# Robust Monitoring of Link Delays and Faults in IP Networks

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4 x () x (3) () (3) () (3) x x x3 x3 () x (3) () (3) () (3) x x x x3 x3 () x (3) () (3) () (3) x x x x3 x3 w w w3 u u u3 () x () x () x3 w w w3 Anchor clique r r () r r (3) () x () () () x x3 w w w3 u u u3 r () r (3) r () x () () () x x3 w w w3 u u u3 r () r (3) r () u u u3 s s (a) The graph induced by the nodes u i, w i and s j. s s (b) The graph G R(I) (V,E). s s (a) The RT of node r () () x (3) () (3) () (3) x x x x3 x3 s s (b) The RT of node x () Fig.. The graph G R(I) (V,E) for the given instance of the SC problem. x () () () x x3 w w w3 r () r (3) r () A. An Efficient Station Selection Algorithm We are interested in solving the problem of covering all graph edges with a small number of RTs. Definition (The Link Monitoring Problem - LM): Given a graph G(V,E) andart,t v, for every node v V, find the smallest subset S V such that v S T v = E. Due to space constraints, we only consider the unweighted version of the LM problem. However, our results can be easily extended to the weighted version of the problem, where each node has an associated cost, and we seek a set S V that minimizes the total cost of the monitoring stations in S. The latter can be used, for instance, to find a station selection when monitoring stations can be installed only at a restricted set of nodes. For restricting the station selection, nodes that cannot support monitoring stations are assigned infinite cost. The LM problem is similar to the set cover (SC) problem, which is a well-known NP-hard problem. In an instance I(Z, Q) of the SC problem, Z = {z,z,,z m } is a universe of m elements and Q = {Q,Q,,Q n } is a collection of n subsets of Z (assume that Q Q Q = Z). The SC problem seeks to find the smallest collection of subsets S Qsuch that their union contains all the elements in Z, i.e., Q S Q = Z. A.. The Hardness of the LM Problem Theorem : The LM problem is NP-hard, even when the RT of each node is restricted to be its shortest path tree. Proof: We show that the LM problem is NP-hard by presenting a polynomial reduction from the set cover problem to the LM problem. Consider an instance I(Z, Q) of the SC problem. Our reduction R(I) constructs the graph G R(I) (V,E) where the RT of each node v V is also its shortest path tree. For determining these RTs, each edge is associated with a weight 4, and the graph contains the following nodes and edges. For each element z i Z, it contains two connected nodes u i and w i. For each set Q j Q, we add a node, labeled by s j, and the edges (s j,u i ) for each element z i Q j. In addition, we use an auxiliary structure, termed an anchor clique x, which is a clique with three nodes, labeled by x (), x () and x (3), and only node x () has additional incident edges. For each element z i Z, the graph G R(I) contains one anchor clique 4 These weights do not represent communication costs. u u u3 s s (c) The RT of node Fig.. The RTs of nodes r (), x () and s. x i whose attachment point, x () i, is connected to the nodes u i and w i. The weights of all the edges described above is. Finally, the graph G R(I) contains an additional anchor clique r that is connected to the remaining nodes and anchor cliques of the graph, and the weights of these edges is +ɛ. An example of such a graph is depicted in Figure for an instance of the SC problem with 3 elements {z,z,z 3 } and two sets Q = {z,z } and Q = {z,z 3 }. Moreover, we assume that the RT, T v, of every node v V is its shortest path tree. We claim that there is a solution of size k to the given SC problem if and only if there is a solution of size k+m+ to the LM instance defined by the graph G R(I) (V,E). Webeginby showing that if there is a solution to the SC problem of size k then there exists a set S of at most k+m+ stations that covers all the edges in G R(I). Let the solution of the SC problem consist of the sets Q i,...,q ik. The set S of monitoring stations contains the nodes r (), x () i (for each element z i Z) and s i,...,s ik. We show that the set S contains k + m + nodes that cover all the graph edges. The tree T r () covers edges (r (),r () ), (r (),r (3) ), all edges (u i,r () ), (w i,r () ), (x () i,r () ), (x () i,x () i ), (x () i,x (3) i ), for each element z i, and the edges (s j,r () ) for every set Q j Q. An example of such a T r () is depicted in Figure -(a). Similarly, for every z i Z, thertt () x covers edges (x () i,x (3) i ), (x () i,x () i ), i (x () i,u i ) and (x () i,w i ). T () x also covers all edges (s j,u i ) i for every set Q j that contains element z i, and edges (r (),r () ) and (r (),r (3) ). An example of the RT T () x is depicted in Figure -(b). Thus, the only remaining uncovered edges are (u i,w i ), for each element z i. Since Q ij, j =,...,k,isa solution to the SC problem, these edges are covered by the RTs T sij, as depicted in Figure -(c). Thus, S is a set of at most k + m + stations that covers all the edges in the graph G R(I). s

