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Motivated by the increasing need to understand the distributed algorithmic foundations of largescale graph computations, we study some fundamental graph problems in a messagepassing model for distributed computing where k ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n >> k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many realworld systems. Communication is pointtopoint, and the goal is to minimize the number of communication rounds of the computation. Our main contribution is the General Lower Bound Theorem , a theorem that can be used to show nontrivial lower bounds on the round complexity of distributed largescale data computations. This result is established via an informationtheoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic, and this theorem can be used in a “cookbook” fashion to show distributed lower bounds for several problems, including nongraph problems. We present two applications by showing (almost) tight lower bounds on the round complexity of two fundamental graph problems, namely, PageRank computation and triangle enumeration . These applications show that our approach can yield lower boundsmore »

We study the communication cost (or message complexity) of fundamental distributed symmetry breaking problems, namely, coloring and MIS. While significant progress has been made in understanding and improving the running time of such problems, much less is known about the message complexity of these problems. In fact, all known algorithms need at least Ω(m) communication for these problems, where m is the number of edges in the graph. We addressthe following question in this paper: can we solve problems such as coloring and MIS using sublinear, i.e., o(m) communication, and if sounder what conditions? In a classical result, Awerbuch, Goldreich, Peleg, and Vainish [JACM 1990] showed that fundamental global problems such asbroadcast and spanning tree construction require at least o(m) messages in the KT1 Congest model (i.e., Congest model in which nodes have initial knowledge of the neighbors' ID's) when algorithms are restricted to be comparisonbased (i.e., algorithms inwhich node ID's can only be compared). Thirty five years after this result, King, Kutten, and Thorup [PODC 2015] showed that onecan solve the above problems using Õ(n) messages (n is the number of nodes in the graph) in Õ(n) rounds in the KT1 Congest model if noncomparisonbased algorithms are permitted. Anmore »

SUMMARY Lamellar magnetism is a source of remanent magnetization in natural rocks different from common bulk magnetic moments in ferrimagnetic minerals. It has been found to be a source for a wide class of magnetic anomalies with extremely high Koenigsberger ratio. Its physical origin are uncompensated interface moments in contact layers of nanoscale ilmenite lamellae inside an hematite host, which also generate unusual lowtemperature (lowT) magnetic properties, such as shifted lowT hysteresis loops due to exchange bias. The atomicmagnetic basis for the exchange bias discovered in the hematiteilmenite system is explored in a series of papers. In this third article of the series, simple models are developed for lamellae interactions of different structures when samples are either cooled in zerofield, or fieldcooled in 5 T to temperatures below the ordering temperature of ilmenite. These models are built on the lowtemperature measurements described earlier in Paper II. The important observations include: (i) the effects of lamellar shapes on magnetic coupling, (ii) the highT acquisition of lamellar magnetism and lowT acquisition of magnetization of ilmenite lamellae, (iii) the intensity of lamellar magnetism and the consequent ilmenite magnetism in populations of randomly oriented crystals, (iv) latticepreferred orientation of the titanohematite host crystal populations andmore »

In this paper, we study faulttolerant distributed consensus in wireless systems. In more detail, we produce two new randomized algorithms that solve this problem in the abstract MAC layer model, which captures the basic interface and communication guarantees provided by most wireless MAC layers. Our algorithms work for any number of failures, require no advance knowledge of the network participants or network size, and guarantee termination with high probability after a number of broadcasts that are polynomial in the network size. Our first algorithm satisfies the standard agreement property, while our second trades a faster termination guarantee in exchange for a looser agreement property in which most nodes agree on the same value. These are the first known faulttolerant consensus algorithms for this model. In addition to our main upper bound results, we explore the gap between the abstract MAC layer and the standard asynchronous message passing model by proving faulttolerant consensus is impossible in the latter in the absence of information regarding the network participants, even if we assume no faults, allow randomized solutions, and provide the algorithm a constantfactor approximation of the network size.