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We proceed now to develop an on-line local routing algorithm for unit wireless communication networks Observe first that UW(Pn) is not necessarily planar For instance if Pn consists of 12 points contained within a circle of radius 1, UW(Pn) is not planar In order to use the results presented in the previous section, we should be able to extract a planar subnetwork from any UW(Pn) Two requirements must be satisfied by the method we use to extract the planar subgraph to fully ensure its functionality for real-life applications: 1 If a cellular communication network is connected, the resulting planar subgraph must be connected 2.

We must have a local protocol so that each node of the network can decide in a consistent manner which neighbor connections to keep, and ensure that, collectively, and without the need to communicate, the set of edges chosen individually by the nodes of the network form a planar graph The necessity for the second condition follows from our desire to have fully distributed protocols that avoid the use of any kind of centralized protocols The problem of extracting or even deciding if a graph contains a planar connected subgraph is a well-known NP-complete problem [16] Fortunately, UW(Pn) networks always have such a subgraph and, in fact, finding it is relatively straightforward The key to our result arises from the use of Gabriel graphs [1].

(2) '(Linearity) .r E JR, A(x), B(.r) (m x n),cI,c2 E JR:

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Given two points p and q on the plane, let C(p, q) be the circle passing through them such that the line segment joining p to q is a diameter of C(p, q) Given a set of n points Pn = {p1, , pn} on the plane, the Gabriel graph of Pn is the graph whose set of vertices is Pn, in which two points u and v of Pn are adjacent iff the C(p, q) contains no other points of Pn Let G (Pn) be the graph with vertex set Pn such that two vertices p and q are adjacent in G (Pn) iff C(p, q) contains no other points of Pn and p and q are adjacent in UW(Pn), that is G (Pn) is the intersection of the Gabriel graph of Pn with UW(Pn) The following result was proved in [3]: Theorem 12.

1 If UW(Pn) is connected then G (Pn) is also connected The easiest proof of this result proceeds as follows Let p and q be such that they are adjacent in UW(Pn) and there is no path connecting them in G (Pn) Suppose further that their distance is the smallest possible among all such pairs of points in Pn Since p and q are not connected in G (Pn), C(p, q) contains at least a third point r Pn Observe that the distances from r to p and q are smaller than the distance from p to q, and thus there is a path P in G (Pn) connecting r to p and a path P connecting r to q The concatenation of these paths produces a path from p to q in G (Pn) Our result follows.

It is obvious that each node p in UW(Pn) can decide locally which of its neighbors in UW(Pn) should be its neighbors in G (Pn) It simply collects the locations from all its neighbors (ie, the elements of Pn at distance at most 1 from p, and tests for each q of them if the circle C(p, q) is empty This can be done using standard algorithms in computational geometry in O(k ln k), where k is the number of neighbors of p in UW(Pn) [21] We now have the general tools to obtain an on-line local routing algorithm on unit.

How much should you worry about speed Worry may be too strong a word. Even though bandwidth is getting cheaper and more available, don t make the mistake of believing everyone accessing your page is on a T1 line (with fast access). You still should be considerate about the amount of data you expect them to download to see your page. Even though this book talks about visitors to your site, in fact, your site is delivered to the desktops of the people who request it in their browsers. When you design your pages with huge graphics background images that fill the screen, large image maps, graphical representations of text rather than text you are telling your visitors you don t respect their time. Is this the message you want to send

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