Given a family @math of graphs and a positive integer @math , a graph @math is called vertex @math -fault-tolerant with respect to @math , denoted by @math -FT @math , if @math contains some @math as a subgraph, for every @math with @math . Vertex-fault-tolerance has been introduced by Hayes [A graph model for fault-tolerant computing systems, IEEE Transactions on Computers, C-25 (1976), pp. 875-884.], and has been studied in view of potential applications in the design of interconnection networks operating correctly in the presence of faults. We define the Fault-Tolerant Complete Matching (FTCM) Problem in bipartite graphs of order @math : to design a bipartite @math , with @math , @math , @math , that has a FTCM, and the tuple @math , where @math and @math are the maximum degree in @math and @math , respectively, is lexicographically minimum. @math has a FTCM if deleting at most @math vertices from @math creates @math that has a complete matching, i.e., a matching of size @math . We show that if @math is integer, solutions of the FTCM Problem can be found among @math -regular bipartite graphs of order @math , with @math , and @math . If @math then all @math -regular bipartite graphs of order @math have a FTCM, and for @math , it is not the case. We characterize the values of @math , @math , @math , and @math that admit an @math -regular bipartite graph of order @math , with @math , and give a simple construction that creates such a graph with a FTCM whenever possible. Our techniques are based on Hall's marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them.
A related line of research is not to design the smallest fault tolerant graphs (in terms of any metric, like the ones given above), but to analyze the level of fault tolerance assured by prescribed topologies @cite_2 @cite_12 . This topic is of particular interest for algorithm design in high performance computing. Supercomputers are comprised of many processing nodes (with some local memory) that use an interconnection network to communicate during the execution of distributed algorithms. An algorithm delegates computational tasks to different nodes, and uses some logical topology for its message passing. This logical topology has to be somehow embedded in the interconnection network provided by the supercomputer. So it is of practical interest to study if the message passing topologies most common in algorithm design (like cycles, and trees of certain types) can still be embedded in interconnection topologies provided by supercomputers (often similar to hypercubes) when the system presents some faults @cite_2 .
@cite_12 @cite_2
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Fault tolerant algorithms for the numerical approximation of elliptic partial differential equations on modern supercomputers play a more and more important role in the future design of exa-scale enabled iterative solvers. Here, we combine domain partitioning with highly scalable geometric multigrid schemes to obtain fast and fault-robust solvers in three dimensions. The recovery strategy is based on a hierarchical hybrid concept where the values on lower dimensional primitives such as faces are stored redundantly and thus can be recovered easily in case of a failure. The lost volume unknowns in the faulty region are re-computed approximately with multigrid cycles by solving a local Dirichlet problem on the faulty subdomain. Different strategies are compared and evaluated with respect to performance, computational cost, and speed up. Especially effective are strategies in which the local recovery in the faulty region is executed in parallel with global solves and when the local recovery is additionally accelerated. This results in an asynchronous multigrid iteration that can fully compensate faults. Excellent parallel performance on a current peta-scale system is demonstrated. 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. As the size of High Performance Computing clusters grows, so does the probability of interconnect hot spots that degrade the latency and effective bandwidth the network provides. This paper presents a solution to this scalability problem for real life constant bisectional-bandwidth fat-tree topologies. It is shown that maximal bandwidth and cut-through latency can be achieved for MPI global collective traffic. To form such a congestion-free configuration, MPI programs should utilize collective communication, MPI-node-order should be topology aware, and the packet routing should match the MPI communication patterns. First, we show that MPI collectives can be classified into unidirectional and bidirectional shifts. Using this property, we propose a scheme for congestion-free routing of the global collectives in fully and partially populated fat trees running a single job. The no-contention result is then obtained for multiple jobs running on the same fat-tree by applying some job size and placement restrictions. Simulation results of the proposed routing, MPI-node-order and communication patterns show no contention which provides a 40 throughput improvement over previously published results for all-to-all collectives. 