Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...

184 G. Calinescu< nodeID, pieceID, coordinates 1 , coordinates 2 , counter >. Any node y receivingsuch a message evaluates if it has neighbors in the rigid piece, and if yes, asabove, it can compute two sets S 1 and S 2 of nodes in the rigid piece which couldbe the common neighbors with v - assuming v has coordinates coordinates 1 orcoordinates 2 with respect to the rigid piece. At least one of S 1 and S 2 is a setof colinear points, and if both are, they coincide. That set of colinear points isthe set of neighbors common to y and v in the rigid piece.All cases are taken care of and we conclude:Theorem 3. There is a distributed algorithm which, under the assumption thatevery node can estimate the distance to every adjacent node, computes for everynode v the set of its 2-hop neighbors N 2 (v) and the links in between N 1 (v) andN 2 (v) with a total of O(n) messages each of size O(log n) bits.5 Updating the 2-Hop NeighborhoodsIn this section we discuss the message complexity of updating the 2-hop neighborhoodsdue to changes in network topology. We do not address updating thevirtual backbone as this was done in [2]. The proposed protocol is straightforwardand does not use the virtual backbone. We assume geographical knowledgeis available in this section.Before leaving the network, a node u uses its knowledge to let its 2-hopneighborhs know the fact it is leaving as described below. First the node ucomputes a maximal independent set (MIS) in the graph induced by its 2-hopneighborhs. Then u computes at most one “connector” node for each MIS node.As before, MIS is a dominating set, and using an area argument, has constantsize. Node u prepares an < ID, position, leaving, relay > message, with its ownID and position, the fact that it is leaving the network, and the full list of relaynodes. Each node, after receiving such a message, make a note that u is leavingand updates its 2-hop neighborhood accordingly, and, if it finds itself in the listof relay nodes, rebroadcast the message once.When a node v joins the network, it will broadcasts its ID and position.Every existing node which receives this message will rebroadcast the ID andposition of v. Every node y receiving such a message, will update its stored 2-hopneighborhood to reflect the presence of v. Ify is adjacent to v, it will broadcastits ID and position. If y is a 2-hop neighbor of v, it selects a common neighborx and asks x to relay to v the position and ID of y. The total bit complexity ofmessage is O(q log n), where q is the size of the 2-hop neighborhood of v, and itcannot be improved by more than a constant factor since v must find out theIDs of the nodes in its 2-hop neighborhood.6 ConclusionsThe virtual backbone of Alzoubi, Wan, and Frieder [2,21] can be constructedwithout any geographical knowledge: their algorithm “operates” directly on the