet.A connector node is used to establish a link in between several pairs ofvirtually-adjacent MIS nodes, and will not retransmit packets which do nottravel in between these pairs of MIS nodes. The connector node will rebroadcastpackets with nonzero counter originated by one of the nodes in a pair of virtuallyadjacentMIS nodes, thus making sure the packet advances towards the otherMIS node in the pair. Recall that the path in between a pair of virtually-adjacentMIS nodes has one or two connector nodes.
180 G. CalinescuWhen receiving a packet of type < ID, position, counter >, an MIS nodechecks whether this is the first message with this ID, and if yes decreases thecounter variable and rebroadcasts the packet.A node listens to the packets broadcasted by all the adjacent MIS nodes (hereit is convenient to assume a MIS is adjacent to itself), and, using its internallist of 1-hop neighbors, checks if the node announced in the packet is a 2-hopneighbor or not - thus constructing the list of 2-hop neighbors.Theorem 1. When finished, the algorithm described above correctly computesthe 2-hop neighborhood for every node in the network, and uses O(n) messagesof size O(log n) each.Proof. The fact that the virtual backbone is a bounded-degree hop-spanner essentiallyimplies the correctness of the algorithm. The precise argument is asfollows. Assume nodes v and u share a neighbor x, and let ¯v, ū, and ¯x be nodesin MIS which are adjacent to v, u, and x. Then ¯v creates a packet with the IDand position of v, and with its counter set to 2. As ¯v and ¯x are virtually-adjacent,¯x will receive the packet and retransmit it with counter set to 1. As ¯x and ū arevirtually-adjacent, ū will also broadcast the packet, and therefore u finds outthe ID and position of v.Regarding the number of messages, we count the packets announcing the IDand position of x. Such packets are being sent by S 1 , the MIS nodes adjacentto x, and we recall that |S 1 |≤5. They are also sent by S 2 , the MIS nodesvirtually-adjacent to S 1 ,byS 3 , the MIS nodes virtually-adjacent to S 2 , andby the connector nodes in between pairs of virtually-adjacent MIS nodes insideS 1 ∪ S 2 , and by the connector nodes in between virtually-adjacent MIS nodes ofS 2 and S 3 . Thus the total number of nodes retransmitting packets announcing IDand position of x is O(∆ 2 ). As ∆, the maximum degree of the virtual backboneis constant, the total number of messages is O(n).⊓⊔We remark that with the counter of a packet being initially set to k (anddecreased by one whenever a MIS node retransmits), the same argument asabove implies that with O(∆ k ) messages every node can compute its k-hopneighborhoods.4 Pairwise Distances AvailableIn this section we assume that neighboring nodes can compute their pairwisedistance, but are not aware of their precise geographical position.Our approach is based on the virtual backbone used before and rigid pieces,which we define as subgraphs containing one MIS node and a subset of its neighborssuch that a system of coordinates can be locally established and in whichthe position of every node of the rigid piece is completely defined. A theory ofgeometric rigidity is well established [23]. We need only simple properties whichare easily proved below.First we describe the distributed algorithm for computing the rigid pieces.Before the actual construction, every node announces all the MIS nodes to whichit is adjacent, and records the information transmitted by all its neighbors.
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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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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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56 M. Diha and S. PierreCmip/Cprop0
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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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118 F. Gamba, J.-F. Wagen, and D. R
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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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