We consider the problem of inferring the topology of a network with M sources and N receivers (an M-by- N network), by sending probes between the sources and receivers. Prior work has shown that this problem can be decomposed into two parts: first, infer smaller subnetwork components (1-by- N's or 2-by-2's) and then merge them to identify the M-by- N topology. We focus on the second part, which had previously received less attention in the literature. We assume that a 1-by- N topology is given and that all 2-by-2 components can be queried and learned using end-to-end probes. The problem is which 2-by-2's to query and how to merge them with the given 1-by- N, so as to exactly identify the 2-by- N topology, and optimize a number of performance metrics, including the number of queries (which directly translates into measurement bandwidth), time complexity, and memory usage. We provide a lower bound, [N 2], on the number of 2-by-2's required by any active learning algorithm and propose two greedy algorithms. The first algorithm follows the framework of multiple hypothesis testing, in particular Generalized Binary Search (GBS). The second algorithm is called the Receiver Elimination Algorithm (REA) and follows a bottom-up approach. It requires exactly N-1 steps, which is much less than all (2N) possible 2-by-2's. Simulation results demonstrate that both algorithms correctly identify the 2-by- N topology and are near-optimal, but REA is more efficient in practice.
In our prior work @cite_10 @cite_12 , we revisited the problem of topology inference using end-to-end probes in networks where intermediate nodes are equipped with multicast and network coding capabilities. We built on @cite_38 and extended it, using network coding at intermediate nodes to deterministically distinguish among all possible quartet topologies, which was not possible before. While in @cite_10 @cite_12 , we focused on inferring the quartets fast and accurately, here we assume that any quartet can be queried and learned, and focus on efficiently selecting and merging the quartets to infer the larger topology. To the best of our knowledge, this work is the first to look at this aspect of the problem.
@cite_38 @cite_10 @cite_12
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In this paper, we combine network coding and tomographic techniques for topology inference. Our goal is to infer the topology of a network by sending probes between a given set of multiple sources and multiple receivers and by having intermediate nodes perform network coding operations. We combine and extend two ideas that have been developed independently. On one hand, network coding introduces topology-dependent correlation, which can then be exploited at the receivers to infer the topology [1]. On the other hand, it has been shown that a traditional (i.e., without network coding) multiple source, multiple receiver tomography problem can be decomposed into multiple two source, two receiver subproblems [2]. Our first contribution is to show that, when intermediate nodes perform network coding, topological information contained in network coded packets allows to accurately distinguish among all different 2-by-2 subnetwork components, which was not possible with traditional tomographic techniques. Our second contribution is to use this knowledge to merge the subnetworks and accurately reconstruct the general topology. Our approach is applicable to any general Internet-like topology, and is robust to the presence of delay variability and packet loss. We consider the problem of inferring the topology of an M-by-N network by sending probes between M sources and N receivers. Prior work has shown that this problem can be decomposed into two parts: first, infer smaller subnetwork components (i.e., 1-by-N's or 2-by-2's) and then merge these components to identify the M-by-N topology. In this paper, we focus on the second part. In particular, we assume that a 1by-N topology is given and that all 2-by-2 components can be queried and learned using end-to-end probes. The problem is which 2-by-2's to query and how to merge them with the 1-byN, so as to exactly identify the 2-by-N topology, and optimize a number of performance metrics including measurement traffic, time complexity, and memory usage. We provide a lower bound, ⌈N 2⌉, on the number of 2-by-2's required by any active learning algorithm and we also propose a greedy algorithm that is nearoptimal and efficient in practice. It follows a bottom-up approach: at every step, it selects two receivers, queries the corresponding 2-by-2, and merges it with the given 1-by-N. The algorithm requires exactly N - 1 steps, which is much less than all (N:2) possible 2-by-2's, and it correctly identifies the 2-by-N topology. We consider the task of topology discovery of sparse random graphs using end-to-end random measurements (e.g., delay) between a subset of nodes, referred to as the participants. The rest of the nodes are hidden, and do not provide any information for topology discovery. We consider topology discovery under two routing models: (a) the participants exchange messages along the shortest paths and obtain end-to-end measurements, and (b) additionally, the participants exchange messages along the second shortest path. For scenario (a), our proposed algorithm results