On Identifying Additive Link Metrics Using Linearly Independent Cycles and Paths Abishek Gopalan and Srinivasan Ramasubramanian, Senior Member, IEEE

906 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 20, NO. 3, JUNE 2012 On Identifying Additive Link Metrics Using Linearly Independent Cycles and Paths Abishek Gopalan and Srinivasan Ramasubramanian, Senior Member, IEEE

Abstract—In this paper, we study the problem of identifying con- where relying on internal nodes/routers to actively monitor and stant additive link metrics using linearly independent monitoring measure link quality is unreasonable. To minimize the cooper- cycles and paths. A monitoring cycle starts and ends at the same ation required from the internal nodes, measurements are typi- monitoring station, while a monitoring path starts and ends at distinct monitoring stations. We show that three-edge connectivity is a cally obtained from end-to-end paths. necessary and sufficient condition to identify link metrics using one The term network tomography, coined by Vardi [1], refers monitoring station and employing monitoring cycles. We develop a to the inference of certain internal characteristics of networks polynomial-time algorithm to compute the set of linearly indepen- based on end-to-end measurements. Network tomography dent cycles. For networks that are less than three-edge-connected, may be classified into passive or active tomography. In pas- we show how the minimum number of monitors required and their placement may be computed. For networks with symmetric di- sive tomography, routers collect information on the normally rected links, we show the relationship between the number of mon- forwarded traffic.Basedonthecollected information, some itors employed, the number of directed links for which metric is network aspects, such as origin–destination trafficmatrix,may known apriori, and the identifiability for the remaining links. To be estimated. In active tomography, the network is specifically the best of our knowledge, this is the first work that derives the probed for information along one or more paths. Based on necessary and sufficient conditions on the network topology for identifying additive link metrics and develops a polynomial-time the path-level observations, individual link behaviors may be algorithm to compute linearly independent cycles and paths. characterized. Most link-level metrics are characterized by a probability dis- Index Terms—Additive link metrics, end-to-end measurements, tribution (such as the queuing delay experienced by a packet, identifiability, independent trees, linear independence, network tomography, statistical inverse. etc.). The link metrics over an end-to-end path can combine in different ways in the network.Someexamplesofwaysin which they could combine are the following: 1) super-linear, I. INTRODUCTION e.g., bit error rates; 2) multiplicative, e.g., reliability; 3) additive, e.g., link delays; and 4) concave, e.g., bottleneck bandwidth. Therefore, the objective of active tomography is to derive CCESS to information in a timely, reliable, and secure the link-level probability distribution of the desired metric by A manner is becoming increasingly critical for the infor- observing the behavior on a certain set of preestablished paths mation-centric lifestyle. As the network transmission speed in- often referred to as the “statistical inverse problem” [2]. Several creases, the bandwidth-delay (propagation delay) product in- works attempt to solve this problem in various contexts such creases, resulting in a large amount of data in transit at any given as identifying additive link metrics (latency [3]–[6] and dis- time. Therefore, any small service disruption, be it due to fail- tances [7], [8]), linearly combining optical characteristics [9], ures or intrusion, leads to a significant loss of data. Thus, tech- network topology [10], [11], and placement of monitors [12]. niques for achieving dependability (reliability, availability, se- For a detailed survey in the field, we refer readers to [13]–[17]. curity, and verifiability) must be proactive in nature. The com- One of the fundamental problems in network tomography is ponents of the network may be constantly monitored for their to identify link metrics when they are assumed to be constant performance in order to proactively reroute trafficthatmaybe and additive. While this assumption simplifies real-world set- affected by a failure. tings in many applications, identifying link metrics even under While it is desirable to monitor the links and nodes of a net- this assumption is not well understood as there are no known work, it is often impractical to make such direct measurements. theoretical guarantees. To this end, we develop the fundamental A typical example is a large-scale network like the Internet, theory on the necessary and sufficient conditions on the network topology and a polynomial-time algorithm to compute all link metrics by establishing linearly independent cycles and paths. Manuscript received February 25, 2011; revised June 30, 2011; ac- cepted September 12, 2011; approved by IEEE/ACM TRANSACTIONS ON The solution developed in this paper is also applicable in sce- NETWORKING Editor P. Van Mieghem. Date of publication November 23, narios where the distribution of the metric on a link is known and 2011; date of current version June 12, 2012. This work was supported in part the parameters of the distribution need to be computed. In such by the National Science Foundation under the Grant CNS-1117274. The authors are with the Department of Electrical and Computer Engineering, cases, one may observe a path for a period of time to deduce the University of Arizona, Tucson, AZ 85721 USA (e-mail: [email protected]. sum of the parameters of the individual link metric distribution edu; [email protected]). along the path. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. The problem of identifying additive link metrics using Digital Object Identifier 10.1109/TNET.2011.2174648 end-to-end measurements is often severely underconstrained,

