Minimizing Probing Cost and Achieving Identifiability in Network Link Monitoring

Transcription

1 Minimizing Proing Cost and Achieving Identifiaility in Network Link Monitoring Qiang Zheng and Guohong Cao Department of Computer Science and Engineering The Pennsylvania State University {quz3, Astract Continuously monitoring the link performance is important to network diagnosis Recently, active proes sent etween end systems are widely used to monitor the link performance In this paper, we address the prolem of minimizing the proing cost and achieving identifiaility in link monitoring Given a set of links to monitor, our ojective is to select as few proing paths as possile to cover all of them, and the selected proing paths can uniquely identify all identifiale links eing monitored We propose an algorithm ased on the linear system model to find out all sets of proing paths that can uniquely identify an identifiale link We etend the ipartite model to reflect the relation etween a set of proing paths and the link that can e uniquely identified Through the etended ipartite model, our optimization prolem is transformed into the classic set cover prolem, which is NP-hard Therefore, we propose a heuristic ased algorithm to greedily select the proing paths Our method eliminates two types of redundant proing paths, ie, those that can e replaced y others and those that cannot e used to achieving identifiaility Simulations ased on real network topologies show that our approach can achieve identifiaility with very low proing cost Compared with prior work, our method is more general and has etter performance Introduction With the rapid growth of the Internet, efficiently monitoring the performance and roustness of the network ecomes more and more important The service providers need to keep tracking of their networks to ensure that their services satisfy the commitments specified in the service level agreements (SLAs) Additionally, applications sensitive to network performance such as online trading, Voice over IP and IPTV, need to know the network status such as link delay, jitter, and loss rate These demands motivate recent research on network monitoring and performance inference Due to various advantages, tomography-ased end-to-end active proe (end-to-end proe for short) has received much attention and has een widely used in network monitoring, such as [] [7] Compared with reply-ased proe such as ping and traceroute, end-to-end proe does not have the prolem of eing ignored y the intermediate routers [8] This work was supported in part y DoD/DTRA under grant DTRA-3- D-- and eing locked y the firewall [9] Furthermore, different from Simple Network Management Protocol (SNMP)-ased polling [], end-to-end proe ased approaches do not need to run agents on routers One of the major considerations in designing a proing scheme is the proing cost, which can e the numer of selected proing paths [], [] [4], the numer of end systems used to send and receive proing packets [7], [], [], and the cost specified y the network component [6] Lower proing cost means less negative effects on the normal data transmission and etter scalaility Therefore, many eisting works [], [6], [7], [] [3] focus on minimizing the proing cost for network monitoring This is usually achieved y carefully selecting the proing paths However, to monitor the performance of a link such as delay and loss rate, selecting proing paths to cover the link may not e sufficient An end-to-end proe measures the performance of the whole proing path, which may consist of multiple links In order to uniquely infer the performance of each link, multiple coordinated proes are needed Unfortunately, in a general network the performance of some links may not e ale to e uniquely inferred from end-to-end proes [] There are two kinds of network links If the performance of a link can e uniquely inferred from a set of end-to-end proes, it is called uniquely identifiale or identifiale for short; otherwise, it is called unidentifiale If the proing paths are not properly selected, the performance of a link cannot e inferred even if it is an identifiale link For eample, as shown in Fig, links e, e, and e 3 are identifiale, ut links e 4 and e are unidentifiale ecause every proing path traversing e 4 also traverses e Selecting the path p (s, s ), p (s, s 3 ), and p 4 (s, s 3 ) can uniquely infer the performance of links e, e, and e 3 However, only choosing p (s, s ) and p (s, s 3 ) cannot uniquely infer the performance of any link In this paper, we aim at minimizing the proing cost and achieving identifiaility in link monitoring More specifically, given a set of links called target links to monitor, our ojective is to select as few proing paths as possile that can cover all the target links and uniquely infer the performance of the identifiale target links under monitoring There are three challenges to achieve our ojective Target links can e links at critical topological location, and their performance usually affects a large area of the network [6]