in a sub-linear edit-distance guarantee using a sub-linear number of uniformly selected participants. For scenario (b), we obtain a much stronger result, and show that we can achieve consistent reconstruction when a sub-linear number of uniformly selected nodes participate. This implies that accurate discovery of sparse random graphs is tractable using an extremely small number of participants. We finally obtain a lower bound on the number of participants required by any algorithm to reconstruct the original random graph up to a given edit distance. We also demonstrate that while consistent discovery is tractable for sparse random graphs using a small number of participants, in general, there are graphs which cannot be discovered by any algorithm even with a significant number of participants, and with the availability of end-to-end information along all the paths between the participants. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013 Many systems require information about the topology of networks on the Internet, for purposes like management, efficiency, testing of new protocols and so on. However, ISPs usually do not share the actual topology maps with outsiders; thus, in order to obtain the topology of a network on the Internet, a system must reconstruct it from publicly observable data. The standard method employs traceroute to obtain paths between nodes; next, a topology is generated such that the observed paths occur in the graph. However, traceroute has the problem that some routers refuse to reveal their addresses, and appear as anonymous nodes in traces. Previous research on the problem of topology inference with anonymous nodes has demonstrated that it is at best NP-complete. In this paper, we improve upon this result. In our previous research, we showed that in the special case where nodes may be anonymous in some traces but not in all traces (so all node identifiers are known), there exist trace sets that are generable from multiple topologies. This paper extends our theory of network tracing to the general case (with strictly anonymous nodes), and shows that the problem of computing the network that generated a trace set, given the trace set, has no general solution. The weak version of the problem, which allows an algorithm to output a "small" set of networks- any one of which is the correct one- is also not solvable. Any algorithm guaranteed to output the correct topology outputs at least an exponential number of networks. Our results are surprisingly robust: they hold even when the network is known to have exactly two anonymous nodes, and every node as well as every edge in the network is guaranteed to occur in some trace. On the basis of this result, we suggest that exact reconstruction of network topology requires more powerful tools than traceroute. We consider the problem of topology recognition in wireless (radio) networks modeled as undirected graphs. Topology recognition is a fundamental task in which every node of the network has to output a map of the underlying graph i.e., an isomorphic copy of it, and situate itself in this map. In wireless networks, nodes communicate in synchronous rounds. In each round a node can either transmit a message to all its neighbors, or stay silent and listen. At the receiving end, a node @math hears a message from a neighbor @math in a given round, if @math listens in this round, and if @math is its only neighbor that transmits in this round. Nodes have labels which are (not necessarily different) binary strings. The length of a labeling scheme is the largest length of a label. We concentrate on wireless networks modeled by trees, and we investigate two problems. What is the shortest labeling scheme that permits topology recognition in all wireless tree networks of diameter @math and maximum degree @math ? What is the fastest topology recognition algorithm working for all wireless tree networks of diameter @math and maximum degree @math , using such a short labeling scheme? We are interested in deterministic topology recognition algorithms. For the first problem, we show that the minimum length of a labeling scheme allowing topology recognition in all trees of maximum degree @math is @math . For such short schemes, used by an algorithm working for the class of trees of diameter @math and maximum degree @math , we show almost matching bounds on the time of topology recognition: an upper bound @math , and a lower bound @math , for any constant @math . Many topology discovery systems rely on traceroute to discover path information in public networks. However, for some routers, traceroute detects their existence but not their address; we term such routers anonymous routers. This paper considers the problem of inferring the network topology in the presence of anonymous routers. We illustrate how obvious approaches to handle anonymous routers lead to incomplete, inflated, or inaccurate topologies. We formalize the topology inference problem and show that producing both exact and approximate solutions is intractable. Two heuristics are proposed and evaluated through simulation. These heuristics have been used to infer the topology of the 6Bone, and could be incorporated into existing tools to infer more comprehensive and accurate topologies. Topology recognition is one of the fundamental