1063-6692/$26.00 © 2011 IEEE GOPALAN AND RAMASUBRAMANIAN: ON IDENTIFYING ADDITIVE LINK METRICS 907 the primary reason being the network topology. We characterize the topology by the presence of a sufficient number of certain cycles/paths that have the property that they are linearly independent. The focus of this paper is to understand the theoretical foundations of computing linearly independent cycles in arbitrary networks and the associated problem/solution characteristics. We acknowledge that there are several link-level metrics (such as bit error rate) that may not be additive over Fig. 1. Example network to illustrate the problem. a path or be additive up a to certain distance threshold. Our goal here is not to identify whether a metric lends itself to an additive treatment over the path or not, or how accurate such as- linearly independent cycles/paths in polynomial time? This sumptions are. Therefore, we do not consider any experimental paper provides the answers to these fundamental questions. study for validating our assumptions on statistical properties of Related Work in Failure Localization: In the area of failure metrics. localization, several researchers have studied the problem of identifying link failures by observing failure of end-to-end paths A. Related Work (or cycles) [20]–[25]. These approaches assume that the link metrics are binary in nature and the path metric is simply a The surveys in [14]–[16] summarize in great detail the gen- boolean OR function (assuming failure is represented by 1 and eral problem of unidentifiability of link metics and why the mea- operational links are represented as 0’s). In this paper, we seek surement matrix is not invertible in most scenarios. a similar approach, except that link metrics are additive and not In [7], the authors estimate distances (time delays) of un- restricted to boolean. known paths by inferring measurements on known paths using Application in Nanoelectronics, Computational Sensing, and tracer stations.1 While they do extract as much as possible from Power Systems Management: The solution developed in this the measurement matrix, they do not attempt to characterize the paper is also applicable in the area of evaluating nanoelectronic conditions under which the matrix will have full rank. In [8], devices. Currently, more processing units are being fabricated the authors exploit the fact that the shortest path has the lowest on a chip that are being connected by an on-chip network, e.g., end-to-end weight and develop a consistent constraint system multicore chips and FPGAs. The on-chip network is comprised for inferring link weights. They then measure how well their of standard CMOS-based switches and metallic wires or carbon solution approximates observed routing. In [13], the author ac- nanowires/nanotubes (in future). Fabrication at such small fea- knowledges the problem of unidentifiable links and tries to iden- ture size leads to several process variations, resulting in some tify the worst performing links in a subnetwork. In [18], the au- links performing poorly (due to increased resistance or due to thors try to estimate link-level loss rates using multicast trees break in connectivity) [26]. The precise resistance values of the and show that there could exist unidentifiable links in the net- links may be measured by computing the pin-to-pin resistances work. The works in [3]–[6] consider an overlay system with over different paths that are linearly independent, obtained using end-hosts and develop a methodology to monitor paths (be- different switching configurations inside the chip. Based on in- tween the end-hosts) that form a basis so that all other paths’ dividual link characterization, the computing elements may be metrics may be identified using the basis. The authors show that connected only using the “good” links [27]. The same problem their method achieves good approximation while bounding as is also of interest to the compressive sensing [28] and power sys- . However, their approach cannot identify the indi- tems [29] communities. In particular, the work in [28] assumes vidual link metrics in the network since they face the problem of that at most link metrics are unknown while the others are as- unidentifiable links due to rank deficiency. They do not address sumed to be zero. They show that in any undirected graph on the problem of when individual link metrics may be identified. nodes that satisfies a minimum degree for every node, they can Xia and Tse [19] consider arbitrary directed networks, where the identify the link metrics by using path measure- link metrics on either direction could be different. They show ments. In contrast, we show that if a network is three-edge-con- that unless every link has a monitor attached on either end, none nected, then we may construct linearly independent cycles, as of the links can be identified. While their result holds, we show many as the total number of links, to uniquely identify all link that knowing a few link metrics aprioriin symmetric directed metrics using only one monitoring node. networks can greatly reduce the number of monitors required to identify all other link metrics. B. Problem Illustration While all the above works provide some insights into the Consider an example network as shown in Fig. 1 with problem of identifying link metrics using end-to-end paths, five nodes and eight links. The numbers on the links represent some fundamental questions on the identifiability remain unan- link IDs. Assume that node is the monitor that can start and swered: 1) Under what topological conditions are additive link terminate probes. metrics identifiable? 2) How many monitoring nodes (nodes We may compute eight cycles originating and terminating at that have the ability to make measurements) are required, and as shown in Fig. 2. We may represent these cycles (denoted where should they be placed in the network? 3) Can we compute by the links present in them) in a matrix form. Let of 1Such techniques are useful in peer-to-peer and overlay networks since a good dimension ,where denotes the number of links and an choice of a server can be made when distances are known. element denotes whether link is present in cycle or not. 908 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 20, NO. 3, JUNE 2012