2 s e s s 3 e s e 4 4 e e 3 Figure An eample of network link monitoring with si proing paths p (s, s ), p (s, s 3 ), p 3 (s, s 4 ), p 4 (s, s 3 ), p (s, s 4 ), and p 6 (s 3, s 4 ), where p i (s j, s k ) denotes the proing path p i etween the end system s j and s k First, a target link may e unidentifiale Using multiple proing paths to monitor an unidentifiale link only wastes andwidth Therefore, it is necessary to differentiate an identifiale link from an unidentifiale one For an unidentifiale target link, we only need to select a proing path traversing it Second, different proing paths have different contriutions to achieving identifiaility, and we need to determine the most useful proing paths Third, a proing path may e ale to e replaced y other proing paths, and we should avoid selecting redundant proing paths The asic idea of our approach is to select proing paths that are the most useful for covering the target links and achieving identifiaility We adopt a linear system to model the relation etween proing paths and links We design a method ased on the matri decomposition and linear replacement to find out irreducile sets of proing paths that can uniquely determine each identifiale target link Moreover, we etend the ipartite model which is commonly used for modeling the relation etween a single proing path and link to reflect the relation etween a set of proing paths and a link Through the ipartite model, our optimization prolem is transformed into the classic set cover prolem, which is NP-hard Therefore, we propose a heuristic ased algorithm to greedily select the proing paths The rest of this paper is organized as follows We present the linear system model and the prolem description in Section Our path selection approach is introduced in Section 3, which includes the algorithm for calculating all solutions to an identifiale link, the etended ipartite model, and the heuristic ased algorithm for selecting proing paths Section 4 presents the performance evaluation of our path selection approach, and Section reviews related work Finally, Section 6 concludes the paper Preliminaries In this section, we first introduce the network model and define our prolem, and then descrie the linear system model used in our approach Prolem Description Similar to prior work on network monitoring [], [], [7] [], we model the network as a connected undirected graph G(V, E) for simplicity, where V is the set of nodes (ie routers) and E = {e i i m} is the set of edges (ie communication links etween routers) Note that the proposed technique also works for asymmetric links In the rest of the paper, we use edge and link interchangealy There are many end systems connecting to the network, and each of them can send and receive proing packets The proing packet from one end system to another traverses the routing path etween them We call such an end-to-end path a proing path If a proing path traverses a link, the path can cover the link In the network, there are n proing paths P = {p i i n} which can e used for monitoring a set of m t target links E t E For simplicity, we assume E t = {e i i m t }, which equals to relaeling the links in G With the router-level topology G, we know which links can e covered y a proing path Generally speaking, network monitoring y means of endto-end proe includes two steps The first step is to select a set of proing paths Then, end systems on the chosen proing paths issue proing packets and send the result to the central Network Operations Center (NOC) In this study, we focus on the first step; ie, selecting proing paths We define the prolem of minimizing the proing cost and achieving identifiaility as follows, where the cost is defined as the numer of proing paths selected for monitoring Definition (Prolem definition): Given a network G, a set of proing paths P, and a set of target links E t, the ojective is to select as few proing paths as possile from P, such that all links in E t are covered and the performance of each identifiale link in E t can e uniquely inferred Linear System Model For the link performance satisfying the additive metric, eg delay, the relation etween the proing paths and links can e naturally modeled as a linear system LS = {ls i i n}, in which ls i is the ith linear equation m j= a ij j = i The variale j denotes the performance of the link e j and the variale i is the performance of the proing path p i The value of a ij is if the proing path p i traverses the link e j ; otherwise it is The linear equation ls i means that the performance of p i is an addition of the performance of all links on p i The linear system LS can also e written as Eq (), where A = (a,, a n ) T is the coefficient matri which is also called the dependency matri [] A = () Take the eample in Fig for instance, the linear system has si equations and five variales,, as shown in Fig (a) The dependency matri is shown in Fig () We use delay as an eample to eplain the meaning of the linear

3 3 3 3 (a) The linear system model LS A () The dependency matri Figure The linear system model and the dependency matri of the eample in Fig equation For the proing path p etween the end system s and s, the overall delay of the path is the sum of the delay on link e and e Correspondingly, in the first equation of the linear system in Fig (a), the variale is equal to the sum of and As mentioned in Section, a link can e either identifiale or unidentifiale, and a proing path may e replaced y other proing paths A link e i is identifiale if and only if the variale i in the linear system is solvale If a row vector a j of the dependency matri A can e linearly epressed y some other row vectors, the proing path p j can e replaced y the proing paths corresponding to the row vectors that linearly epress a j In Fig (), since the row vector