distributed tasks in networks. Each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph, with all ports correctly marked. This task is usually unfeasible without any a priori information. Such information can be provided to nodes as advice. An oracle knowing the network can give a possibly different string of bits to each node, and all nodes must reconstruct the network using this advice, after a given number of rounds of communication. During each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations. The time of completing topology recognition is the number of rounds it takes, and the size of advice is the maximum length of a string given to nodes. We investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes. We provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time, in the class of all graphs of size n and diameter D≤αn, for any constant α<1. In most cases, our bounds are asymptotically tight. More precisely, if the allotted time is D-k, where 0<k≤D, then the optimal size of advice is i¾?n 2 logn D-k+1. If the allotted time is D, then this optimal size is i¾?n logn. If the allotted time is D+k, where 0<k≤D 2, then the optimal size of advice is i¾?1+logn k. The only remaining gap between our bounds is for time D+k, where D 2<k≤D. In this time interval our upper bound remains O1+logn k, while the lower bound that holds for any time is 1. This leaves a gap if D∈ologn. Finally, we show that for time 2D+1, one bit of advice is both necessary and sufficient. Our results show how sensitive is the minimum size of advice to the time allowed for topology recognition: allowing just one round more, from D to D+1, decreases exponentially the advice needed to accomplish this task. In this paper we study the problem of building a constant-degree connected dominating set (CCDS), a network structure that can be used as a communication backbone, in the dual graph radio network model ( in J Parallel Distrib Comput 64:89–96, 2004; in Proceedings of the international symposium on principles of distributed computing 2009, Distrib Comput 24(3–4):187–206 2011, Proceedings of the international symposium on principles of distributed computing 2010). This model includes two types of links: reliable, which always deliver messages, and unreliable, which sometimes fail to deliver messages. Real networks compensate for this differing quality by deploying low-layer detection protocols to filter unreliable from reliable links. With this in mind, we begin by presenting an algorithm that solves the CCDS problem in the dual graph model under the assumption that every process (u ) is provided with a local link detector set consisting of every neighbor connected to (u ) by a reliable link. The algorithm solves the CCDS problem in (O ( ^2 n b + ^3 n ) ) rounds, with high probability, where ( ) is the maximum degree in the reliable link graph, (n ) is the network size, and (b ) is an upper bound in bits on the message size. The algorithm works by first building a Maximal Independent Set (MIS) in ( ^3 n ) time, and then leveraging the local topology knowledge to efficiently connect nearby MIS processes. A natural follow-up question is whether the link detector must be perfectly reliable to solve the CCDS problem. With this in mind, we first describe an algorithm that builds a CCDS in (O( )polylog ((n)) ) time under the assumption of (O(1) ) unreliable links included in each link detector set. We then prove this algorithm to be (almost) tight by showing that the possible inclusion of only a single unreliable link in each process’s local link detector set is sufficient to require ( ( ) ) rounds to solve the CCDS problem, regardless of message size. We conclude by discussing how to apply our algorithm in the setting where the topology of reliable and unreliable links can change over time. Computing the topology of a network in the Internet is a problem that has attracted considerable research interest. The usual method is to employ Traceroute, which produces sequences of nodes that occur along the routes from one node (source) to another (destination). In every trace thus produced, a node occurs by either its unique identifier, or by the anonymous identifier "*". We have earlier proved that there exists no algorithm that can take a set of traces produced by running Traceroute on network N and compute one topology which is guaranteed to be the topology of N. This paper proves a much stronger result: no algorithm can produce a small set of topologies that is guaranteed to contain the topology of N, as the size of the solution set is exponentially large. This result holds even when every edge occurs in a trace, all the unique identifiers of all the nodes are known, and the number of nodes that are irregular (anonymous in some traces) is given. On the basis of this strong result, we suggest that efforts to exactly reconstruct network topology should focus on special cases where the solution set is small. Information about the topology and link-level characteristics of a network is critical for many applications including network diagnostics and management. However, this information is not always directly accessible; subnetworks may not cooperate in releasing information and widespread local measurement can be prohibitively expensive. Network tomographic techniques