Robert Elsässer

Diffusion Schemes for Load Balancing on Heterogeneous Networks

Several different diffusion schemes have previously been developed for load balancing on homogeneous processor networks. We generalize existing schemes, in order to deal with heterogeneous networks.Generalized schemes may operate efficiently on networks where each processor can have arbitrary computing power, i.e., the load will be balanced proportionally to these powers. The balancing flow that is calculated by schemes for homogeneous networks is minimal with regard to the l2 -norm and we prove this to hold true for generalized schemes, too. We demonstrate the usability of generalized schemes by a number of experiments on several heterogeneous networks.

On the communication complexity of randomized broadcasting in random-like graphs

Broadcasting algorithms have a various range of applications in different fields of computer science. In this paper we analyze the number of message transmissions generated by efficient randomized broadcasting algorithms in random-like networks. We mainly consider the classical random graph model, i.e., a graph <i>G_p</i> with <i>n</i> nodes in which any two arbitrary nodes are connected with probability <i>p</i>, independently. For these graphs, we present an efficient broadcasting algorithm based on the random phone call model introduced by Karp et al. [21], and show that the total number of message transmissions generated by this algorithm is bounded by an asymptotically optimal value in almost all connected random graphs. More precisely, we show that if <i>p</i> ≥ log^δ <i>n</i>/<i>n</i> for some constant δ > 2, then we are able to broadcast any information <i>r</i> in a random graph <i>G_p</i> of size <i>n</i> in <i>O</i>(log <i>n</i>) steps by using at most <i>O</i>(<i>n</i> max{log log <i>n</i>, log <i>n</i>/ log <i>d</i>}) transmissions related to <i>r</i>, where <i>d</i> = <i>pn</i> denotes the expected average degree in <i>G_p</i>. We also show that for these kind of graphs there is a a matching lower bound on the number of transmissions generated by any efficient broadcasting algorithm which works within the limits of the random phone call model. Please note that the main result holds with probability 1-1/<i>n</i>^Ω(1), even if <i>n</i> and <i>d</i> are unknown to the nodes of the graph.The algorithm we present in this paper is based on a simple communication model [21], is scalable, and robust. It can efficiently handle restricted communication failures and certain changes in the size of the network, and can also be extended to certain types of truncated power law graphs based on the models of [1, 2, 5]. In addition, our methods and results might be useful for further research on this field.

Tight bounds for the cover time of multiple random walks

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=

On the runtime and robustness of randomized broadcasting

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.

Broadcasting vs. mixing and information dissemination on Cayley graphs

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.

Diffusive load balancing schemes on heterogeneous networks

Up to now, diffusive load balancing schemes have only been developed for homogeneous networks. We generalize existing diffusion schemes, in order to deal with heterogeneous networks. In these networks, every processor can have arbitrary computing power, and the load has to be balanced proportionally to these weights. The balancing flow that is calculated by the schemes for homogeneous networks is minimal with regard to the l2-norm and we prove this to hold true for the generalized schemes, too. By means of a number of experiments we demonstrate the usability of the generalized schemes on heterogeneous networks.

The Power of Two Choices in Distributed Voting

Distributed voting is a fundamental topic in distributed computing. In pull voting, in each step every vertex chooses a neighbour uniformly at random, and adopts its opinion. The voting is completed when all vertices hold the same opinion. On many graph classes including regular graphs, pull voting requires Ω(n) expected steps to complete, even if initially there are only two distinct opinions.

Distributing Unit Size Workload Packages in Heterogeneous Networks

The task of balancing dynamically generated work load occurs in a wide range of parallel and distributed applications. Diffusion based schemes, which belong to the class of nearest neighbor load balancing algorithms, are a popular way to address this problem. Originally created to equalize the amount of arbitrarily divisible load among the nodes of a static and homogeneous network, they have been generalized to heterogeneous topologies. Additionally, some simple diffusion algorithms have been adapted to work in dynamic networks as well. However, if the load is not divisible arbitrarily but consists of indivisible unit size tokens, diffusion schemes are not able to balance the load properly. In this paper we consider the problem of balancing indivisible unit size tokens on heterogeneous systems. By modifying a randomized strategy invented for homogeneous systems, we can achieve an asymptotically minimal expected overload in l1, l2 and l1 norm while only slightly increasing the run-time by a logarithmic factor. Our experiments show that this additional factor is usually not required in applications.

Load balancing in dynamic networks

Efficient load balancing algorithms are the key to many efficient parallel applications. Until now, research in this area mainly focused on static networks. However, observations show that diffusive algorithms, originally designed for these networks, can also be applied in nonstatic scenarios. In this paper we prove that the general diffusion scheme can be deployed on dynamic networks and show that its convergence rate depends on the average value of the quotient of the second smallest eigenvalue and the maximum vertex degree of the networks occurring during the iterations. In the presented experiments we illustrate that even if communication links of static networks fail with high probability, load can still be balanced quite efficiently. Simulating diffusion on ad-hoc networks we demonstrate that diffusive schemes provide a reliable and efficient load balancing strategy also in mobile environments.

Tight Bounds for the Cover Time of Multiple Random Walks

We study the cover time of multiple random walks. Given a graph G of n vertices, assume that k independent random walks start from the same vertex. The parameter of interest is the speed-up defined as the ratio between the cover time of one and the cover time of k random walks. Recently Alon et al. developed several bounds that are based on the quotient between the cover time and maximum hitting times. Their technique gives a speed-up of *** (k ) on many graphs, however, for many graph classes, k has to be bounded by