a can e linearly epressed as a = a + a 3 + a 4, the proing path p, p 3, and p 4 together can replace the proing path p Due to the linear dependence among row vectors, there may e several linear equations to solve a solvale variale i ; ie, there are multiple sets of proing paths to uniquely determine an identifiale link For an identifiale link e i, we define its solution as follows Definition (Solution to an identifiale link): A solution to an identifiale link e i is a set of proing paths satisfying: () they can uniquely determine e i, and () each proing path in the set cannot e replaced y other proing paths in the set With this definition, each proing path in a solution cannot e replaced y other proing paths in the same solution This is consistent with our ojective of minimizing the overall selected proing paths Consider a set of proing paths that are ale to uniquely determine e i, if a proing path can e replaced y others in the set, after removing this proing path the set can still uniquely determine e i Hence, for determining e i, containing such replaceale proing paths only increases the proing cost Thus, a solution to an identifiale link e i should not contain any replaceale proing path This is the asis of our path selection algorithm The solution to a solvale variale i in the linear system does not necessarily match a solution to e i A solution to i is a linear epression n j= c ij j The variale in this linear epression may e replaced y other variales in the same epression Actually, a solution to e i corresponds to the solution to i satisfying that every variale j in it is Tale Tale of notations Symols Meaning G network under study m numer of links in G e i the ith link in G n numer of proing paths in G p i the ith proing path in G E t set of target links m t numer of target links LS linear system model of G ls i the ith linear equation of LS i measured performance of the ith proing path A dependency matri S i set of all solutions to the variale i in LS Sj i the jth solution to the variale i in LS L set of linear epressions, L = {l i i t} l i the linear epression for the ith row vector in A N not replaceale y other variales in the same solution In the rest of the paper, when we say a solution to a solvale variale i, it means a solution whose variales are not replaceale y other variales in it Since a j = j, a variale j is replaceale y other i s if and only if a j can e linearly epressed y the row vectors corresponding to these i s For eample, a solution to the solvale variale in Fig (a) is + 4 It contains three variales,, and 4 Each of them is not replaceale y other two variales, ecause the corresponding row vectors a, a, and a 4 are linearly independent Accordingly, the proing path p, p, and p 4 together can uniquely determine the identifiale link e, and each of the three proing paths cannot e replaced y another two proing paths Therefore, the set of the proing paths {p, p, p 4 } is a solution to e Tale summarizes the symols used in the linear system model and the symols that will e used in the path selection algorithm in the net section 3 Path Selection Algorithm A solution to an identifiale link is a set of proing paths that can uniquely determine this link Since a proing path may e replaceale y others, an identifiale link may have multiple solutions and a proing path may e useful for uniquely determining multiple identifiale links Consequently, different proing paths have different contriutions to determining the identifiale links In this section, we propose an algorithm to calculate all solutions to each identifiale target link Among these solutions, we determine the contriution of each proing path to achieving identifiaility and then choose the most useful proing paths 3 Decomposition of Linear System The simplest way to determine if a link is identifiale or not is to solve the linear system LS Many techniques in the linear algera, such as Gaussian elimination, can achieve this The linear dependence in row vectors of the dependency matri only affects how many solutions an identifiale link has, ut does not affect whether a link is identifiale or not

4 Therefore, the first step for calculating all solutions to each identifiale target link is to decompose the linear system so as to determine whether a target link is identifiale and produce a solution to each of identifiale target links We divide the row vectors of the dependency matri into two groups As a result, the dependency matri A is decomposed into the form shown as Eq () In this decomposition, the matri A R contains r linearly independent row vectors of A, and A N contains other n r row vectors Each row vector of A N can e linearly epressed y the row vectors of A R For simplicity, we renumer the proing paths such that A R = (a,, a r ) T and A N = (a r+,, a n ) T ( ) AR A = () A N Accordingly, the linear system LS can e epressed as Eq (3), in which the vector is partitioned into two vectors R = (,, r ) T and N = ( r+,, n ) T Since each row vector of A N is a linear comination of row vectors of A R, a solvale/unsovlale variale in the linear system A R = R is also solvale/unsolvale in the linear system in Eq () ( ) ( ) AR R = (3) A N N To construct the decomposition of the matri A, we use the algorithm introduced in [3], which is ased on standard rank-revealing decomposition techniques [] It is possile that a dependency matri has multiple decompositions of Eq () In the net