3) Given the measurements on the linearly independent cycles and paths, we may obtain the individual link metrics in linear time without having to compute matrix inverse. 4) For networks that do not satisfy the necessary conditions to identify metrics with one monitor, we identify the minimum number of monitors, their placement, and compute linearly independent cycles and paths for a given place- Fig. 2. (left) List of cycles and (right) the corresponding cycle-link matrix. ment of monitors. 5) For networks with symmetric directed links, we prove unidentifiability with one monitor, show the relationship If each cycle accumulates the link metrics on its path, then the between the number of monitors employed, the number of value observed at is the sum of the link metrics. Let denote directed links for which the metrics are known apriori, the column matrix of the accumulated metrics corre- and identifiability of metrics on all other links. sponding to the cycles. Let denote the column matrix The polynomial-time algorithm for constructing the set of lin- of the link variables that we are trying to identify. Our goal is early independent cycles that traverses a given monitoring sta- to solve the system of linear equations represented by . tion in a three-edge-connected network requires the construc- In order to uniquely determine , has to be invertible. Such tion of three link-independent trees—a problem of considerable a matrix is also called identifiable since it has full rank. Cycles importance even beyond the scope of network tomography. In- that make up such a matrix are referred to as linearly indepen- dependent trees are employed for fault-tolerant and multipath dent cycles. The eight cycles computed in Fig. 2 are linearly routing [30]–[34]. Proving the existence and developing poly- independent, thus all link metrics may be identified. nomial-time algorithms to compute (node/link) independent The matrix has full rank, hence all link metrics can be trees rooted at a given node is considered a fundamental result in uniquely identified by solving for . The example il- graph theory literature [35]. Khuller and Schieber [36] provide lustrates that in certain topologies, all link metrics may be iden- an algorithm that constructs link-independent trees for any , tified using one monitor employing only cycles. In this paper, provided vertex-independent trees exist. For the case of , we study the topological constraints under which the link met- Cheriyan and Maheswari [37] have constructed vertex-indepen- rics are identifiable using one monitor employing monitoring dent trees. We have proved that the algorithm proposed in [36] cycles only. In addition, when the network does not satisfy the is incorrect. We refer the interested reader to [38] for further de- topological conditions, we identify the number of monitors re- tails. To the best of our knowledge, our work in [38] provides quired and their placement in order to identify link metrics using the first known algorithm that constructs three edge-indepen- monitoring cycles and paths. dent trees in a three-edge-connected network. Note that in the above example, any arbitrary set of eight cycles may not give full rank, even though every link appears D. Organization in at least one cycle.2 To the best of our knowledge, there is no prior work on the computation of linearly independent cy- The rest of this paper is organized as follows. Section II cles traversing a given node in the network. One approach to discusses in detail the aspects of identifiability in undirected compute the set of linearly independent set of cycles is to con- graphs by providing a detailed analysis of the necessary consider one cycle at a time. Given a set of linearly independent ditions, the underlying method required to identify all link cycles, where the cardinality of the set is less than ,weneed metrics, describes some properties of three link-independent to compute another linearly independent cycle that traverses the trees, and shows that the number of distinct cycles that are also monitor. This problem, however, is not studied in the literature. linearly independent equals the number of links. Section III An alternate approach to construct the set of linearly indepen- discusses some implementation issues and challenges. In dent cycles is to develop a systematic method to construct these Section IV, we study the identifiability problem in symmetric cycles, which we develop in this paper. We show that by con- directed networks. Section V concludes the paper. structing three edge-independent trees rooted at the monitor, we may construct the desired linearly independent cycles. II. IDENTIFYING METRICS ON UNDIRECTED NETWORKS We consider a network3 denoted by ,where de- C. Contributions notes the set of nodes and denotes the set of undirected links. The contributions of this paper are the following. Let denote the unknown constant weight on link .Weas- 1) We show that three-edge connectivity is a necessary sume that the network will employ monitors at some nodes in and sufficient condition for uniquely identifying additive the network. Monitoring paths that start and end at distinct mon- metrics on all links using one monitoring station and itoring stations may be established. Alternatively, monitoring employing monitoring cycles. cycles may be established that start and end at the same mon- 2) We develop a polynomial-time algorithm to compute itor. We allow the use of nonsimple cycles and paths. In this linearly independent cycles using one monitor and thus paper, nonsimple cycles (paths) are those where a node may ap- uniquely identify additive metrics on all the links. pear more than once, however a link will not.

2One such set—124, 476, 1276, 1386, 1354, 4586, 12586, 67531—has rank 3The terms “graph” and “network” are used interchangeably unless otherwise six. speciﬁed. GOPALAN AND RAMASUBRAMANIAN: ON IDENTIFYING ADDITIVE LINK METRICS 909

Fig. 3. Cutset in a two-edge-connected network involving two links and .

