Advances in Greedy Algorithms by Witold Bednorz

By Witold Bednorz

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Also, for every edge ( , ) in , we add the edge (wi,wj) to G. The weight of each edge (wi,wj) in G is set to 1 + ε, while the remaining edges have a weight of 1. Finally, we assume that there are monitoring stations at node r1 and nodes ui for each vertex ∈ . For example, consider the graph ( , ) that contains nodes ={ , , } and edges ( ) = { , ), ( , )}. Figure 10 shows the corresponding graph G as well as the routing trees of the nodes r1 and u1. Note that edge (wi,wj) is only contained in the RTs of ui and uj.

In practice, the topology of RTs can be calculated by querying the routing tables of nodes. In our solution, the routing tree of node s may be its SPT but this is not an essential requirement. We associate a positive cost cu,v with sending a message between any pair of nodes u, v ∈ V . For every intermediate node w ∈ Pu,v both cu,w and cv,w are at most cu,v and cu,w + cv,w ≥ cu,v. Typical examples of this cost model are the fixed cost, where all messages have the same cost, and hop count, where the message cost is the number of hops in its route.

Computers and Intractability: A Guide to the Theory of NP-Completeness”. H. Freeman Publishing Company, 1979. 38 Advances in Greedy Algorithms [25] Y. Bejerano abd R. Rastogi, “Robust monitoring of link delays and faults in IP networks”. IEEE/ACM Trans. on Networking, Vol. 14, No. 5, pp 1092-1103, 2006. [26] U. Feige, “A threshold of ln n for approximating set-cover”, Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 314-318, 1996. [27] B. M. Waxman. “Routing of Multipoint Connections”.

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