part, we will see that our algorithm for calculating all solutions to an identifiale target link does not have any requirement on the decomposition of the linear system Therefore, we can use any tie reaking strategy to choose a decomposition By solving the linear system A R = R, we can determine whether a target link is identifiale or not and calculate a solution S i to the solvale variale i The solution calculated from the linear system A R = R is referred to as the ase solution There is only one ase solution to each solvale variale i ecause the row vectors of A R are linearly independent Since the row vector of A N can e linearly epressed y the row vectors of A R, we calculate the linear epression for each row vector of A R as shown in Eq (4), where i =,, n r r c ij a j + a r+i = (4) j= Multiplying vector to oth sides of Eq (4) results in r j= c ija j + a r+i = Since a i = i eists for i n, we can get the linear epression in Eq () We use L = {l i i n r} to denote the set of these linear epressions, in which l i is the epression containing r+i r c ij j + r+i = () j= Take the eample in Fig () for instance, a decomposition of the matri A is A R = (a, a, a 3, a 4 ) T and A N = (a, a 6 ) T The corresponding partition of is R = (,, 3, 4 ) T and N = (, 6 ) T By solving the linear system A R = R, we discover that e, e, and e 3 are identifiale ut e 4 and e are unidentifiale The ase solution to is = + 4 It corresponds to a solution to e, ie the set {p, p, p 4 } The set L contains two linear epressions, in which l is = and l is = 3 Calculation of Solutions By decomposing the linear system LS, we find out all identifiale target links, otain the ase solution to each solvale variale, and calculate the linear epression for r+,, n In this susection, we propose an algorithm to calculate all solutions to a solvale variale, through which we can otain all solutions to the corresponding identifiale link The asic idea of calculating all solutions to a solvale variale i is to replace the variales in the solutions with the linear epression in the set L Initially, we have only the ase solution S i otained from the linear system A R = R For each linear epression l j L, if it has a common variale k with S, i replacing k in S i with l j results in a linear epression that can solve i Then we check if this linear epression has any variale that can e replaced y other variale in it If not, this linear epression is a solution to i We apply similar linear replacements to all solutions until no new solution can e produced Later we will see that this algorithm can find all solutions to a solvale variale i The algorithm for calculating all solutions to a solvale variale i is shown in Algorithm The input is the ase solution S i otained from A R = R and the set L of the linear epressions We use a queue to store the solutions As shown in line 4, each time we take out a solution from the queue and apply all possile linear replacements to it The linear replacement generates a linear epression St i (line 8) Then, in line 9, the algorithm checks if each variale in St i is replaceale y other variales in St i If St i does not have any replaceale variale, it is a solution to i For a solution not in Q and S i, it is a new solution Thus we put it into the queue After applying all linear replacements to the current solution Sc, i we put it into the solution set S i As a result, at any moment the set S i contains all solutions that have een applied linear replacements, and the queue Q contains all solutions that will e applied linear replacements When the queue Q ecomes empty, the algorithm stops and all solutions to i are in the set S i Theorem : The algorithm SolutionCalculation can find out all solutions to a solvale variale in the linear system Proof: Suppose there is a solution Sq i to the solvale variale i that is not found out y the algorithm This

5 Algorithm SolutionCalculation Input: The ase solution S i, and the set of the linear epressions L Output: A set S i containing all solutions to i Procedure: : INIT(Q) //Initialize a queue Q : S i = 3: ENQUEUE(Q, S i ) 4: while Q do : Sc i = DEQUEUE(Q) 6: for each linear epression l j L do 7: for each k in oth Sc i and l j do 8: St i replace k in Sc i with l j 9: if each variale in St i cannot e replaced y other variales in St i, and Si t / Q Si t / Si then : ENQUEUE(Q, St i) : end if : end for 3: end for 4: S i = S i Sc i : end while solution has the following form n i = d qj j (6) j= From the algorithm SolutionCalculation we can see that all solutions in S i is a linear cominations of S i and the linear epressions in L Therefore, this missed solution Sq i cannot e linearly epressed y S i and linear epressions in L Consider the linear system in Eq (7) that includes S, i Sq, i and all linear epressions in L r j= d j j i = n j= d qj j i = r j= c j j + r+ = (7) r j= c n r,j j + n = The coefficient matri of this linear system is as Eq (8) r r+ n i d d r d q d qr d q,r+ d qn c c r (8) c n r, c n r,r By sutracting d q,r+k times the ( + k)th row from the second row (k =,, n r), the matri turns into the form as in Eq (9), where d qj = d qj n r k= d q,r+kc kj for j =,, r r r+ n i d d r d q d qr c c r c n r, c n r,r (9) Since Sq i cannot e linearly epressed y S i and linear epressions in L, the first two row vectors in Eq (9) are linearly independent It means that d j d qj for j =,, r Therefore, there are two solutions to i in the linear system