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. Traditionally, networks such as datacenter interconnects are designed to optimize worst-case performance under arbitrary traffic patterns. Such network designs can however be far from optimal when considering the actual workloads and traffic patterns which they serve. This insight led to the development of demand-aware datacenter interconnects which can be reconfigured depending on the workload. Motivated by these trends, this paper initiates the algorithmic study of demand-aware networks, and in particular the design of bounded-degree networks. The inputs to the network design problem are a discrete communication request distribution, ( D ), defined over communicating pairs from the node set V, and a bound, ( ), on the maximum degree. In turn, our objective is to design an (undirected) demand-aware network (N=(V,E) ) of bounded-degree ( ), which provides short routing paths between frequently communicating nodes distributed across (N ). In particular, the designed network should minimize the expected path length on (N ) (with respect to ( D )), which is a basic measure of the efficiency of the network. We derive a general lower bound based on the entropy of the communication pattern ( D ), and present asymptotically optimal demand-aware network design algorithms for important distribution families, such as sparse distributions and distributions of locally bounded doubling dimensions. In this paper we consider problems that arise in a shared memory multiprocessor in which memory is physically distributed among a number of memories local to each processor or cluster of processors. The issue we address is that of deciding which local memories should contain copies of pages of data. In the migration problem we operate under the constraint that a page must be kept in exactly one local memory. In the replication problem we allow a page to be kept in any subset of the local memories, but do not allow a local memory to drop a page once it has it. For interconnection topologies that are complete graphs, or trees we have obtained efficient on-line algorithms for these problems. Our migration algorithms also extend to interconnections that are products of these topologies (e.g. a hypercube is a product of simple trees). An on-line algorithm decides how to process each request (which is a read or write request from a processor to a page) without knowing future requests. Our algorithms are also said to be competitive because their performance is within a small constant factor of that of any other algorithm, including algorithms that make use of knowledge of future requests. This research was supported in part by the National Science Foundation under grant CCR8658139. This research was sponsored by the Defense Advanced Research Projects Agency (DOD), monitored by the Space and Naval Warfare Systems Command under Contract N00039-87-C-0251. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Defense Advanced Research Projects Agency or the US Government. Many real-world domains can be represented as large node-link graphs: backbone Internet routers connect with 70,000 other hosts, mid-sized Web servers handle between 20,000 and 200,000 hyperlinked documents, and dictionaries contain millions of words defined in terms of each other. Computational manipulation of such large graphs is common, but previous tools for graph visualization have been limited to datasets of a few thousand nodes. Visual depictions of graphs and networks are external representations that exploit human visual processing to reduce the cognitive load of many tasks that require understanding of global or local structure. We assert that the two key advantages of computer-based systems for information visualization over traditional paper-based visual exposition are interactivity and scalability. We also argue that designing visualization software by taking the characteristics of a target user's task domain into account leads to systems that are more effective and scale to larger datasets than previous work. This thesis contains a detailed analysis of three specialized systems for the interactive exploration of large graphs, relating the intended tasks to the spatial layout and visual encoding choices. We present two novel algorithms for specialized layout and drawing that use quite different visual metaphors. The H3 system for visualizing the hyperlink structures of web sites scales to datasets of over 100,000 nodes by using a carefully chosen spanning tree as the layout backbone, 3D hyperbolic geometry for a Focus+Context view, and provides a fluid interactive experience through guaranteed frame rate drawing. The Constellation system features a highly specialized 2D layout intended to spatially encode domain-specific information for computational linguists checking the plausibility of a large semantic network created from dictionaries. The Planet Multicast system for displaying the tunnel topology of the Internet's multicast backbone provides a literal 3D geographic layout of arcs on a globe to help MBone maintainers find misconfigured long-distance tunnels. Each of these three systems provides a very different view of the graph structure, and we evaluate their efficacy for the intended task. We generalize these findings in our analysis of the importance of interactivity and specialization for graph visualization systems that are effective and scalable. A principle task in parallel and distributed systems is to reduce the communication load in the interconnection network, as this is usually the major bottleneck for the performance of distributed applications. We introduce a framework for solving online problems that aim to minimize the congestion (i.e. the maximum