5 S, C Z While C do Q arg max Qv Q S Q v C S S {Q} C C Q End While Return S u u3 u 3 u u u 3 33 u u3 u 3 u u 3 33 u u u 3 u u u 3 w +e w +e w 3 +e w +e w w 3 r r u Fig. 3. A formal description of the Greedy Algorithm. Fig. 4. The RTs of nodes r and u 3. Next, we show that if there is a set of at most k+m+ stations that covers all the graph edges then there is a solution for the SC problem of size at most k. Note that there needs to be a monitoring station in each anchor clique and suppose w.l.o.g for each element z i. None of these m + stations covers edges (u i,w i ) for elements z i Z. The other k monitoring stations are placed in the nodes u i,w i and s j. In order to cover edge (u i,w i ), there needs to be a station at one of the nodes u i, w i or s j for some set Q j containing element z i. Also, observe that the RTs of u i and w i cover only edge (u i,w i ) for element z i and no other element edges. Similarly, the RT of s j covers only edges (u i,w i ) for elements z i contained in set Q j.lets be a collection of sets defined as follows. For every monitoring station at any node s j add the set Q j Q to S, and for every monitoring station at any node u i or w i we add to S an arbitrary set Q j Qsuch that z i Q j. Since the set of monitoring stations cover all the element edges, the collection S covers all the elements of Z, and is a solution to the SC that the selected stations are r () and x () i problem of size at most k. The above reduction R(I) from the SC problem can be extended to derive a lower bound for the best approximation ratio achievable by any algorithm (see [7] for proof). Theorem : The lower bound of any approximation algorithm for the LM problem is ln( V ). A.. A Greedy Algorithm for the LM Problem We now present an efficient algorithm for solving the LM problem. The algorithm maps the given instance of the LM problem, involving graph G(V,E), to an instance of the SC problem, and then uses a greedy heuristic for solving the SC instance. In the mapping, the set of edges E defines the universe of elements Z, and the collection of sets Q includes the subsets Q v = {e e T v } for every node v V.The greedy heuristic, depicted in Figure 3, is an iterative algorithm that selects, in each iteration, the set Q with the maximum number of uncovered elements. According to [3], the greedy algorithm is a (ln( ) + )-approximation algorithm for the SC problem, where is the size of the biggest subset. Thus, since for the LM problem, every subset includes all the edges of the corresponding RT and its size is exactly V, we have the following result. Theorem 3: The greedy algorithm computes a (ln( V )+)- approximation for the LM problem. Note that the worst-case time complexity of the greedy algorithm can be shown to be O( V 3 ). B. An Efficient Probe Assignment Algorithm Once we have selected a set S of monitoring stations, we need to compute a probe assignment A for measuring the latency of network links. Recall from Section III that a feasible probe assignment is a set of probe messages {m(s, u) s S, u V }, where each m(s, u) represents a probe message that is sent from station s to node u. Further, for every edge e =(u, v) E, there is a station s S such that e T s and A contains the probes 5 m(s, u) and m(s, v). The cost of a probe assignment A is COST A = m(s,u) A c s,u and the optimal probe assignment is the one with the minimum cost. B.. The Hardness of the Probe Assignment Problem In the following, we show that computing the optimal assignment is NP-hard. Theorem 4: Given a set of stations S, the problem of computing the optimal probe assignment is NP-hard. Proof: We show a reduction from the vertex cover problem, which is defined as follows: Given k Z + and a graph Ĝ = ( ˆV,Ê), find a subset V ˆV containing at most k vertices such that each edge in Ê is incident on a node in V.For a graph Ĝ, wedefine an instance of the probe assignment problem, and we show that there is a vertex cover of size at most k for Ĝ if and only if there exists a feasible probe assignment A with cost at most COST A =5 ˆV + Ê + k. We assume that the cost of any probe c s,u =, thus, COST A is the number of probes in A (the proof for c s,u = h s,u is similar). For a graph Ĝ, we construct the network graph G(V,E) and set of stations S for the probe assignment problem as follows. In addition to a root node r, graph G contains for each node ˆv i in Ĝ four nodes, w i, u i, u i and u i3, connected by the edges (w i,r), (w i,u i ), (u i,u i ), (u i,u i3 ) and (u i,u i3 ). Also, for every edge (ˆv i, ˆv j ) in Ĝ, we add the edge (w i,w j ) to G. Figure 4 shows an example of the constructed G for the graph Ĝ containing nodes ˆv, ˆv and ˆv 3, and edges (ˆv, ˆv ) and (ˆv, ˆv 3 ). Finally, we assign a weight +ɛ to each edge (w i,w j ) in G, while the remaining edges are assigned a weight of. These weights are used only for determining the RTs that in our reduction are the shortest path trees, and we assume that there are monitoring stations at node r and nodes u i3 for each vertex ˆv i Ĝ. Figure 4 illustrates the RTs of nodes r and 5 If s is one of the edge endpoints, say node v, then the probe m(s, v) is omitted from A.

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