A R = R, ie r j= d j j and r j= d qj j It contradicts with the fact that A R = R has only one solution to each solvale variale as mentioned efore Therefore, no such S i q eists In conclusion, SolutionCalculation can find out all solutions to the solvale variale i As mentioned in Section, each solution to a solvale variale i corresponds to a solution to the identifiale link e i For each variale j contained in a solution to i, the corresponding solution to e i contains the proing path p j Therefore, after calculating all solutions to i, we can easily otain all solutions to e i To make it more clear, we use an eample to show how to calculate all solutions to the identifiale link e in Fig Initially, we have a solution S that is + 4 The linear epression set L contains two equations l : = and l : = The solution S has two common variales and 4 with the linear epression l Replacing in S with results in +3, which is a new solution We name it as S Similarly, replacing 4 in S with 3 + produces +3 Then we use l to replace and 4 in S, oth resulting in +3 6 named as S3 Until now, we have already applied all possile linear replacement to S Net, we apply linear replacements toward S and S3 The generated new solutions are put into the queue and the linear replacement continues until the queue is empty When applying l to the solution 4 +6 to replace 4, we get a linear epression Since the corresponding row vectors {a, a, 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 , 4 + 6, +3, +3 6,, and 4+ 6 Accordingly, there are si solutions to the link e : {p, p, p 4 }, {p, p 3, p }, {p, p 3, p 6 }, {p 3, p 4, p, p 6 }, {p, p 4, p, p 6 }, and {p, p 4, p, p 6 } 33 Etended Bipartite Model As in the literature [], the relation etween the proing paths and the target links is usually modeled as a ipartite graph G B = (U, V, E), where U and V are two sets of vertices and E is a set of edges A verte in the set U and V denotes a proing path and a target link, respectively If a proing path p i traverses a target link e j, there is an edge etween the verte u i U and v j V Thus the edge reflects the coverage relation etween the proing path and the target link However, this ipartite model cannot deal with our path selection prolem, ecause it is unale to reflect the relation etween an identifiale target link and its solutions Therefore, we propose an etended ipartite model In our ipartite

6 model G B = (U, V, E), the verte set U has two suset U and U, and the verte set V also has two susets V and V, shown in Fig 3 A verte in the set V and V represents an identifiale and unidentifiale target link, respectively Each verte in U denotes a solution to an identifiale target link Hence it corresponds to a set of proing paths It is possile that a proing path is not in any solution We use a verte in U to denote each of such proing paths There are three kinds of edges in the etended ipartite mode as follows V V U U Figure 3 An illustration of the etended ipartite model ) The edge etween V and U For an identifiale target link denoted y the verte vi V, if the solution represented y the verte u j U is a solution to this identifiale link, there is an edge connecting verte vi and u j Hence the edge reflects the identifiaility relation etween a solution and an identifiale target link ) The edge etween V and U For an unidentifiale target link represented y verte vi V, if a proing path in the solution represented y verte u j U covers it, there is an edge etween vi and u j The edge reflects the coverage relation etween a solution and an unidentifiale target link 3) The edge etween V and U For an unidentifiale target link represented y the verte vi V, if a proing path represented y the verte u j U traverses it, there is an edge connecting vi and u j The edge reflects the coverage relation etween a single proing path and an unidentifiale target link There is no edge etween set V and U For identifiale target links, we need to select proing paths to uniquely determine them, which equals to the prolem of choosing vertices from U such that their neighors include all vertices in V The edges etween V and U are sufficient to model the prolem Thus, we do not add any edge etween an identifiale target link and a single proing path The commonly used ipartite model G B is a special case of our ipartite model If we do not consider the identifiaility, we can intentionally mark all target links as unidentifiale to make the set V empty Accordingly, the set U ecomes empty ecause there is no solution Hence the etended ipartite model turns into the commonly used ipartite model We can construct the ipartite graph G B in the following three steps First, we uild sets V, V, and U, and let U e empty Then we construct the edges etween U and V, and the edges etween U and V Finally, for each proing path that is not in the set U, we add a verte to U to denote it and construct the edges etween it and the verte in V numnewv ertices numnewp aths 34 Path Selection Algorithm Through the etended ipartite model, our path selection prolem equals to selecting a set of vertices from U U, so that all vertices in V V are their neighors This prolem is NP-hard, ecause it can e easily transformed into the set cover prolem Thus we design a heuristic ased algorithm to solve it The asic idea of the path selection algorithm is to greedily select proing paths until all identifiale target links can e uniquely determined and all unidentifiale