Problem Statement: Given a network with undirected links: 1) What are the necessary and sufficient conditions on the topology of the network such that the constant additive link metrics may be identified with only one monitor employing only monitoring cycles? 2) If the network topology satisfies the necessary and sufficiency conditions, develop an algorithm to construct the linearly independent cycles. 3) If the network does not satisfy the necessary conditions for identifying the link metrics with only one monitor, what is the minimum number of monitors required and their placement such that all the link metrics may be identified using linearly independent cycles and paths?

A. Necessary and Sufficient Conditions Fig. 4. Procedure to construct linearly independent cycles in a three-edge-con- Theorem 2.1: Three-edge connectivity is a necessary and suf- nected network given the position of a monitoring station. ficient condition for identifying all additive link metrics using one monitor employing monitoring cycles. node to the monitor on tree ,where .Let de- Proof: We prove the necessary part of the claim by con- note the sum of the link metrics on path .If , ,and tradiction. First, in a one-edge-connected network, there exists denote the sum of the link metrics in these cycles, respec- a link whose removal would disconnect the graph. Thus, no cy- tively, we have three linear equations cles may be constructed through that link. Hence, the metric on that link is not identifiable. Now, assume that the given network is two-edge-connected. Then, there exists at least one cutset of size two, as shown in Fig. 3. As the monitor is present in only one of the components, any cycle that traverses link will also traverse link . Hence, the metrics on links and cannot From the above equations, we may obtain the value of , be uniquely identified. Thus, three-edge connectivity is a nec- . For a node at layer 1, simply denotes the link metric essary condition. as the node is directly connected to the monitor on .Thus,by We prove the sufficiency part by demonstrating a method to considering the nodes in the breadth-first manner, based on , identify the additive link metrics. Fig. 4 shows the procedure to the metrics on all the links in may be identified successively. construct a set of linearly independent cycles in any three-edge- By repeating the same procedure for and , all link metrics connected network that has a monitoring station. The procedure may be identified. Thus, contains a sufficient number of lin- reduces the given network into a minimally three-edge-con- early independent cycles to identify all link metrics. nected network .In , we construct three link-independent To compute the metrics on links that are pruned from the spanning trees rooted at the monitoring station. The existence original graph, we consider the pruned links one at at time. For of three independent trees, rooted at a node, are guaranteed in a pruned link , we consider the graph obtained adding link any three-edge-connected graph [38]. Thepropertyoflink-in- to the minimally three-edge-connected graph. In the resultant dependent trees is that the paths from any node to the root graph, we compute a cycle traversing and . The cycle on the trees are mutually link-disjoint.Let , ,and de- involves only one unknown metric, the metric on link , thus it note the three link-independent trees in rooted at the monitor may be identified uniquely. node . Since the network is minimally three-edge-connected, The set of linearly independent cycles is simply obtained as every link appears in at least one of the trees. For every node , the set of all the cycles computed in the two steps above. we compute three cycles by combining all combinations of two Note that we compute three cycles per node by combining of the three paths from node to on the three trees. The set two link-disjoint paths at a time in a minimally three-edge-con- of all cycles computed is denoted by .Wefirst show that using nected network. Thus, we consider a total of cycles the cycles in , we may uniquely identify all the link metrics to be added to . Since we are able to compute all the link met- in . rics from , there are at least linearly independent cycles Consider the tree . Arrange the nodes in the tree in the in . In fact, there are exactly linearly independent cycles breadth-first manner based on its distance from the root-referred as there are only variables in the system of equations, thus to as “level.” A node at level is hops away from the monitor. the rank cannot exceed . Interestingly, we can show that the Consider a node at level ,say .Let denote the path from number of distinct cycles in is exactly ,i.e., . 910 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 20, NO. 3, JUNE 2012

Fig. 6. Example two-vertex- and minimally three-edge-connected network.

Fig. 5. Outline to construct three link-independent trees rooted at monitoring station in a minimally three-edge-connected graph.

