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In sensor network localization, since only nodes that are within communication range can measure their relative distances, the graphs formed by connecting each pair of nodes that can measure each other s distance are better modeled as unit disk graphs Such constraints could have the potential of allowing computationally ef cient localization algorithms to be designed The localization problem considered here is to reconstruct the positions of a set of sensors given the distances between any pair of sensors that are within some unitdisk radius r of each other Some of the sensors may be beacons, sensors with known positions, but our impossibility results are not affected much by whether beacons are available To avoid precision issues involving irrational distances, it is assumed that the input to the problem is presented with the distances squared.

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If we make the further assumption that all sensors have integer coordinates, all distances will be integers as well For the main result, we consider a decision version of the localization problem, which we call UNIT-DISK GRAPH RECONSTRUCTION This problem essentially asks if a particular graph with given edge lengths can be physically realized as a unit-disk graph with a given disk radius in two dimensions The input is a graph G where each edge uv of G is labeled with an integer lu v2 , the square of its length, together with an integer r 2 that is the square of the radius of a unit disk.

The output is yes or no, depending on whether there exists a set of points in R2 such that the distance between u and v is lu v whenever uv is an edge in G and exceeds r whenever uv is not an edge in G The main result, is that UNIT-DISK GRAPH RECONSTRUCTION is NP-hard, based on a reduction from CIRCUIT SATISFIABILITY The constructed graph for a circuit with m wires has O(m2 ) vertices and O(m2 ) edges, and the number of sol-.

T1: R1(X), W1(Y) T2: R2(Y), W2(X)

(nothing to do with Token Ring), tracing a route, timestamping the packet as it traverses routers, and security entries. These fields may or may not be in the header (which allows for the variable length header). It was found that most of these features were not used or were better implemented in other protocols, so IPv6 does not implement them as a function of the IP header. Source routing is the ability of the originating station to place route information into the datagram to be interpreted by routers. Router will forward the datagram based on information in the source route fields, and in some cases, it will be blind. The originator indicates the path it wishes to take, and the routers must obey, even if there is a better route. There are two types: loose source route (LSR) and strict source route (SSR). The difference between the two is relatively simple. Routes (IP addresses) are placed in a field of the IP header. The IP addresses indicate the route the datagram would like to take to the destination. Loose source route allows a router to forward the datagram to any router it feels is correct to service the next route indicated in the source route field. A complete list of IP addresses from the source to the destination is probably not in the IP header, but some points in the Internet should be used to forward the datagram. For example, IP multicast uses LSR for tunneling its IP multicast datagrams over the nonmulticast enabled IPv4 Internet. Strict source routing forces a router to forward a datagram to its destination completely based on the routes indicated by the source route field.

Let f be a parameter satisfying 1/f = e (t/e). Then, we have t = e ln f. Therefore, we have the following lemma: Lemma 3.1 An oblivious protocol Election ( 1/n, 1/n, 1/n, . . . ) elects a leader in e ln f time slots with probability at least 1 1/f for any f 1. Note that the value of n must be known to every station in order to perform Election ( 1/n, 1/n, 1/n, . . . ). 10.3.2 Oblivious Leader Election for Scenario 2 The main purpose of this subsection is to discuss a randomized leader election protocol for an n-station RN under the assumption that an upper bound u of the number n of stations is known beforehand. However, the actual value of n itself is not known. Let Di (1 1) be the sequence of probabilities of length i defined as Di = 1 1 1 , 2,..., i 1 2 2 2