Consensus problems occur in many contexts and have therefore been intensively studied in the past. In the standard consensus problem there are n processes with possibly different input values and the goal is to eventually reach a point at which all processes commit to exactly one of these values. We are studying a slight variant of the consensus problem called the stabilizing consensus problem [1]. In this problem, we do not require that each process commits to a final value at some point, but that eventually they arrive at a common value without necessarily being aware of that. This should work irrespective of the states in which the processes are starting. Coming up with a self-stabilizing rule is easy without adversarial involvement, but we allow some T -bounded adversary to manipulate any T processes at any time. In this situation, a perfect consensus is impossible to reach, so we only require that there is a time point t and value v so that at any point after t, all but up to O(T ) processes agree on v, which we call an almost stable consensus. As we will demonstrate, there is a surprisingly simple rule for the standard message passing model that just needs O(log n log log n) time for any √ n-bounded adversary and just O(log n) time without adversarial involvement, with high probability, to reach an (almost) stable consensus from any initial state. A stable consensus is reached, with high probability, in the absence of adversarial involvement.Dagstuhl Seminar Proceedings 09371 Algorithmic Methods for Distributed Cooperative Systems

We empirically analyze two versions of the well-known "randomized rumor spreading" protocol to disseminate a piece of information in networks. In the classical model, in each round, each informed node informs a random neighbor. In the recently proposed quasirandom variant, each node has a (cyclic) list of its neighbors. Once informed, it starts at a random position of the list, but from then on informs its neighbors in the order of the list.While for sparse random graphs a better performance of the quasirandom model could be proven, all other results show that, independent of the structure of the lists, the same asymptotic performance guarantees hold as for the classical model.In this work, we compare the two models experimentally. Not only does this show that the quasirandom model generally is faster, but it also shows that the runtime is more concentrated around the mean. This is surprising given that much fewer random bits are used in the quasirandom process.These advantages are also observed in a lossy communication model, where each transmission does not reach its target with a certain probability, and in an asynchronous model, where nodes send at random times drawn from an exponential distribution. We also show that typically the particular structure of the lists has little influence on the efficiency.

We consider the problem of balancing load items (tokens) in networks. Starting with an arbitrary load distribution, we allow nodes to exchange tokens with their neighbors in each round. The goal is to achieve a distribution where all nodes have nearly the same number of tokens.For the continuous case where tokens are arbitrarily divisible, most load balancing schemes correspond to Markov chains, whose convergence is fairly well-understood in terms of their spectral gap. However, in many applications, load items cannot be divided arbitrarily, and we need to deal with the discrete case where the load is composed of indivisible tokens. This discretization entails a non-linear behavior due to its rounding errors, which makes this analysis much harder than in the continuous case.We investigate several randomized protocols for different communication models in the discrete case. As our main result, we prove that for any regular network in the matching model, all nodes have the same load up to an additive constant in (asymptotically) the same number of rounds as required in the continuous case. This generalizes and tightens the previous best result, which only holds for expander graphs [17], and demonstrates that there is almost no difference between the discrete and continuous cases. Our results also provide a positive answer to the question of how well discrete load balancing can be approximated by (continuous) Markov chains, which has been posed by many researchers, e.g., [23,29,32]. * A preliminary version of this work appeared in

a b s t r a c tWe study the cover time of multiple random walks on undirected graphs G = (V , E). We consider k parallel, independent random walks that start from the same vertex. The speedup is defined as the ratio of the cover time of a single random walk to the cover time of these k random walks. Recently, Alon et al. (2008) [5] derived several upper bounds on the cover time, which imply a speed-up of Ω(k) for several graphs; however, for many of them, k has to be bounded by O(log n). They also conjectured that, for any 1 ⩽ k ⩽ n, the speed-up is at most O(k) on any graph. We prove the following main results:• We present a new lower bound on the speed-up that depends on the mixing time. It gives a speed-up of Ω(k) on many graphs, even if k is as large as n. • We prove that the speed-up is O(k log n) on any graph. For a large class of graphs we can also improve this bound to O(k), matching the conjecture of Alon et al.• We determine the order of the speed-up for any value of 1 ⩽ k ⩽ n on hypercubes, random graphs and degree restricted expanders. For d-dimensional tori with d > 2, our bounds are tight up to logarithmic factors.• Our findings also reveal a surprisingly sharp threshold behaviour for certain graphs, e.g., the d-dimensional torus with d > 2 and hypercubes: there is a value T such that the speed-up is approximately min{T , k} for any 1 ⩽ k ⩽ n.

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