obviate the need for network cooperation, but the majority assume probing from a single source, which imposes scalability limitations because sampling traffic is concentrated on network links close to the source. We describe a multiple source, end-to-end sampling architecture that uses coordinated transmission of carefully engineered multipacket probes to jointly infer logical topology and estimate link-level performance characteristics. We commence by demonstrating that the general multiple source, multiple destination tomography problem can be formally reduced to the two source, two destination case, allowing the immediate generalization of any sampling techniques developed for the simpler, smaller scenario. We then describe a method for testing whether links are shared in the topologies perceived by individual sources, and describe how to fuse the measurements in the shared case to generate more accurate estimates of the link-level performance statistics In topology recognition, each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph, with all ports correctly marked. This task is usually unfeasible without any a priori information. Such information can be provided to nodes as advice. An oracle knowing the network can give a (possibly different) string of bits to each node, and all nodes must reconstruct the network using this advice, after a given number of rounds of communication. During each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations. The time of completing topology recognition is the number of rounds it takes, and the size of advice is the maximum length of a string given to nodes.We investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes. We provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time, in the class of all graphs of size n and diameter D ? α n , for any constant α < 1 . In most cases, our bounds are asymptotically tight. Despite many efforts over the past decade, the ability to generate topological maps of the Internet at the router-level accurately and in a timely fashion remains elusive. Mapping campaigns commonly involve traceroute-like probing that are usually non-adaptive and incomplete, thus revealing only a portion of the underlying topology. In this paper we demonstrate that standard probing methods yield datasets that implicitly contain information about much more than just the directly observed links and routers. Each probe, in addition to the underlying domain knowledge, returns information that places constraints on the underlying topology, and by integrating a large number of such constraints it is possible to accurately infer the existence of unseen components of the Internet. We describe DomainImpute, a novel data analysis methodology designed to accurately infer the unseen hop-count distances between observed routers. We use both synthetic and a large empirical dataset to validate the proposed methods. On our empirical real world dataset, we show that our methods can estimate over 55 of the unseen distances between observed routers to within a one-hop error. We consider the problem of computing a maximal independent set (MIS) in an extremely harsh broadcast model that relies only on carrier sensing. The model consists of an anonymous broadcast network in which nodes have no knowledge about the topology of the network or even an upper bound on its size. Furthermore, it is assumed that an adversary chooses at which time slot each node wakes up. At each time slot a node can either beep, that is, emit a signal, or be silent. At a particular time slot, beeping nodes receive no feedback, while silent nodes can only differentiate between none of its neighbors beeping, or at least one of its neighbors beeping. We start by proving a lower bound that shows that in this model, it is not possible to locally converge to an MIS in sub-polynomial time. We then study four different relaxations of the model which allow us to circumvent the lower bound and find an MIS in polylogarithmic time. First, we show that if a polynomial upper bound on the network size is known, it is possible to find an MIS in ( O ( ^3 n) ) time. Second, if we assume sleeping nodes are awoken by neighboring beeps, then we can also find an MIS in ( O ( ^3 n) ) time. Third, if in addition to this wakeup assumption we allow sender-side collision detection, that is, beeping nodes can distinguish whether at least one neighboring node is beeping concurrently or not, we can find an MIS in ( O ( ^2 n) ) time. Finally, if instead we endow nodes with synchronous clocks, it is also possible to find an MIS in ( O ( ^2 n) ) time. Many inference problems that arise in sensor networks require the computation of a global conclusion that is consistent with local information known to each node. A large class of these problems---including probabilistic inference, regression, and control problems---can be solved by message passing on a data structure called a junction tree. In this paper, we present a distributed architecture for solving these problems that is robust to unreliable communication and node failures. In this architecture, the nodes of the sensor network assemble themselves into a junction tree and exchange messages between neighbors to solve the inference problem efficiently and exactly. A key part of the architecture is an efficient