The counting of the number of distinct cycles obtained by merging disjoint paths is complicated due to the following two factors: 1) the number of occurrences of a cycle in the set of cycles differs from one cycle to another; and 2) although every link is present in at least one of the trees and a link may appear in at most two trees, not all links appear in two trees. In order to show that , an understanding of the Fig. 7. Segments on the example network. structure involved in computing the three link independent trees is required. understanding of the tree construction procedure, hence we B. Constructing Three Link-Independent Trees choose not to present it in this paper. Segments and Their Properties: Given and a node ,an In order to understand the computation of the number of dis- arbitrary neighbor of ,say , is selected and the link – is tinct cycles in , it is necessary to understand the three-tree removed. The network is then decomposed into a sequence of construction procedure. We, therefore, briefly discuss the algo- segments numbered from 0 to .The th segment is rithm to construct three independent trees. Interested readers are denoted by . referred to [38] for a detailed description of the construction The segments have the following properties. mechanism. 1) Segment is a cycle that starts and ends at . We consider a minimally three-edge-connected network, de- 2) At every stage ,thesegment starts and ends at noted by ,where and denote the set of nodes and two (not necessarily distinct) nodes that are already part links, respectively. Every link is assumed to be bidirec- of earlier segments and traverses only new nodes (at least tional. In addition, we are given a monitoring station . one). Fig. 5 shows the outline to compute the three link-indepen- 3) For each , the following property holds. Consider dent trees rooted at in a minimally three-edge-connected the graph obtained by removing all the links in . graph. The firststepistodividethegivennetworkinto The vertices and all their other links are retained. In this three-edge- and two-vertex-connected components, referred to network, all nodes in remain connected as 3E-2V components. Every link in the graph belongs to a to node . unique component. Any two 3E-2V components may share at 4) contains all vertices in . most one common node.4 If the decomposition results in more Consider an example 3E-2V network shown in Fig. 6. Fig. 7 than one component, then we identify a “root” node, referred shows the sequence of segments computed on the example net- to as , in every component that does not contain node . work. The monitor is at node and is node in this ex- The root node of a component is the node through which ample. The solid lines denote the links that are part of the seg- any path from a node in the component to must traverse. ments. The dotted lines indicate other links in but not in the For the component that contains the monitor, the root node is set of segments. In particular, the link – is the link that was the monitor. The second step is to consider each of the 3E-2V removed between – . components and compute three link-independent trees, referred Computing the Red and Blue Trees: The red and blue trees to as red, blue, and green trees, that are rooted at the root node are computed using the path augmentation approach [30], [31], of that component. As the final step, the corresponding trees where the sequence of segments is used for augmentation. In from different components are merged. order to construct the red and blue trees, a global order [30] or The computation of three trees in a 3E-2V component in- partial order [31] among the links in the network is maintained volves computation of a sequence of “segments” that have spe- to ensure the disjointedness of paths on the two trees. We outline cial properties. In the following, we discuss the properties of the construction of red and blue trees from the segments. We initialize to be empty. Let the first cycle consist segments and how they are usedtocomputetheredandblue of the nodes . We let the red chain be trees. The computation of green edges involves an in-depth and the blue chain be 4Referred to as an articulation node. . These chains are added to the respective trees. The GOPALAN AND RAMASUBRAMANIAN: ON IDENTIFYING ADDITIVE LINK METRICS 911 partial order maintained is denoted by . Let segment consist of nodes ,where are nodes that were added in earlier segments and nodes are not present in any segment ; .For any node ,let and denote the parent on the red and blue trees, respectively.5 Then, if 6 in the partial order, the red chain is and the blue chain is . The red and blue chains are added to the red and blue trees, respectively. The partial order for these edges are updated Fig. 8. Counting cycles on the example network. as . If , the chain is reversed, and the par- C. Counting Distinct Cycles tial order is Theorem 2.2: The number of distinct cycles obtained by com- . In this fashion, all segments are bining all possible two of the three independent paths at every processed to compute the red and blue neighbors for all nodes. node in is . Observe that for every segment , the links Proof: For every node other than , we compute three cy- are the only links that have a single color assigned on them (red cles by combining two of the three independent (disjoint) paths. or blue). All other links in have both the red and blue colors The total number of cycles thus obtained is .How- defined on them in opposite directions. Hence, the green edge ever, in such a counting, several cycles are counted more than for a node could only be on the following links: once. In order to compute the number of distinct cycles, we an- 1) a link that has only one color definedonitafter are alyze every segment that is augmented and count the number of constructed; or distinct cycles it provides. 2) a link that is not part of the sequential ordering of segments Every link in the network is present in at most two indepen- and hence has no color defined. dent trees. Thus, every link has at most two colors assigned to The actual computation of the green neighbors is omitted here it—one in each direction. Let denote the number of links in since it requires knowledge of how the segments themselves which one direction is colored green and the other edge is uncol- are computed. It is sufficient to understand which links may ored. Let denote the number of links in which one direction be present on the green tree to compute the number of distinct is colored green and the other direction is colored either red or cycles. blue. Since and together count all the links that are on Example: We now illustrate how the segments are processed the green tree, we have on the example network in Fig. 8. When the first cycle is added, the red chain is . The partial order is (1) .For ,since in the order, becomes the red end of the chain. The par- Consider segment .Let denote the number of newly tial order among the newly added edges is added nodes in this segment. Let the nodes in the segment .For ,since be numbered through ,where and denote , the red neighbor of is . The partial order for the the two end-nodes (previously added) to which this segment edges is . Finally for , connects.8 since ,7 becomes the red end of the chain. Based on the above notation, we may derive a relationship The red, blue, and green neighbors for all nodes are shown in between the number of links, number of nodes, and the number Fig. 8. of segments added. Segment with new nodes has Observe that the sequential ordering of segments only guar- links. Besides these links, there exist links that have only a antees the inclusion of all nodes but not necessarily all links in green directed edge on them in . These links do not appear in . Hence, there are some links that have a color defined on only the segments themselves. Thus, we have one direction while other links have colors on both edges. Fur- thermore, since the network is minimally three-edge-connected, (2) all links have at least one color defined on them (part of at least one tree). The three link-independent trees obtained on the example network are redrawn in Fig. 9. (3) Having understood the structure behind the link-independent trees, we can now enumerate the number of distinct cycles that (4) could be obtained by merging all possible two of the three paths at all nodes, i.e., the number of distinct cycles in in Step 4 of For each segment, we count the number of distinct cycles in the Fig. 4. set of cycles, which we would obtain when the paths for 5 are already part of some ; . the newly added nodes are merged. 6 If and are the same node, this condition always holds. 8Note that and need not be distinct since the paths from any node 7Because is transitive. to the root are only guaranteed to be link-disjoint. 912 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 20, NO. 3, JUNE 2012