load of a network link) in general topology networks. We apply this framework to the problem of online routing of virtual circuits and to a dynamic data management problem. For both scenarios we achieve a competitive ratio of O(log sup 3 n) with respect to the congestion of the network links. Our online algorithm for the routing problem has the remarkable property that it is oblivious, i.e., the path chosen for a virtual circuit is independent of the current network load. Oblivious routing strategies can easily be implemented in distributed environments and have therefore been intensively studied for certain network topologies as e.g. meshes, tori and hypercubic networks. This is the first oblivious path selection algorithm that achieves a polylogarithmic competitive ratio in general networks. We construct and analyze a fast broadcasting algorithm working in the presence of Byzantine component faults. Such faults are particularly difficult to deal with, as faulty components may behave arbitrarily (even maliciously) as transmitters, by either blocking, rerouting, or altering transmitted messages in a way most detrimental to the broadcasting process. We assume that links and nodes of a communication network are subject to Byzantine failures, and that faults are distributed randomly and independently, with link failure probability p and node failure probability q, these parameters being constant and satisfying the inequality (1 - p)2(1 - q) > 1 2. A broadcasting algorithm, working in an n-node network, is called almost safe if the probability of its correctness is at least 1 - 1 n, for sufficiently large n. Thus the robustness of the algorithm grows with the size of the network. Our main result is the design and analysis of an almost safe broadcasting algorithm working in time O(log2 n) and using O(n log n) messages in n-node networks. Under a stronger assumption on failure probability parameters, namely (1 - p)2(1 - q)2 > 1 2, our algorithm can be modified to work in time O(log2 n log log n), also using O(n log n) messages. The novelty of our algorithm is that it can cope with the most difficult type of faults, potentially affecting all components of the network (both its links and nodes), and that it is simultaneously robust and efficient. We develop and implement an optimal broadcast algorithm for fully connected processor networks under a bidirectional communication model in which each processor can simultaneously send a message to one processor and receive a message from another, possibly different processor. For any number of processors p the algorithm requires [email protected][email protected]? communication rounds to broadcast N blocks of data from a root processor to the remaining processors, meeting the lower bound in the model. For data of size m, assuming that sending and receiving data of size m^' takes time @[email protected]^', the best running time that can be achieved by the division of m into equal-sized blocks is ((@[email protected]?-1)@[email protected])^2. The algorithm uses a regular, circulant graph communication pattern, and degenerates into a binomial tree broadcast when the number of blocks to be broadcast is one. The algorithm is furthermore well suited to fully connected clusters of SMP (Symmetric Multi-Processor) nodes. The algorithm is implemented as part of an MPI (Message Passing Interface) library. We demonstrate significant practical bandwidth improvements of up to a factor 1.5 over several other, commonly used broadcast algorithms on both a small SMP cluster and a 72 node NEC SX vector supercomputer. We consider broadcasting a message from one node of a tree to all other nodes. In the presence of up to k link failures the tree becomes disconnected, and only nodes in the connected component C containing the source can be informed. The maximum ratio between the time used by a broadcasting scheme B to inform C and the optimal time to inform C, taken over all components C yielded by configurations of at most k faults, is the k-vulnerability of B. This is the maximum slowdown incurred by B due to the lack of a priori knowledge of fault location, for at most k faults. This measure of fault tolerance is similar to the competitive factor of on-line algorithms: in both cases, the performance of an algorithm lacking some crucial information is compared to the performance of an “off-line” algorithm, one that is given this information as input. It is also the first known tool to measure and compare fault tolerance of broadcasting schemes in trees. We seek broadcasting schemes with low vulnerability, working for tree networks. It turns out that schemes that give the best broadcasting time in a fault-free environment may have very high vulnerability, i.e., poor fault tolerance, for some trees. The main result of this paper is an algorithm that, given an arbitrary tree T and an integer k, computes a broadcasting scheme B with lowest possible k-vulnerability among all schemes working for T. Our algorithm has running time O(kn2+n2logn), where n is the size of the tree. We also give an algorithm to find a “universally fault-tolerant” broadcasting scheme in a tree T: one that approximates the lowest possible k-vulnerability, for all k simultaneously. The task of moving data (i.e., the routing problem) in large-scale sensor networks has to contend with several obstacles, including severe power constraints at each node and temporary, but random, failures of