distributed algorithm for optimizing the choice of junction tree to minimize the communication and computation required by inference. We present experimental results from a prototype implementation on a 97-node Mica2 mote network, as well as simulation results for three applications: distributed sensor calibration, optimal control, and sensor field modeling. These experiments demonstrate that our distributed architecture can solve many important inference problems exactly, efficiently, and robustly. We study the problem of learning the topology of an undirected network by observing a random subsample. Specifically, the sample is chosen by randomly selecting a fixed number of vertices, and for each we are allowed to observe all edges it is incident with. We analyze a general formalization of learning from such samples, and derive confidence bounds on the number of differences between the true and learned topologies, as a function of the number of observed mistakes and the algorithm’s bias. In addition to this general analysis, we also analyze a variant of the problem under a stochastic block model assumption. Selective families, a weaker variant of superimposed codes [KS64, F92, 197, CR96], have been recently used to design Deterministic Distributed Broadcast (DDB) protocols for unknown radio networks (a radio network is said to be unknown when the nodes know nothing about the network but their own label) [CGGPR00, CGOR00]. We first provide a general almost tight lower bound on the size of selective families. Then, by reverting the selective families - DDB protocols connection, we exploit our lower bound to construct a family of “hard” radio networks (i.e. directed graphs). These networks yield an O(n log D) lower bound on the completion time of DDB protocols that is superlinear (in the size n of the network) even for very small maximum eccentricity D of the network, while all the previous lower bounds (e.g. O(D log n) [CGGPR00]) are superlinear only when D is almost linear. On the other hand, the previous upper bounds are all superlinear in n independently of the eccentricity D and the maximum in-degree d of the network. We introduce a broadcast technique that exploits selective families in a new way. Then, by combining selective families of almost optimal size with our new broadcast technique, we obtain an O(Dd log3 n) upper bound that we prove to be almost optimal when d = O(n D). This exponentially improves over the best known upper bound [CGR00) when D, d = O(polylogn). Furthermore, by comparing our deterministic upper bound with the best known randomized one [BGI87] we obtain a new, rather surprising insight into the real gap between deterministic and randomized protocols. It turns out that this gap is exponential (as discovered in [BGI87]), but only when the network has large maximum in-degree (i.e. d = O(na), for some constant a > O). We then look at the multibroadcast problem on unknown radio networks. A similar connection to that between selective families and (single) broadcast also holds between superimposed codes and multibroadcast. We in fact combine a variant of our (single) broadcast technique with superimposed codes of almost optimal size available in literature [EFF85, HS87, I97, CHI99]. This yields a multibroadcast protocol having completion time O(Dd2 log3 n). Finally, in order to determine the limits of our multibroadcast technique, we generalize (and improve) the best known lower bound [CR96] on the size of superimposed codes. Consider a set of mobile robots placed on distinct nodes of a discrete, anonymous, and bidirectional ring. Asynchronously, each robot takes a snapshot of the ring, determining the size of the ring and which nodes are either occupied by robots or empty. Based on the observed configuration, it decides whether to move to one of its adjacent nodes or not. In the first case, it performs the computed move, eventually. This model of computation is known as Look-Compute-Move. The computation depends on the required task. In this paper, we solve both the well-known Gathering and Exclusive Searching tasks. In the former problem, all robots must simultaneously occupy the same node, eventually. In the latter problem, the aim is to clear all edges of the graph. An edge is cleared if it is traversed by a robot or if both its endpoints are occupied. We consider the exclusive searching where it must be ensured that two robots never occupy the same node. Moreover, since the robots are oblivious, the clearing is perpetual, i.e., the ring is cleared infinitely often. In the literature, most contributions are restricted to a subset of initial configurations. Here, we design two different algorithms and provide a characterization of the initial configurations that permit the resolution of the problems under very weak assumptions. More precisely, we provide a full characterization (except for few pathological cases) of the initial configurations for which gathering can be solved. The algorithm relies on the necessary assumption of the local-weak multiplicity detection. This means that during the Look phase a robot detects also whether the node it occupies is occupied by other robots, without acquiring the exact number. For the exclusive searching, we characterize all (except for few pathological cases) aperiodic