Fig. 9. Example network and three link-independent trees rooted at node B. (a) Example network. (b) Red tree. (c) Blue tree. (d) Green tree.

We ﬁrst consider the cycles obtained by merging the red and Now, observe that these distinct cycles are also linearly blue paths. The newly added segment will be employed for the independent as we may employ the methodology outlined in red tree in one direction and the blue tree in the other direction Theorem 2.1 to identify the link metrics.9 for all the nodes. The cycles obtained by combining the red Example: We will illustrate the counting on the example net- and blue paths from each of the nodes from through work considered earlier. The three link-independent trees along will be identical. Thus, the segment contributes only one distinct with the segments augmented in are shown in Fig. 8. We have cycle when when the red and blue paths for any of the newly links C–A, D–A, B–A that appear only on the green tree, thus added nodes are combined. . Links G–H, F–A, K–E, H–D, E–D appear on two trees, We now consider the cycles formed by combining the red and of which one of them is green, thus . green paths. Since new nodes are added in the segment, the The number of distinct cycles obtained from segment is cycles obtained by merging the red and green paths of nodes given by and computed as follows for through will all be distinct from each other. However, the 0, 1, 2, and 3: cycle obtained at node will be the same as that obtained for (counted as part of the segment in which was added), thus may result in counting it twice. Let denote if the link – has green label on : 1 if true, 0 otherwise. The number of distinct cycles obtained by merging red and green paths from the newly added nodes is . The number of distinct cycles is 15, which is the total number We now consider the cycles formed by combining the blue of links in the network. and green paths. By following the same line of argument as Theorem 2.3: The number of linearly independent cycles pro- we did for the cycles obtained by combining the red and green duced by the procedure described in Fig. 4 is ,thenumber paths, we consider the link – .Let denote if the of links in the given network. link – has green label on or not: 1 Proof: The number of cycles computed by the procedure if true, 0 otherwise. The number of distinct cycles obtained by in Fig. 4 is . By Theorem 2.2, .Thesetof merging blue and green paths from the newly added nodes is cycles in is constructed by considering one link at a time. . Thus, . Thus, the total number of cycles con- Thus, the total number of distinct cycles provided by the structed by the procedure is , the number of links in the given th segment, denoted by , is computed as network. Note that the computation of the number of distinct cycles does not depend on the algorithm employed to construct the (5) three independent trees. We may use the notion of segment to (6) denote a chain of nodes such that: 1) is red; 2) is blue; 3) is red; The total number of distinct cycles obtained from all the seg- 4) is blue. Given any three independent trees, we may ments, denoted by , is computed as compute these segments in a straightforward manner. We may use the same computation outlined above disregarding whether these segments form augmenting paths or not. (7) The computation above assumes that every link is present in at least one tree, which is readily satisﬁed if we consider a minimally three-edge-connected network. However, given three in- (8) dependent trees rooted at a node, the network obtained by considering all the links in them need not be minimally three-edge- (9) connected. Thus, given a network and three independent trees, we may simply assume that is the set of links that appear in at Subtracting (4) from (9), and using (1), we have least one tree, while denotes the set of all links in the network.

9Note that linear independence of cycles implies distinct cycles, however the (10) converse is not true. GOPALAN AND RAMASUBRAMANIAN: ON IDENTIFYING ADDITIVE LINK METRICS 913

