Thomas Sauerwald

Quasirandom Rumor Spreading: Expanders, Push vs. Pull, and Robustness

Randomized rumor spreading is an efficient protocol to distribute information in networks. Recently, a quasirandom version has been proposed and proven to work equally well on many graphs and better for sparse random graphs. In this work we show three main results for the quasirandom rumor spreading model. We exhibit a natural expansion property for networks which suffices to make quasirandom rumor spreading inform all nodes of the network in logarithmic time with high probability. This expansion property is satisfied, among others, by many expander graphs, random regular graphs, and Erdős-Renyi random graphs. For all network topologies, we show that if one of the push or pull model works well, so does the other. We also show that quasirandom rumor spreading is robust against transmission failures. If each message sent out gets lost with probability f , then the runtime increases only by a factor of

Stabilizing consensus with the power of two choices

\O(1/(1-f))

Tight bounds for the cover time of multiple random walks

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On the runtime and robustness of randomized broadcasting

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 [2]. 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, stable value without necessarily being aware of that. This should work irrespective of the states in which the processes are starting. Our main result is a simple randomized algorithm called median rule that, with high probability, just needs O(log m log log n + log n) time and work per process to arrive at an almost stable consensus for any set of m legal values as long as an adversary can corrupt the states of at most √n processes at any time. Without adversarial involvement, just O(log n) time and work is needed for a stable consensus, with high probability. As a by-product, we obtain a simple distributed algorithm for approximating the median of n numbers in time O(log m log log n + log n) under adversarial presence.

Broadcasting vs. mixing and information dissemination on Cayley graphs

We 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 speed-up 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 @W(k) for several graphs; however, for many of them, k has to be bounded by O(logn). They also conjectured that, for any 1=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=

Counting arbitrary subgraphs in data streams

In this paper, we study the following randomized broadcasting protocol. At some time t an information r is placed at one of the nodes of a graph. In the succeeding steps, each informed node chooses one neighbor, independently and uniformly at random, and informs this neighbor by sending a copy of r to it. We begin by developing tight lower and upper bounds on the runtime of the algorithm described above. First, it is shown that on ?-regular graphs this algorithm requires at least log2?1?n+log(???1)?n?o(logn)?1.69log2n rounds to inform all n nodes. Together with a result of Pittel B. Pittel, On spreading a rumor, SIAM Journal on Applied Mathematics, 47 (1) (1987) 213?223 this bound implies that the algorithm has the best performance on complete graphs among all regular graphs. For general graphs, we prove a slightly weaker lower bound of log2?1?n+log4n?o(logn)?1.5log2n, where ? denotes the maximum degree of G. We also prove two general upper bounds, (1+o(1))nlnn and O(n??), respectively, where ? denotes the minimum degree.The second part of this paper is devoted to the analysis of fault-tolerance. We show that if the informed nodes are allowed to fail in some step with probability 1?p, then the broadcasting time increases by at most a factor 6/p. As a by-product, we determine the performance of agent based broadcasting in certain graphs and obtain bounds for the runtime of randomized broadcasting on Cartesian products of graphs.

Near-perfect load balancing by randomized rounding

One frequently studied problem in the context of information dissemination in communication networks is the broadcasting problem. In this paper, we study the following randomized broadcasting protocol: At some time t an information r is placed at one of the nodes of a graph G. In the succeeding steps, each informed node chooses one neighbor, independently and uniformly at random, and informs this neighbor by sending a copy of r to it. First, we consider the relationship between randomized broadcasting and random walks on graphs. In particular, we prove that the runtime of the algorithm described above is upper bounded by the corresponding mixing time, up to a logarithmic factor. One key ingredient of our proofs is the analysis of a continuous-type version of the afore mentioned algorithm, which might be of independent interest. Then, we introduce a general class of Cayley graphs, including (among others) Star graphs, Transposition graphs, and Pancake graphs. We show that randomized broadcasting has optimal runtime on all graphs belonging to this class. Finally, we develop a new proof technique by combining martingale tail estimates with combinatorial methods. Using this approach, we show the optimality of our algorithm on another Cayley graph and obtain new knowledge about the runtime distribution on several Cayley graphs.

Quasirandom load balancing

We study the subgraph counting problem in data streams. We provide the first non-trivial estimator for approximately counting the number of occurrences of an arbitrary subgraph H of constant size in a (large) graph G. Our estimator works in the turnstile model, i.e., can handle both edge-insertions and edge-deletions, and is applicable in a distributed setting. Prior to this work, only for a few non-regular graphs estimators were known in case of edge-insertions, leaving the problem of counting general subgraphs in the turnstile model wide open. We further demonstrate the applicability of our estimator by analyzing its concentration for several graphs H and the case where G is a power law graph.

Tight Bounds for the Cover Time of Multiple Random Walks

We consider and analyze a new algorithm for balancing indivisible loads on a distributed network with n processors. The aim is minimizing the discrepancy between the maximum and minimum load. In every time-step paired processors balance their load as evenly as possible. The direction of the excess token is chosen according to a randomized rounding of the participating loads. We prove that in comparison to the corresponding model of Rabani, Sinclair, and Wanka (1998) with arbitrary roundings, the randomization yields an improvement of roughly a square root of the achieved discrepancy in the same number of time-steps on all graphs. For the important case of expanders we can even achieve a constant discrepancy in O(log n (log log n)3) rounds. This is optimal up to loglog-factors while the best previous algorithms in this setting either require ©(log2 n) time or can only achieve a logarithmic discrepancy. Our new result also demonstrates that with randomized rounding the difference between discrete and continuous load balancing vanishes almost completely.

Tight Bounds for Randomized Load Balancing on Arbitrary Network Topologies

We propose a simple distributed algorithm for balancing indivisible tokens on graphs. The algorithm is completely deterministic, though it tries to imitate (and enhance) a randomized algorithm by keeping the accumulated rounding errors as small as possible. Our new algorithm, surprisingly, closely approximates the idealized process (where the tokens are divisible) on important network topologies. On