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The function f is (i) scale invariant, if for every d DS and every > 0 we have f (d) = f ( d); (ii) rich, if Ran(f ) = PART(S); (iii) consistent, if for every d, d DS and PART(S) such that f (d) = and d is a -transformation of d we have f (d ) = , Unfortunately, as we shall see in Theorem 14, established in the work by Kleinburg [8], there is no clustering function that enjoys all three properties The following de nition will be used in the proof of Lemma 2 De nition 11 A dissimilarity d DS is (a, b)-conformant to a clustering if x y implies d(x, y) a and x y implies d(x, y) b A dissimilarity is conformant to a clustering if it is (a, b)-conformant to for some pair of numbers (a, b).

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The obstacles could correspond to events and will have negative values whereas the destination will have a positive value Finally, joining the two algorithms, a distributed motion planning protocol could be computed by the sensor network where different path computation algorithms could be run as distributed protocols on top of the distributed map, updating the path dynamically according to different events in the environment On the other hand, this path stored in the sensor network can be used for the navigation of a robot (which has communication with the nodes of the sensor network) In the same way as the path message is propagated, the process has two phases: rst, the robot has to reach the location where the path starts, and then the robot has to be guided along the path The rst phase contains the following steps: 1.

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Note that if d is a -transformation of d, and d is (a, b)-conformant to , then d is also (a, b)-conformant to De nition 12 Let PART(S) be a partition on S and f be a clustering function on S A pair of positive numbers (a, b) is -forcing with respect to f if for every d DS that is (a, b)-conformant to we have f (d) = Lemma 2 If f is a consistent clustering function on a set S, then for any partition Ran(f ) there exist a, b R>0 such that the pair (a, b) is -forcing..

One (or all) of the sensors aware that they are near to the start of the path send out a message that contains the location of the start of the path 2 That message is forwarded throughout the sensor network 3 The robot sends its location by a message in three different directions (120 dispersal angle) 4 The sensor node that receives the message from the robot knows the start location, the robot location and the direction which the message came from Using this information, the node can send a directional message to the robot that directs it to the start point After the initialization phase which places the robot on the path, the navigation guidance algorithm is used to control the motion direction of the robot This algorithm can be summarized as follows [163]: 1.

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The robot starts by sending out a QueryOnPath message which includes the sender s identi cation and location 2 If it is received by a sensor on the path, this sensor replies with a QueryAck message which includes the path section, some consecutive way points, and an indication of where these way points t into the path..