D. Algorithm Complexity and Correctness be three-edge-connected. In such scenarios, we need to employ The correctness follows from the constructive proofs in multiple monitors. From the proof of Theorem 2.1, the scenarios Theorems 2.1 and 2.3. We now analyze the running time com- in which we may not be able to identify linearly independent cy- plexity of the algorithm in Fig. 4. The input to the algorithm cles become obvious. is at least three-edge-connected. Lemma 2.4: In order to uniquely identify the metrics on all Step 2 of the procedure reduces to a minimally three-edge- links, it is necessary and sufficient that every component ob- connected network, which may be achieved in two steps. First, tained by removing any two links in the network must each have we compute a three-edge-connected sparse spanning subgraph a monitoring station. of [39]. The number of edges in the sparse graph is guar- Proof: See [25, Lemma 1]. Although the lemma in [25] anteed to be at most . Second, we reduce the sparse is used in the context of distinguishing two link failures, the spanning subgraph to a minimally three-edge-connected graph. argument used in the lemma applies here as well. To achieve this, we consider one edge at a time and check if As the placement of monitors to uniquely identify the the edge may be removed without affecting the three-edge con- link metrics follows the same conditions as that required for nectivity of the spanning subgraph, which may be achieved in uniquely localizing single link failures, we may use the place- time for every edge [40]. As the number of edges in the ment of monitoring stations in [25] for our problem. For sake sparse subgraph is , the minimally three-edge-connected of completeness, we outline the procedure briefly. Given a graph is obtained in time. network, it is divided into three-edge-connected components. Step 3 of the procedure, which constructs three link-in- Every three-edge-connected component that has a degree10 of dependent trees on the minimally three-edge-connected two or fewer requires a monitor inside the component. The graph , takes time [37], [38]. network is then decomposed into two-edge-connected com- Step 4, which involves computing three equations (cycles) ponents. For every two-edge-connected component that has per node, may be computed in constant time per node from the degree two or fewer, and that does not have a monitor already three trees, thus taking time. placed using the previous step, a monitor is required. Step 5 involves computing one cycle for every link in . Once the monitors are placed, the monitors are simply merged The cycle computation for each link takes a constant time, by together. The resultant network is at least three-edge-connected making use of the partial order of the nodes to which the link with a single monitor. The algorithms developed in the previous connects. For example, consider link that connects nodes sections of this paper may be applied to obtain the cycles. Ob- and .If traverses on the red path or there does not exist an serve that some cycles in the graph where monitors are merged ordering between them , we may compute will become monitoring paths (from one distinct monitor to an- the cycle through bymergingtheredpathfrom and blue path other) in the original network. Thus, given any network, we may from with .Otherwise , we may compute identify the minimum number of monitors required, their place- the desired cycle by merging the red path from and blue path ment, and the linearly independent cycles and paths to identify from with . Thus, the remaining cycles may be obtained in the additive link metrics on all undirected links. time. III. DISCUSSION Hence, the overall complexity of the procedure in Fig. 4 is . In the previous sections of the paper, we developed the nec- Once the linearly independent cycles are obtained, we may essary theoretical foundations for identifying link metrics using construct a matrix of size and compute its inverse to obtain linearly independent cycles and paths. We now discuss how the individual link metrics. The complexity of inverting a matrix cycles/paths may be realized in practice and some of the chal- of size is . However, we may use the structure of lenges involved for undirected networks. the graph to reduce the complexity of solving for individual link A. Measurements in IP Networks metrics (as outlined in Theorem 2.1) to . For every node, we solve a system of three equations in constant time, which Network monitoring and identifying link metrics can prove provides the metrics along the red, blue, and green paths from useful to a service provider. As a natural choice, we consider the node to the monitoring station. From these path metrics, the here how these algorithms may be implemented in an IP net- individual link metrics are obtained by considering the links in work. In order to enable a monitor to send packets along a cycle, the breadth-first order on each tree. Therefore, computing the wemayemployIP-in-IPtunneling[41].Forexample,inorder metrics on all the links in the minimally three-edge-connected to measure the metric on a cycle obtained by combining the red graph takes time. Computing the metrics on all other and blue paths at node , the monitoring station must send a links (in ) takes time—constant time per link as the packet on the red (blue) tree to node and node must forward path metrics are already known. Thus, the time complexity to that packet back to node on the blue (red) tree. To achieve this compute individual link metrics from cycle metrics is . routing, we employ three undirected spanning trees—obtained from the three link-independent trees rooted at and treating E. Application to Networks That are Less Than all the links on the tree as undirected. For each of these three Three-Edge-Connected spanning trees, every node maintains a routing table entry for While three-edge connectivity is a necessary and sufficient every other node. Thus, the monitor may create a packet that condition for identifying additive link metrics using one mon- 10The degree of a component is the number of links that connect a node in itor employing only monitoring cycles, some networks may not the component to another node outside the component. 914 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 20, NO. 3, JUNE 2012 is destined to itself to be routed over the blue (red) tree, which is then encapsulated in another header that is destined to node to be routed on the red (blue) tree. Such a packet may be used to measure the metric along the cycle obtained by combining the red and blue paths at node .

