An analysis of diffusive load-balancing

Subramanian, Raghu; Scherson, Isaac D.

doi:10.1145/181014.181361

Cited by 70 publications

(55 citation statements)

References 9 publications

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“…Convergence conditions have also been established for distributed consensus on dynamically changing graphs (e.g., [13,20,26,34]) and with asynchronous communication and computation ( [2]; see also the early work in [31,32]). Other work gives bounds on the convergence factor W − J for a particular choice of weights, in terms of various geometric quantities such as conductance (e.g., [30]). When W ij are nonnegative, the model (1) corresponds to a symmetric Markov chain on the graph, and W − J is the second largest eigenvalue magnitude (SLEM) of the Markov chain, which is a measure of mixing time (see, e.g., [6,8,10]).…”

Section: X(t + 1) − J X(0) W − J X(t) − J X(0) mentioning

confidence: 99%

Distributed average consensus with least-mean-square deviation

Xiao

Boyd

Kim

2007

Journal of Parallel and Distributed Computing

976

732

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We consider a stochastic model for distributed average consensus, which arises in applications such as load balancing for parallel processors, distributed coordination of mobile autonomous agents, and network synchronization. In this model, each node updates its local variable with a weighted average of its neighbors' values, and each new value is corrupted by an additive noise with zero mean. The quality of consensus can be measured by the total mean-square deviation of the individual variables from their average, which converges to a steady-state value. We consider the problem of finding the (symmetric) edge weights that result in the least mean-square deviation in steady state. We show that this problem can be cast as a convex optimization problem, so the global solution can be found efficiently. We describe some computational methods for solving this problem, and compare the weights and the mean-square deviations obtained by this method and several other weight design methods.

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Section: X(t + 1) − J X(0) W − J X(t) − J X(0) mentioning

confidence: 99%

Distributed average consensus with least-mean-square deviation

Xiao

Boyd

Kim

2007

Journal of Parallel and Distributed Computing

976

732

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“…Cybenko [3] (see also [13,15]) shows a tight connection between the convergence rate of the diffusion algorithm and the absolute value of the second largest eigenvalue λ max of the diffusion matrix P with P ij = 1/(d + 1) if {i, j} ∈ E. Subramanian and Scherson [15] observe similar relations between convergence time and certain properties of the underlying network like electrical and fluid conductance.…”

Section: Continuous Diffusionmentioning

confidence: 92%

“…Note that this method balances the load perfectly if the number of steps is sufficiently large. Here we consider the (arguably more realistic [15]) case of discrete diffusion where tokens are indivisible. Quantifying by how much the integrality assumption decreases the efficiency of load balancing is an interesting question and has been posed by many authors (e.g., [8,[11][12][13][14][15]).…”

Section: Introductionmentioning

confidence: 99%

Randomized diffusion for indivisible loads

Berenbrink

Cooper

Friedetzky

et al. 2015

Journal of Computer and System Sciences

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe present a new randomized diffusion-based algorithm for balancing indivisible tasks (tokens) on a network. Our aim is to minimize the discrepancy between the maximum and minimum load. The algorithm works as follows. Every vertex distributes its tokens as evenly as possible among its neighbors and itself. If this is not possible without splitting some tokens, the vertex redistributes its excess tokens among all its neighbors randomly (without replacement).In this paper we prove several upper bounds on the load discrepancy for general networks. These bounds depend on some expansion properties of the network, that is, the second largest eigenvalue, and a novel measure which we refer to as refined local divergence. We then apply these general bounds to obtain results for some specific networks. For constant-degree expanders and torus graphs, these yield exponential improvements on the discrepancy bounds compared to the algorithm of Rabani, Sinclair, and Wanka [14]. For hypercubes we obtain a polynomial improvement. In contrast to previous papers, our algorithm is vertex-based and not edge-based. This means excess tokens are assigned to vertices instead to edges, and the vertex reallocates all of its excess tokens by itself. This approach avoids nodes having "negative loads" (like in [8,10]), but causes additional dependencies for the analysis.

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“…A very natural question is how much this integrality assumption decreases the efficiency of load balancing. In fact, finding a precise quantitative relationship between the discrete and the idealized case is an open problem posed by many authors, e.g., [9,11,15,16,28,31,33,36].…”

mentioning

confidence: 99%

“…In the natural diffusion paradigm, an arbitrary amount of load can be sent along each edge at each step [31,33]. For the idealized case of divisible load, a popular diffusion algorithm is the first-order-scheme by Subramanian and Scherson [36] whose convergence rate is fairly well captured in terms of the spectral gap [27].…”

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confidence: 99%

Quasirandom Load Balancing

Friedrich¹,

Gairing²,

Sauerwald

2012

SIAM J. Comput.

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Abstract. 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 d-dimensional torus graphs with n nodes it deviates from the idealized process only by an additive constant. In contrast, the randomized rounding approach of Friedrich and Sauerwald [11] can deviate up to Ω(polylog(n)) and the deterministic algorithm of Rabani, Sinclair and Wanka [33] has a deviation of Ω(n 1/d ). This makes our quasirandom algorithm the first known algorithm for this setting which is optimal both in time and achieved smoothness. We further show that on the hypercube as well, our algorithm has a smaller deviation from the idealized process than the previous algorithms.To prove these results, we derive several combinatorial and probabilistic results that we believe to be of independent interest. In particular, we show that first-passage probabilities of a random walk on a path with arbitrary weights can be expressed as a convolution of independent geometric probability distributions.1. Introduction. Load balancing is a requisite for the efficient utilization of computational resources in parallel and distributed systems. The aim is to reallocate the load such that afterward, each node has approximately the same load. Load balancing problems have various applications, e.g., for scheduling [37], routing [5], and numerical computation [38,39].Typically, load balancing algorithms iteratively exchange load along edges of an undirected connected graph. In the natural diffusion paradigm, an arbitrary amount of load can be sent along each edge at each step [31,33]. For the idealized case of divisible load, a popular diffusion algorithm is the first-order-scheme by Subramanian and Scherson [36] whose convergence rate is fairly well captured in terms of the spectral gap [27].However, for many applications the assumption of divisible load may be invalid. Therefore, we consider the discrete case where the load can only be decomposed into indivisible unit-size tokens. A very natural question is how much this integrality assumption decreases the efficiency of load balancing. In fact, finding a precise quantitative relationship between the discrete and the idealized case is an open problem posed by many authors, e.g., [9,11,15,16,28,31,33,36].A simple method for approximating the idealized process was analyzed by Rabani, Sinclair, and Wanka [33]. Their approach (which we will call "RSW algorithm") is to round down the fractional flow of the idealized process. They introduce a very useful parameter of the graph called local divergence and prove that it gives tight upper bounds on the deviation between the idealized process and their discrete process. However, one drawback of the RSW algorithm is that it can ...

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An analysis of diffusive load-balancing

Cited by 70 publications

References 9 publications

Distributed average consensus with least-mean-square deviation

Distributed average consensus with least-mean-square deviation

Randomized diffusion for indivisible loads

Quasirandom Load Balancing

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