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Figure 10.28. Re ection spectrum (1), uorescent emission spectrum (2) by excitation at 532 nm, and laser emission spectrum (3) by excitation with a second harmonic light of Nd-YAG laser of a Ch mixture of E44, S811, m-azo-8, and DCM (73.1/22.2/4.3/0.4 in wt%). Source: Kurihara et al., 2006.

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done in the worst case to find the least cost paths from the source to all destinations In the first iteration, we need to search through all n nodes to determine the node, w, not in N that has the minimum cost In the second iteration, we need to check n-1 nodes to determine the minimum cost; in the third iteration n-2 nodes and so on Overall, the total number of nodes we need to search through over all the iterations is n*(n+1)/2, and thus we say that the above implementation of the link state algorithm has worst case complexity of order n squared: O(n2) (A more sophisticated implementation of this algorithm, using a data structure known as a heap, can find the minimum in line 9 in logarithmic rather than linear time, thus reducing the complexity) Before completing our discussion of the LS algorithm, let us consider a pathology that can arise with the use of link state routing Figure 422 shows a simple network topology where link costs are equal to the load carried on the link, eg, reflecting the delay that would be experienced In this example, link costs are not symmetric, ie, c(A,B) equals c(B,A) only if the load carried on both directions on the AB link is the same In this example, node D originates a unit of traffic destined for A, node B also originates a unit of traffic destined for A, and node C injects an amount of traffic equal to e, also destined for A The initial routing is shown in Figure 42-2a, with the link costs corresponding to the amount of traffic carried





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Figure 42-2: Oscillations with Link State routing When the LS algorithm is next run, node C determines (based on the link costs shown in Figure 42-2a) that the clockwise path to A has a cost of 1, while the counterclockwise path to A (which it had been using) has a cost of 1+e Hence C's least cost path to A is now clockwise Similarly, B determines that its new least cost path to A is also clockwise, resulting in the routing and resulting path costs shown in Figure 42-2b When the LS algorithm is run next, nodes B, C and D all detect that a zero cost path to A in the counterclockwise direction and all route their traffic to the counterclockwise routes The next time the LS algorithm is run, B, C, and D all then route their traffic to the clockwise routes What can be done to prevent such oscillations in the LS algorithm One solution would be to mandate that link costs not depend on the amount of traffic carried -- an unacceptable solution since one goal of routing is to avoid highly congested (eg, high delay) links Another solution is to insure that all routers do not run the LS algorithm at the same time This seems a more reasonable solution, since we would hope that even if routers run the LS algorithm with the same periodicity, the execution instants of the algorithm would not be the same at each node Interestingly, researchers have recently noted that routers in the Internet can self-synchronize among themselves [Floyd 1994], ie, even though they initially execute the algorithm with the same period but at different instants of time, the algorithm execution instants can eventually become, and remain, synchronized at the routers One way to avoid such self-synchronization is to purposefully introduce randomization into the period between execution instants of the algorithm at each node Having now studied the link state algorithm, let's next consider the other major routing algorithm that is used in practice today - the distance vector routing algorithm.

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While the LS algorithm is an algorithm using global information, the distance vector (DV) algorithm is iterative, asynchronous, and distributed It is distributed in that each node receives some information from one or more of its directly attached neighbors, performs a

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