# Detecting Anomalies Using End-to-End Path Measurements

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4 4 Our sample-based heuristic uses the samples to formulate an integer program over a finite set of candidate paths to monitor. The objective function seeks to minimize the expected communication cost, while additional constraints are used to enforce the correctness criterion. A key advantage of the sample-based heuristic is that it adopts a global approach to path selection - however, it may have high computational complexity because accurately capturing the probability distribution in high-dimensional delay space could require large sample sizes. Let S be a set of sample points in the link delay space that captures the joint probability distribution of link delays. Since link delays are independent, each sample point can be generated by drawing link delay values independently based on their individual probability density functions. Let S in (similarly S out ) be the sample points in S which are inside (outside) of R. Note that since anomalies are rare, almost all the sample points in S are inside R. Thus, S S in.now, let S in be the set of sample points in M. Clearly, if M R, then S in S in (but not vice-versa). Now, using the sample points, Ĉ =(k S in + m ( S in S in ))/ S in gives an estimate of the expected cost C = k Pr[M]+m (1 Pr[M]). Our sample-based heuristic has two phases. In the first phase, the objective is to select paths and thresholds using the sample points so as to minimize the expected cost Ĉ subject to the constraint S in S in. However, the solution obtained does not necessarily satisfy our correctness criterion since S in S in does not guarantee that M R. To ensure that no anomalies are missed, we have a second phase of the heuristic where we use a greedy strategy to add more paths with corresponding thresholds to the set of monitored paths until M R. a) Phase I: Consider an arbitrary path p i. Clearly, the only interesting thresholds γ for p i are the ones for which some sample point in S lies on the hyperplane e j p i d j = γ defined by p i and γ. Now, represent each such interesting threshold by the set of sample points in S which do not satisfy the threshold. Clearly, for each path, there are at most S interesting thresholds. Thus, there are a total of at most n S subsets for the n paths. Each of these subsets corresponds to a unique (path, threshold) pair. Let us call these subsets S= {S 1,S 2,...}. Now, our problem can be stated as the integer program given in Figure 2. In this program, x Si denotes whether a subset S i Sis included in the solution or not. If x Si = 1, then the corresponding (path, threshold) pair is included for monitoring. Thus, S in = S ( x Si =1S i ). x e denotes if a particular sample point e S in is part of any selected subset. Thus, if x e =1, then x e / S in. Now, constraint (1) ensures that for each selected subset S i, all sample points in it have x e =1 and are thus excluded from S in. Constraint (2) ensures that each sample point in S out is included in at least one selected subset. k is a variable which captures the number of subsets which have been selected, or in other words, the number of minimize (k ( S in x e )+m x e )/ S in e S in e S in subject to: x e x Si, S i S,e S in S i (1) x Si 1, e S out (2) S i e x Si = k (3) S i S Fig. 2. x Si {0, 1}, S i S (4) x e {0, 1}, e S in (5) Integer Program for path selection problem. Procedure SAMPLINGPHASE1 (R, S) Input: Normal region R = {(p i,τ i):p i P }, and sample S. Output: Set of paths to monitor and corresponding thresholds. begin 1. M = ; 2. cost = m; 3.for k =1to m { 4. Solve LP to obtain optimal fractional x e,x Si values; 5. while e S out s.t. x Si < 1 S i e 6. Round fractional x Si values using randomized rounding; 7. if x Si =1and e S i then x e =1else x e =0; 8. Ĉ =(k ( S in x e)+m x e)/ S in ; e S in e S in 9. if Ĉ<costthen { 10. M = (path, threshold) pairs for sets S i s.t. x Si =1; 11. cost = Ĉ; 12. } 13. } 14. return M; end Fig. 3. Phase I of the sampling-based heuristic. paths which are to be monitored 3. With this interpretation, it is easy to see that the objective function is exactly equal to Ĉ =(k S in + m ( S in S in ))/ S in. To make the objective function linear, we iterate over different values of k ranging from 1 to m fixing k to a constant value in any particular iteration. This gives an integer linear program for this problem. We also relax the integrality constraints to get a linear program. Figure 3 depicts the procedure for phase 1 of our heuristic. We use an LP solver to obtain a solution to our LP for all values of k ranging from 1 to m, round all the solutions we obtain to get corresponding integral solutions and compare the objective values for all these solutions. The solution having the smallest value of the objective function gives the final set of paths and thresholds that we monitor. In our rounding procedure, for each fractional variable x between 0 and 1, the 3 Note that any two subsets corresponding to the same path share a subsetsuperset relationship. Hence, in the optimal solution, only one of these two subsets will be selected. This ensures that the number of selected subsets, k is equal to the number of monitored paths.

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