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where pi = P (Xi = 1) is not constant but a function of i, chosen carefully so that the limit in (4.10) does not exist. For example, let pi = 0.5 if 2k < log log i 2k + 1, 0 if 2k + 1 < log log i 2k + 2 (4.13)

Consumer authentication is a blanket term to discuss emerging tools intended to validate that the authorized credit card holder is the one actually attempting to make a purchase. Visa calls their consumer authentication service Veri ed by Visa, and MasterCard calls their service MasterCard SecureCode. American Express does not offer any similar service today, but has indicated that they are looking into the program.

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direction If C is hit for the second time and none of the visited nodes is closer to D than U, face exploration is continued as if started at node U, but the radius of circle C is exponentially increased (rC rrC ) In order to avoid a complete face exploration, the algorithm applies an elaborate early fallback technique to return to greedy routing as soon as possible However, it is proved in ref [18] that algorithms will lose their asymptotic optimality when resuming greedy routing as soon as they arrive at the rst node closer to the destination D than the concave node U GOAFR maintains two counters to keep track of the number of nodes closer to and the number of nodes not closer to the destination than the starting node U of the current face traversal.

The square of the rst factor integrates to the Fisher information and hence must be bounded as y . The second factor goes to zero since x ln x 0 as x 0 and gt (y) 0 as y . Hence, the rst term in

If face exploration has visited up to a constant factor s more nodes closer to D, GOAFR will interrupt face traversal, advance to the node seen so far that is closest to the destination D, and the packet will be handled in greedy mode again Thus, GOAFR does not explore the complete face in general, but on the other hand, greedy routing is not resumed at the rst node closer to destination D than concave node U The latter property of GOAFR is nally used in ref [19] in order to prove its asymptotic optimality Finally, from simulation results it turned p out that r0 1:4, r 2, and s 1=100 are good choices for practical purposes A simpli ed example of GOAFR is depicted in Figure 124, where source node S will forward in greedy mode to node M, which has no neighbor closer to destination D.

Thus, the recovery strategy of GOAFR will begin exploration of face F1 in the clockwise direction At node L, face traversal hits the circle centered at node D, and the algorithm switches to exploration in the opposite direction Each routing step updates the number p of nodes closer and the number q of nodes not closer to the destination D When arriving at node P, a certain threshold condition p 1=3q holds ( p 2, q 3, and s 1=3) and the message will be handled in greedy mode again For arbitrary unit-disk graphs (ie, no restrictions regarding minimum node distance and maximum node degree), cost metrics divide into two classes, linearly bounded and superlinear cost functions The rst ones are lowerbound by a linear function, while for the latter there exists no such function A theoretical result from ref.

[19] reveals that for any localized memoryless routing algorithm A, there exists a node con guration where the cost of the path produced by A is unbounded with respect to the path produced by the shortest weighted-path algorithm if a superlinear cost function is considered Thus, discussion of asymptotic optimality is reasonable only if restricted minimum node distance, maximum node degree, or linearly bounded cost metrics are considered Standard cost metrics like hop count or Euclidean distance are linearly bound from below, while energy metrics de ned as a polynomial da , with a 1 and d being the distance between sender and receiver, fall into the class of superlinear functions However, as discussed in Section 125, Routing with Energy-Aware Cost Metrics, from a practical point of view even energy-aware metrics are often considered to be of the form d a c, with c .

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