B. Identification Using Only Paths Fig. 10. Cutset to show lack of identifiability in directed networks. Thus far, we have assumed that we can use: 1) cycles only, or 2) cycles and paths. However, it is possible that we may not The matrix representation of these cycles (each row repre- be able to employ cycles in some networks. For example, if an sents a cycle; columns through represent the variables IP network does not implement IP-in-IP tunneling, then routing , respectively) will be of the form over a desired cycle may not be possible. In such scenarios, we may need to utilize only paths that start and end at distinct moni- . . . . toring stations. Thus, we need to identify the minimum number . . .. . of monitors such that all link metrics may be identified using only paths, assuming nonsimple paths are allowed...... We may show that the link metrics may be identified using ...... only paths11 if the following two conditions are satisfied: 1) the placement of monitoring stations satisfy the requirements in . .. . . Lemma 2.4; and 2) when the monitoring stations are connected . . . . to a virtual node with virtual links, the resultant network is three-edge-connected. If these conditions are met, we can es- The first rows represent the cycles that pass through tablish linearly independent cycles through the virtual node. two links, one of which is . Each identity submatrix is of Each such cycle is a path that would start and end at the dis- size .The since the minimum oftherow tinct monitors on the original network. However, it is not clear and column ranks dictate the maximum rank of any matrix. We if monitoring paths alone is sufficient if the network obtained show that the rank is strictly lessthan by applying the linear by connecting the monitoring stations to a virtual node is not transformation to columns and in matrix . three-edge-connected.

IV. IDENTIFYING METRICS ON DIRECTED LINKS

In this section, we study the problem of identifying additive Now, columns are identical, and hence link weights on symmetric directed networks. In particular, we . Observe that we have assumed that the components and consider the scenario when bidirectionality of links is achieved provide rich connectivity. If the connectivity is sparse, then using two unidirectional links. Thus, if a link exists, then not all combinations of summations of and may be exists as well. However, the metrics on these two directed evaluated, hence the rank may be significantly lower than . links could be different. We consider a graph ,where now denotes the set of directed links. We assume that a cycle In the proof above, the rank of the matrix is in fact exactly may traverse link and . if and components provide rich connectivity such We now show that no amount of connectivity will help in that all summations may be computed. This is true because identifying metrics on directed links with only one monitor. the metric on any one directed link, say , helps uniquely iden- Theorem 4.1: Given a directed network, it is not possible tify all other variables. Now, observe that if a network has any to identify any link metric with one monitor employing cycles cutset where all the link metrics are unknown, then none of them only. may be identified. Therefore, the natural follow-up is to identify Proof: Consider some arbitrary cutset of the graph as the minimum number of directed links for which the metric must shown in Fig. 10 involving links. Without loss of generality be known so that the metric on all other directed links may be the monitor is in component . Further assume that all link identified with only one monitor. metrics on either side of the cutset are known. Even under Theorem 4.2: Given a symmetric directed network with this highly relaxed assumption, it is not possible to identify nodes, the minimum number of directed links for which the any link metric on the cutset. Consider all possible cycles metrics need to be known such that the metrics on other links involving (two) links in the cutset. We have a total of equa- may be identified using one monitor and monitoring cycles is tions involving variables. All other cycles passing through . more than two links of the cutset can be written as a linear Proof: We show the necessary part of the proof using con- combination of these cycles and hence provide no additional tradiction. Assume that the metrics on directed links are information. known. Let denote the set of undirected links for which the 11The paths may have loops, however the paths will start and end at two dis- metric on at least one direction is known. Then, . tinct monitor nodes. Now, view the network as an undirected graph with nodes GOPALAN AND RAMASUBRAMANIAN: ON IDENTIFYING ADDITIVE LINK METRICS 915 and links. Clearly, the set of nodes cannot form a con- three-edge-connected network. For networks that do not satisfy nected spanning network with or less links. Thus, the the necessary conditions, we showed that the minimum number original network must have at least two components separated of monitors, their placement, and the linearly independent cy- by a set of unknown links. Thus, any link traversing between cles and paths may be identified. For networks with directed these two components is part of a cutset, where all link metrics links, we showed that the sum of the number of monitors and the are unknown. From Theorem 4.1, none of the links metrics on number of links for which the edges need to be known equals the cutset is identifiable. the number of nodes in the network, irrespective of the connec- We prove the sufficiency part by construction. Consider the tivity in the network. Future research could involve extending undirected version of the network. Compute a spanning tree, these results to networks with directed links, where links may rooted at the monitor. This spanning tree contains exactly not be bidirectional, i.e., if exists, does not neces- undirected edges, or equivalently directed edges. sarily exist. For each undirected edge in the spanning tree, assume that the link metric on one direction is known, thus requiring the knowl- ACKNOWLEDGMENT edge of exactly link metrics. We can now show that any Any opinions, findings, conclusions, or recommendations ex- other link metric may be identified with these known metrics. pressed in this paper are those of the author(s) and do not nec- First, given that only one direction of the link metric is known, essarily reflect the views of the National Science Foundation the metric on the link in the other direction may be identified (NSF). by considering the links in the breadth-first manner and establishing a path from the monitor traversing the link and back to REFERENCES the monitor. By successively computing these metrics over the [1] Y. Vardi, “Estimating source-destination traffic intensities from link spanning tree, all the directed link metrics may be data,” J. Amer. Statist. Assoc., vol. 91, pp. 365–377, 1996. [2] E. Lawrence, G. Michailidis, and V. N. Nair, “Statistical inverse prob- known. 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