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

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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 dis- tinct 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) addi- tive, e.g., link delays; and 4) concave, e.g., bottleneck band- width. 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 charac- terize 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 the- oretical foundations of computing linearly independent cycles in arbitrary networks and the associated problem/solution char- acteristics. 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.
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