target links are covered For each identifiale target link, we select a set of proing paths ased on its solutions Currently, we give the identifiale target link a priority higher than the unidentifiale target link, ecause uniquely determining an identifiale link is more comple and needs more proing paths than covering an unidentifiale link Hence, the algorithm first deals with the identifiale target link and then the unidentifiale target link An important point is that the selected proing paths may contain replaceale proing paths Therefore, after finishing the path selection, the algorithm removes all replaceale ones from the selected proing paths The linear system ased method for finding out all replaceale proing paths is introduced in Section For a large scale network, there can e thousands of proing paths Accordingly, an identifiale target link may have a large numer of solutions In order to enhance the scalaility of our method, we introduce a parameter α to control how many solutions to an identifiale target link are used in path selection The path selection algorithm is shown as Algorithm It takes the linear system LS and the parameter α as input, and returns a set of proing paths that can uniquely determine all identifiale target links and cover all unidentifiale target links Firstly, the algorithm calculates solutions to each identifiale target link with the algorithm SolutionCalculation, and then select α solutions for each identifiale target link Net, the algorithm uilds the etended ipartite graph G B = (U, V, E) ased on the linear system LS and selected solutions After uilding the etended ipartite graph, the algorithm starts to use a heuristic to select proing paths From line 8 to 6, the algorithm greedily selects vertices from U until all vertices in V are covered Each time, the algorithm chooses an uncovered verte with the maimal value of Intuitively, it means that we like to add as few proing paths as possile and uniquely determine as many identifiale target links as possile When all identifiale target links are handled, the unselected proing paths are contained in

7 oth U and U Thus the algorithm selects vertices from U U for the uncovered unidentifiale target links until all of them are covered Finally, the algorithm removes all replaceale proing paths from the set Algorithm PathSelection Input: The linear system LS and the parameter α Output: A set of proing paths P s Procedure: : for each identifiale target link do : calculate solutions to it y means of the algorithm SolutionCalculation, and then select α solutions 3: end for 4: Construct the etended ipartite graph G B = (U, V, E) with the selected solutions : P s = 6: Mark all vertices in V, V, U, and U as uncovered 7: Mark all proing paths as unused 8: while V has uncovered vertices do 9: Select an uncovered verte u m from U with the largest numnewv ertices, where numnewv ertices is the numer of numnewp aths uncovered verte in V connected to u m, and numnewp aths is the numer of unused proing path contained in u m If there are multiple candidates, select the one that connects to the maimal uncovered vertices in V : : Mark u m as covered Mark all uncovered vertices in V V connected to u m as covered : for each unused proing path p i in u m do 3: P s = P s p i 4: Mark p i as used : end for 6: end while 7: while V has uncovered vertices do 8: Select an uncovered verte u m from U U that has the largest numnewv ertices, where numnewv ertices is the numer of numnewp aths uncovered verte in V connected to u m, and numnewp aths is the numer of unused proing path contained in u m 9: : Mark u m as covered Mark all uncovered vertices in V connected to u m as covered : for each unused proing path p j in u m do : P s = P s p j 3: Mark p j as used 4: end for : end while 6: Remove all replaceale proing paths from the set P s Theorem : The upper ound of the numer of proing paths selected y the algorithm PathSelection is the row rank of the dependency matri Proof: Consider the set P s efore removing the replaceale paths in it, we have P s P For the linear system LS s formed y the proing paths in P s, the row rank of the coefficient matri A s is no larger than that of the dependency matri A in Eq () After removing all replaceale proing paths, the numer of proing paths in P s is equal to the row rank of A s, which is unchanged after removing replaceale paths Therefore, the numer of proing paths returned y the algorithm is no larger than the row rank of the dependency matri Finally, we use an eample to show the algorithm for selecting proing paths for the target link e and e 4 in Fig In the eample, we use all solutions for path selection The corresponding ipartite model is shown in Fig 4 The set U has si vertices, each of which represents a solution to the identifiale link e The set U is empty, ecause all proing paths traversing the unidentifiale link e 4 are in the set U The verte e 4 in V has no edge to the verte {p, p, p 4 } in U ecause these three paths do not traverse e 4 The algorithm first chooses a path set for e Among all si vertices in U, the verte {p, p, p 4 }, {p, p 3, p }, numnewv ertices and {p, p 3, p 6 } have the same value of numnewp aths Since {p, p 3, p } and {p, p 3, p 6 } can also cover an unidentifiale link, the algorithm selects a path set from them, eg {p, p 3, p } After choosing it, all vertices in V V are marked as covered Since the selected set does not contain