configurations from which the problem is feasible. We also provide some impossibility results for the case of periodic configurations. This paper studies a multi-facility network synthesis problem, called the Two-level Network Design (TLND) problem, that arises in the topological design of hierarchical communication, transportation, and electric power distribution networks. We are given an undirected network containing two types of nodes---primary and secondary---and fixed costs for installing either a primary or a secondary facility on each edge. Primary nodes require higher grade interconnections than secondary nodes, using the more expensive primary facilities. The TLND problem seeks a minimum cost connected design that spans all the nodes, and connects primary nodes via edges containing primary facilities; the design can use either primary or secondary edges to connect the secondary nodes. The TLND problem generalizes the well-known Steiner network problem and the hierarchical network design problem. In this paper, we study the relationship between alternative model formulations for this problem (e.g., directed and undirected models), and analyze the worst-case performance for a composite TLND heuristic based upon solving embedded subproblems (e.g., minimum spanning tree and either Steiner tree or shortest path subproblems). When the ratio of primary to secondary costs is the same for all edges and when we solve the embedded subproblems optimally, the worst-case performance ratio of the composite TLND heuristic is 4 3. This result applies to the hierarchical network design problem with constant primary-to-secondary cost ratio since its subproblems are shortest path and minimum spanning tree problems. For more general situations, we express the TLND heuristic worst-case ratio in terms of the performance of any heuristic used to solve the embedded Steiner tree subproblem. A companion paper develops and tests a dual ascent procedure that generates tight upper and lower bounds on the optimal value of a multi-level extension of this problem. In this paper, we develop methods to ''sample'' a small realistic graph from a large Internet topology. Despite recent activity, modeling and generation of realistic graphs resembling the Internet is still not a resolved issue. All previous work has attempted to grow such graphs from scratch. We address the complementary problem of shrinking an existing topology. In more detail, this work has three parts. First, we propose a number of reduction methods that can be categorized into three classes: (a) deletion methods, (b) contraction methods, and (c) exploration methods. We prove that some of them maintain key properties of the initial graph. We implement our methods and show that we can effectively reduce the nodes of an Internet graph by as much as 70 while maintaining its important properties. Second, we show that our reduced graphs compare favorably against construction-based generators. Finally, we successfully validate the effectiveness of our best methods in an actual performance evaluation study of multicast routing. Apart from its practical applications, the problem of graph sampling is of independent interest. Since ad hoc and sensor networks can be composed of a very large number of devices, the scalability of network protocols is a major design concern. Furthermore, network protocols must be designed to prolong the battery lifetime of the devices. However, most existing routing techniques for ad hoc networks are known not to scale well. On the other hand, the so-called geographical routing algorithms are known to be scalable but their energy efficiency has never been extensively and comparatively studied. In a geographical routing algorithm, data packets are forwarded by a node to its neighbor based on their respective positions. The neighborhood of each node is constituted by the nodes that lie within a certain radio range. Thus, from the perspective of a node forwarding a packet, the next hop depends on the width of the neighborhood it perceives. The analytical framework proposed in this paper allows to analyze the relationship between the energy efficiency of the routing tasks and the extension of the range of the topology knowledge for each node. A wider topology knowledge may improve the energy efficiency of the routing tasks but increases the cost of topology information due to signaling packets needed to acquire this information. The problem of determining the optimal topology knowledge range for each node to make energy efficient geographical routing decisions is tackled by integer linear programming. It is shown that the problem is intrinsically localized, i.e., a limited topology knowledge is sufficient to make energy efficient forwarding decisions. The leading forwarding rules for geographical routing are compared in this framework, and the energy efficiency of each of them is studied. Moreover, a new forwarding scheme, partial topology knowledge forwarding (PTKF), is introduced, and shown to outperform other existing schemes in typical application scenarios. A probe-based distributed protocol for knowledge range adjustment (PRADA) is finally introduced that allows each node to efficiently select online its topology knowledge range. PRADA is shown to rapidly converge to a near-optimal solution.