# Minimizing Probing Cost and Achieving Identifiability in Probe Based Network Link Monitoring

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6 6 find it with linear replacement Since solutions to i are linear combinations of variables b j, S i q is in the following form i = n d qj b j (7) j= The above equation together with base solution S i and linear epressions in L form a linear system as shown in Eq (8) The first equation is base solution S i Since it is computed from linear system A R = b R, the value of i is a linear combination of b j s, where j r The second equation is the same as Eq (7) For the other n r equations, each of them is a linear epression in set L, which is shown in Eq (6) r j= d jb j i = n j= d qjb j i = r j= c jb j + b r+ = r j= c n r,jb j + b n = In this linear system, b,,b n and i are unknown variables Hence, the coefficient matri is shown in Eq (9) b b r b r+ b n i d d r d q d qr d q,r+ d qn c c r (9) c n r, c n r,r we subtract d q,r+k times of the (2 +k)th row from the second row, where k =,,n r For eample, the second row is subtracted by d q,r+ times of the third row, d q,r+2 times of the fourth row, and so on The matri turns into the form as shown in Eq () Compared with the matri in Eq (9), the difference is only at the second row For j =,,r, variable d qj is shown in Eq () b b r b r+ b n i d d r d q d qr c c r () c n r, c n r,r (8) n r d qj = d qj c kj d q,r+k () k= Since row vectors in matri A R are linear independent, linear system A R = b R has only one solution to solvable variable i, ie, base solution S i Therefore, the first two row vectors in Eq () must be the same Otherwise, there are two different linear combinations of b,,b n to epress i, ie, A R = b R has two different solutions to i It shows that the second row vector of Eq (9) can be linearly epressed by other row vectors This contradicts with the assumption that Sq i cannot be linearly epressed by S i and linear epressions in set L Therefore, no such missed solution Sq i eists, which indicates that SolutionCalculation can find out all solutions to each solvable variable A solution to variable i is a linear combinations of b,,b n Since b j corresponds to probing path p j,wecan easily obtain the corresponding solution to identifiable link e i Based on Theorem, our algorithm can find all solutions to each identifiable link The computational compleity of this algorithm is determined by the number of linear replacements, which may not be a linear function of the size of dependency matri For a large-scale network, an identifiable link may have a large number of solutions, and thus the algorithm needs to run quite long Actually, the algorithm for selecting probing paths does not need to use all solutions For an identifiable link, we can choose some of its solutions for probing path selection, and hence the algorithm only needs to calculate some solutions rather than all solutions More details will be introduced in Section 3 To make it more clear, we use an eample to show how to calculate all solutions to identifiable link e in Fig Initially, we have base solution S : 2 b + 2 b 2 2 b 4 Set L contains two linear equations l : b 2 b 3 b 4 + b = and l 2 : b b 3 b 4 + b 6 = The base solution has two common variables b 2 and b 4 with linear epression l Replacing b 2 in S with b 3 +b 4 b results in 2 b + 2 b 3 2 b, which is a new solution to We name it as S2 Similarly, replacing b 4 in S with b 2 b 3 +b produces 2 b + 2 b 3 2 b Then we use l 2 to replace b and b 4 in S Both of them result in 2 b 2+ 2 b 3 2 b 6, which is named as S3 Until now, we have already applied all possible linear replacements to S Net, we apply linear replacements to S2 and S3 The generated new solutions are put into the queue, and linear replacements continue until the queue is empty When applying l to solution b 2 b 4 2 b + 2 b 6 to replace b 4, we get linear epression b 2 b 2+ 2 b 3 b + 2 b 6 Since the corresponding row vectors {a, a 2, a 3, a, a 6 } are not linearly independent, we do not take it as a solution to i The algorithm stops after finding out si solutions listed in the first column of Table 2 Accordingly, identifiable link e has si solutions listed in the second column of Table 2 TABLE 2 Solutions to variable in linear system and corresponding solutions to link e Solution to Solution to e 2 b + 2 b 2 2 b 4 {p,p 2,p 4 } 2 b + 2 b 3 2 b {p,p 3,p } 2 b b 3 2 b 6 {p 2,p 3,p 6 } b b 4 2 b 2 b 6 {p 3,p 4,p,p 6 } b 2 b 4 2 b + 2 b 6 {p,p 4,p,p 6 } b 2 2 b b 2 b 6 {p 2,p 4,p,p 6 } 34 Etended Bipartite Model As in the literature [6], the relationship between probing paths and target links is usually modeled as a bipartite graph G B =

12 (a) AS29 2% probers 4% probers 6% probers (b) AS7 2% probers 4% probers 6% probers (c) AS294 2% probers 4% probers 6% probers (d) AS332 2% probers 4% probers 6% probers (e) AS336 2% probers 4% probers 6% probers (f) AS349 2% probers 4% probers 6% probers (g) AS36 2% probers 4% probers 6% probers (h) AS4323 2% probers 4% probers 6% probers (i) AS78 2% probers 4% probers 6% probers Fig 8 The cost per identifiable target link of the path selection algorithm with α = The lower bound is TABLE 4 Comparison of overall probing cost of our algorithm and the SelectPath algorithm (percentage of probers: 2%) α Topology SelectPath AS AS AS AS AS AS AS AS AS TABLE Comparison of overall probing cost of our algorithm and the SelectPath algorithm (percentage of probers: 4%) α Topology SelectPath AS AS AS AS AS AS AS AS AS The SelectPath algorithm selects probing paths corresponding to the maimal independent set of row vectors of the dependency matri Although vectors in this set are linearly independent, it does not necessarily mean there is no redundancy Fig 2(b) is a simple eample to show this fact There are three identifiable links e, e 2, and e 3 The maimal

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