any replaceale proing path, the algorithm stops and returns a set of proing paths {p, p 3, p } V e V p p, p, p 4 p, p 3, p p, p 3, p, p 4, p, p, p 4, p, p 3, p 4, p, 6 p 6 p 6 p 6 U Figure 4 The etended ipartite model for selecting proing paths for the target link e and e 4 in Fig 4 Performance Evaluations In this section, we evaluate the performance of our algorithm that can select a minimal set of proing paths to cover a set of target links and uniquely determine all identifiale target links We compare the performance of our method with others [3], and study how network topologies and the percentages of target links affect the performance 4 Simulation Setup The simulation is ased on eight ISP topologies derived y the Rocketfuel project [] Tale summarizes these topologies The first column shows the total numer of nodes and the second column shows the numer of leaf nodes for each topology Similar to [4], we connect end systems to the leaf nodes to send and receive proing packets All topologies adopt the shortest path routing calculated ased on the link cost in the topologies Since the link is symmetrical in these topologies, the routing path etween two nodes is symmetrical Thus the numer of proing paths is n l(n l ), where n l is the numer of leaf nodes Due to shortest path routing, each topology has some links that cannot e covered y the proing paths We list the numer of links that can e covered y proing paths in the fourth column Among these links, some can e uniquely determined y proing paths The last column shows the numer of identifiale links and the percentage of links covered y the proing paths In si topologies, more than e 4

8 Tale Summary of Topologies Used in Simulation Topology Total Leaf Total Covered Identifiale Nodes Nodes Links Links Links (%) AS (667%) AS (63%) AS (84%) AS (379%) AS (679%) AS (6%) AS (8%) AS (47%) half of the links covered y proing paths are identifiale In topology AS7, it is as high as 84% For each topology, we randomly select a certain percent of the link that can e covered y proing paths as the target link The percentage of the target link ranges from % to % We run each simulation for times and take the statistical average as the result For each test case, we run the algorithm PathSelection with α =,,,, In our path selection algorithm, we do not specify how to select α solutions, thus we randomly select solutions in the evaluation Different strategies for selecting solutions to uild etended ipartite graph are left for future work The major performance metric in our evaluation is the numer of the selected proing paths, which is referred to as the overall proing cost We also use two other metrics to show the amortized proing cost ) Cost per target link: It is the ratio of the overall proing cost to the numer of target links This metric quantifies the amortized cost on each target link ) Overhead per identifiale target link: The algorithm introduced in [] can select a minimal set of proing paths to cover all target links The difference etween the overall proing cost and the numer of proing paths for covering all target links can e considered as additional proing paths used for achieving identifiaility The overhead per identifiale target link is the ratio of such additional cost to the numer of identifiale target links 4 Overall Proing Cost The overall proing cost of our path selection algorithm on eight topologies is shown in Fig On all these topologies, the overall proing cost of the algorithm with α = is almost the same as that of the algorithm with α = under each percentage of target links Hence, in Fig we omit the case of α = From the figure we can see that increasing α, namely using more solutions for path selection, can reduce the overall proing cost As the percentage of target links increases, more and more non-redundant proing paths are selected Accordingly, the overall proing cost gradually converges to the theoretical upper ound shown in Theorem Therefore, when the percentage of target links reaches %, the overall proing cost under different α ecomes similar 43 Amortized Proing Cost In this part, we show the amortized proing cost of the path selection algorithm with α = As shown in Fig 6, the proing cost per target link decreases as the percentage of target links increases The overhead per identifiale target link shown in Fig 7 has similar trend When the percentage of target links increases, the amortized overhead decreases For oth of them, when the percentage of target links reaches %, the amortized cost reduces to aout /3 of the case when % links are target links This indicates that our path selection algorithm can effectively choose the proing paths that are most useful for identifying multiple target links and eliminate redundant proing paths Cost per target link AS AS9 AS7 AS99 Cost per target link AS668 AS48 AS88 AS94 (a) AS, AS9, AS7, AS99 () AS668, AS48, AS88, AS94 Figure 6 The cost per target link of the path selection algorithm with α = Overhead per identifiale target link AS AS9 AS7 AS99 Overhead per identifiale target link AS668 AS48 AS88 AS94 (a) AS, AS9, AS7, AS99 () AS668, AS48, AS88, AS94 Figure 7 The overhead per identifiale target link of the path selection algorithm with α = 44 Comparisons The closest work in this area is the SelectPath algorithm [3], which can select a minimal set of proing paths to uniquely determine all identifiale links in the network However, the SelectPath algorithm is only a special case of what our algorithm can do, ie when all links are target links Even for this special case, we show that the performance of our solution is guaranteed to e etter or equal to theirs in this susection