nodes, rendering routing schemes designed for traditional communication networks ineffective. We consider the open problem of finding optimum routes between any fixed source-destination pair in a large-scale network, such that the communication load (i.e., the required power) is distributed among all the nodes, the overall latency is minimized, and the algorithm is decentralized and robust. A recent work addressed this problem in the context of a grid topology and showed how to obtain load-balanced routing, but transmissions are restricted to be among near-neighbors and the overall latency grows linearly with the number of nodes. We show how one can route messages between source and destination nodes along random small-world topologies using a decentralized algorithm. Specifically, nodes make connections independently (based only on the source and destination information in the packets), according to a distribution that guarantees an average latency of O(log sup 2 (N)), while preventing hotspot regions by providing an almost uniform distribution of traffic load over all nodes. Surprisingly, the randomized nature of the network structure keeps the average per-node power consumption almost the same as in the case of a grid topology (i.e., local transmissions), while providing an exponential reduction in latency, resulting in a highly fault-tolerant and stable design capable of working in very dynamic environments. Structural fault tolerance (SFT) is the ability of a multiprocessor to reconfigure around faulty processors or links in order to preserve its original processor interconnection structure; In this paper, we focus on the design of SFT multiprocessors that have low switch and link overheads, but can tolerate a very large number of processor faults on the average. Most previous work has concentrated on deterministic k-fault-tolerant (k-FT) designs in which exactly k spare processors and some spare switches and links are added to construct multiprocessors that can tolerate any k processor faults. However, after k faults are reconfigured around, much of the extra links and switches can remain unutilized. It is possible within the basic node-covering framework, which was introduced by Dutt and Hayes as an efficient k-FT design method, to design FT multiprocessors that have the same amount of switches and links as, say, a two-FT deterministic design, but have s spare processors, where s spl Gt 2, so that, on the average, k= spl Theta (s) (k spl les s) processor failures can be reconfigured around. Such designs utilize the spare link and switch capacity very efficiently, and are called probabilistic FT designs. An elegant and powerful method to construct covering graphs or CG's, which are key to obtaining the probabilistic FT designs, is to use linear error-correcting codes (ECCs). We show how to construct probabilistic designs with very high average fault tolerance but low wiring and switch overhead using ECCs like the 2D-parity, full-two, 3D-parity, and full-three codes. Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of (n ) vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed (K 1 ), all components of size greater than (K ) are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous’s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is ‘nearly’ Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erdős–Renyi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from (The barely subcritical regime. Arxiv preprint, 2012) on the size of the largest component in the barely subcritical regime. In this paper, a mathematical model for the beacon-enabled mode of the IEEE 802.15.4 medium-access control (MAC) protocol is provided. A personal area network (PAN) composed of multiple nodes, which transmit data to a PAN coordinator through direct links or multiple hops, is considered. The application is query based: Upon reception of the beacon transmitted by the PAN coordinator, each node tries to transmit its packet using the superframe structure defined by the IEEE 802.15.4 protocol. Those nodes that do not succeed in accessing the channel discard the packet; at the next superframe, a new packet is generated. The aim of the paper is to develop a flexible mathematical tool able to study beacon-enabled 802.15.4 networks organized in different topologies. Both the contention access period (CAP) and the contention-free period defined by the standard are considered. The slotted carrier-sense multiple access with collision avoidance (CSMA CA) algorithm used in the CAP portion of the superframe is analytically modeled. The model describes the probability of packet successful reception and access delay statistics. Moreover, both star and tree-based topologies are dealt with; a suitable comparison between these topologies is provided. The model is a useful tool for the design of MAC parameters and to select the better topology. The mathematical model is validated through simulation results. The model differs from those previously published by other authors in the literature as it precisely follows the MAC procedure defined by the standard in the context of the application scenario described. The aim of this paper is to study communication in networks where nodes fail in a random dependent way. In order to capture fault dependencies, we introduce the neighborhood fault model, where damaging events, called spots, occur randomly and independently with probability p at nodes of a network, and cause faults in the given node and all of its neighbors. Faults at distance at most 2 become dependent in this model and are positively correlated. We investigate the impact of spot probability on feasibility and time of communication in the fault-free part of the network. We show a network which supports fast communication with high probability, if p = 1 c log n. We also show that communication is not feasible with high probability in most classes of networks, for constant spot probabilities. For smaller spot probabilities, high probability communication is supported even by bounded degree networks. It is shown that the torus supports communication with high probability when p decreases faster than 1 n1 2, and does not when p ? 1 O(n1 2). Furthermore, a network built of tori is designed, with the same faulttolerance properties and additionally supporting fast communication. We show, however, that networks of degree bounded by a constant d do not support communication with high probability, if p ? 1 O(n1 d). While communication in networks with independent faults was widely studied, this is the first analytic paper which investigates network communication for random dependent faults. 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. This paper addresses the problem of resilient in-network consensus in the presence of misbehaving nodes. Secure and fault-tolerant consensus algorithms typically assume knowledge of nonlocal information; however, this assumption is not suitable for large-scale dynamic networks. To remedy this, we focus on local strategies that provide resilience to faults and compromised nodes. We design a consensus protocol based on local information that is resilient to worst-case security breaches, assuming the compromised nodes have full knowledge of the network and the intentions of the other nodes. We provide necessary and sufficient conditions for the normal nodes to reach asymptotic consensus despite the influence of the misbehaving nodes under different threat assumptions. We show that traditional metrics such as connectivity are not adequate to characterize the behavior of such algorithms, and develop a novel graph-theoretic property referred to as network robustness. Network robustness formalizes the notion of redundancy of direct information exchange between subsets of nodes in the network, and is a fundamental property for analyzing the behavior of certain distributed algorithms that use only local information. The data center network is increasingly a cost, reliability and performance bottleneck for cloud computing. Although multi-tree topologies can provide scalable bandwidth and traditional routing algorithms can provide eventual fault tolerance, we argue that recovery speed can be dramatically improved through the co-design of the network topology, routing algorithm and failure detector. We create an engineered network and routing protocol that directly address the failure characteristics observed in data centers. At the core of our proposal is a novel network topology that has many of the same desirable properties as FatTrees, but with much better fault recovery properties. We then create a series of failover protocols that benefit from this topology and are designed to cascade and complement each other. The resulting system, F10, can almost instantaneously reestablish connectivity and load balance, even in the presence of multiple failures. Our results show that following network link and switch failures, F10 has less than 1 7th the packet loss of current schemes. A trace-driven evaluation of MapReduce performance shows that F10's lower packet loss yields a median application-level 30 speedup. In the standard CONGEST model for distributed network computing, it is known that "global" tasks such as minimum spanning tree, diameter, and all-pairs shortest paths, consume large bandwidth, for their running-time is @math rounds in @math -node networks with constant diameter. Surprisingly, "local" tasks such as detecting the presence of a 4-cycle as a subgraph also requires @math rounds, even using randomized algorithms, and the best known upper bound for detecting the presence of a 3-cycle is @math rounds. The objective of this paper is to better understand the landscape of such subgraph detection tasks. We show that, in contrast to , which are hard to detect in the CONGEST model, there exists a deterministic algorithm for detecting the presence of a subgraph isomorphic to @math running in a number of rounds, for every tree @math . Our algorithm provides a distributed implementation of a combinatorial technique due to Erd o for sparsening the set of partial solutions kept by the nodes at each round. Our result has important consequences to , i.e., to randomized algorithms whose aim is to distinguish between graphs satisfying a property, and graphs far from satisfying that property. In particular, we get that, for every graph pattern @math composed of an edge and a tree connected in an arbitrary manner, there exists a distributed testing algorithm for @math -freeness, performing in a constant number of rounds. Although the class of graph patterns @math formed by a tree and an edge connected arbitrarily may look artificial, all previous results of the literature concerning testing @math -freeness for classical patterns such as cycles and cliques can be viewed as direct consequences of our result, while our algorithm enables testing more complex patterns.