9 4 3 3 = = = = (a) AS = = = = = = = = () AS = = = = = 4 = 3 = = (c) AS = = 3 = = (d) AS99 (e) AS668 (f) AS48 (g) AS88 (h) AS94 Figure The overall proing cost of the path selection algorithm with α =,,, = = = = = = = = Tale 3 Largest Overall Proing Cost of The PathSelection Algorithm with Different α and Overall Proing Cost of The SelectPath Algorithm Topology α SelectPath [3] AS AS AS AS AS AS AS AS Since the SelectPath algorithm cannot e modified for an aritrary set of target links, we only compare the performance on the case that all links are target links, although this is not fair to our algorithm which is more general As shown in [3], the overall proing cost of the SelectPath algorithm is equal to the row rank of the dependency matri According to Theorem, the row rank of the dependency matri is the upper ound of the overall proing cost of our algorithm As a result, our method is theoretically no worse than the SelectPath algorithm For each topology and α, we choose the largest overall proing cost among simulation runs Tale 3 shows the largest overall proing cost of our algorithm with different α and the overall proing cost of the SelectPath algorithm On five topologies, our algorithm with each α outperforms the SelectPath algorithm On the other three topologies AS, AS7 and AS48, the performance of our algorithm is equal to the SelectPath algorithm The SelectPath algorithm selects the proing paths corresponding to the asis of row vectors of the dependency matri Although the vectors in the asis are linearly independent, it does not mean there is no redundance Fig () is a simple eample to eplain this fact There are three identifiale links e, e, and e 3 The asis of the row vectors contains 4 linearly independent vectors However, only three proing paths p, p, and p 4 are adequate to uniquely determining these three identifiale links Adding any other proing paths does not contriute to achieving identifiaility, no matter it is replaceale y p, p, and p 4 or not Our algorithm outperforms the SelectPath algorithm ecause we try to avoid not only the replaceale proing path ut also the proing path without any contriution to achieving identifiaility Related Work Minimizing the proing cost is usually achieved y carefully selecting proing paths This prolem has een studied without resource limitation [], [3], [4], and in scenarios with eplicit operational requirement [7] and resource constraint [3] Various kinds of proing costs and monitoring ojectives have een studied However, As pointed out in [4], many network inference prolems are ill-posed, ecause the numer of measurements are not sufficient to uniquely determine the result Our work jointly addresses minimizing the proing cost and achieving identifiaility Achieving identifiaility in network link monitoring is also considered in [3], [] [3] However, our work is quite different from them Zhao et al [3] proposed a method for determining the shortest sequence of links whose properties can e uniquely identified y end-to-end proes It is similar to differentiating the identifiale from unidentifiale links in our work However, their method does not consider how

10 to minimize the proing cost The failure localization in [] aims at accurately pinpointing a failed link, which in essence is to achieve identifiaility However, it only focuses on locating a single link Our method is ale to uniquely determine every identifiale link Both [] and [3] deal with the prolem of selecting a minimal set of proing paths that can uniquely determine all identifiale links in the network It is only a special case of what our algorithm can address, ie when all links are target links Even for this special case, the performance of our solution is guaranteed to e etter or equal to theirs Another major difference is the method we adopt Their methods are ased on eliminating the replaceale path; ie, the linearly dependent row vector in the dependency matri We take a different approach, and only select the proing paths that are the most useful to our ojective Besides eliminating the replaceale proing path, we also try to avoid selecting the proing path without any contriution to achieving identifiaility 6 Conclusions and Future Work The end-to-end proe has received considerale attention in network link monitoring In this paper, we propose an approach to minimize the proing cost and achieve identifiaility The asic idea is to select proing paths that are the most useful for covering the target links and achieving identifiaility Furthermore, our method eliminates two types of redundant proing paths, ie, those that can e replaced y others and those without any contriution to achieving identifiaility With our approach, the numer of selected proing paths is proved to e ounded Eperiments ased on eight ISP topologies demonstrate that our approach can achieve identifiaility with a minimal set of proing paths Compared with prior works, our method is more general and has etter performance The algorithm proposed for calculating all solutions to an identifiale link may generate a large numer of path sets when the network scale is large It is possile that 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