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A simple approach for adapting continuous load balancing processes to discrete settings
We consider the neighbourhood load balancing problem. Given a network of processors and an arbitrary distribution of tasks over the network, the goal is to balance load by exchanging tasks between neighbours. In the continuous model, tasks can be arbitrarily divided and perfectly balanced state can always be reached. This is not possible in the discrete model where tasks are non-divisible. In this paper we consider the problem in a very general setting, where the tasks can have arbitrary weights and the nodes can have different speeds. Given a continuous load balancing algorithm that balances the load perfectly in (Formula presented.) rounds, we convert the algorithm into a discrete version. This new algorithm is deterministic and balances the load in (Formula presented.) rounds so that the difference between the average and the maximum load is at most (Formula presented.) , where d is the maximum degree of the network and (Formula presented.) is the maximum weight of any task. For general graphs, these bounds are asymptotically lower compared to the previous results. The proposed conversion scheme can be applied to a wide class of continuous processes, including first and second order diffusion, dimension exchange, and random matching processes. For the case of identical tasks, we present a randomized version of our algorithm that balances the load up to a discrepancy of (Formula presented.) provided that the initial load on every node is large enough.Hoda Akbari, Petra Berenbrink and Thomas Sauerwald work was supported by an NSERC Discovery Grant âAnalysis of Randomized Algorithmsâ
Improved Analysis of Deterministic Load-Balancing Schemes
We consider the problem of deterministic load balancing of tokens in the
discrete model. A set of processors is connected into a -regular
undirected network. In every time step, each processor exchanges some of its
tokens with each of its neighbors in the network. The goal is to minimize the
discrepancy between the number of tokens on the most-loaded and the
least-loaded processor as quickly as possible.
Rabani et al. (1998) present a general technique for the analysis of a wide
class of discrete load balancing algorithms. Their approach is to characterize
the deviation between the actual loads of a discrete balancing algorithm with
the distribution generated by a related Markov chain. The Markov chain can also
be regarded as the underlying model of a continuous diffusion algorithm. Rabani
et al. showed that after time , any algorithm of their
class achieves a discrepancy of , where is the spectral
gap of the transition matrix of the graph, and is the initial load
discrepancy in the system.
In this work we identify some natural additional conditions on deterministic
balancing algorithms, resulting in a class of algorithms reaching a smaller
discrepancy. This class contains well-known algorithms, eg., the Rotor-Router.
Specifically, we introduce the notion of cumulatively fair load-balancing
algorithms where in any interval of consecutive time steps, the total number of
tokens sent out over an edge by a node is the same (up to constants) for all
adjacent edges. We prove that algorithms which are cumulatively fair and where
every node retains a sufficient part of its load in each step, achieve a
discrepancy of in time . We
also show that in general neither of these assumptions may be omitted without
increasing discrepancy. We then show by a combinatorial potential reduction
argument that any cumulatively fair scheme satisfying some additional
assumptions achieves a discrepancy of almost as quickly as the
continuous diffusion process. This positive result applies to some of the
simplest and most natural discrete load balancing schemes.Comment: minor corrections; updated literature overvie
Discrete Load Balancing in Heterogeneous Networks with a Focus on Second-Order Diffusion
In this paper we consider a wide class of discrete diffusion load balancing
algorithms. The problem is defined as follows. We are given an interconnection
network and a number of load items, which are arbitrarily distributed among the
nodes of the network. The goal is to redistribute the load in iterative
discrete steps such that at the end each node has (almost) the same number of
items. In diffusion load balancing nodes are only allowed to balance their load
with their direct neighbors.
We show three main results. Firstly, we present a general framework for
randomly rounding the flow generated by continuous diffusion schemes over the
edges of a graph in order to obtain corresponding discrete schemes. Compared to
the results of Rabani, Sinclair, and Wanka, FOCS'98, which are only valid
w.r.t. the class of homogeneous first order schemes, our framework can be used
to analyze a larger class of diffusion algorithms, such as algorithms for
heterogeneous networks and second order schemes. Secondly, we bound the
deviation between randomized second order schemes and their continuous
counterparts. Finally, we provide a bound for the minimum initial load in a
network that is sufficient to prevent the occurrence of negative load at a node
during the execution of second order diffusion schemes.
Our theoretical results are complemented with extensive simulations on
different graph classes. We show empirically that second order schemes, which
are usually much faster than first order schemes, will not balance the load
completely on a number of networks within reasonable time. However, the maximum
load difference at the end seems to be bounded by a constant value, which can
be further decreased if first order scheme is applied once this value is
achieved by second order scheme.Comment: Full version of paper submitted to ICDCS 201
Competitive function approximation for reinforcement learning
The application of reinforcement learning to problems with continuous domains requires representing the value function by means of function approximation. We identify two aspects of reinforcement learning that make the function approximation process hard: non-stationarity of the target function and biased sampling. Non-stationarity is the result of the bootstrapping nature of dynamic programming where the value function is estimated using its current approximation. Biased sampling occurs when some regions of the state space are visited too often, causing a reiterated updating with similar values which fade out the occasional updates of infrequently sampled regions.
We propose a competitive approach for function approximation where many different local approximators are available at a given input and the one with expectedly best approximation is selected by means of a relevance function. The local nature of the approximators allows their fast adaptation to non-stationary changes and mitigates the biased sampling problem. The coexistence of multiple approximators updated and tried in parallel permits obtaining a good estimation much faster than would be possible with a single approximator. Experiments in different benchmark problems show that the competitive strategy provides a faster and more stable learning than non-competitive approaches.Preprin
Simulator adaptation at runtime for component-based simulation software
Component-based simulation software can provide many opportunities to compose and configure simulators, resulting in an algorithm selection problem for the user of this software. This thesis aims to automate the selection and adaptation of simulators at runtime in an application-independent manner. Further, it explores the potential of tailored and approximate simulators - in this thesis concretely developed for the modeling language ML-Rules - supporting the effectiveness of the adaptation scheme.Komponenten-basierte Simulationssoftware kann viele Möglichkeiten zur Komposition und Konfiguration von Simulatoren bieten und damit zu einem Konfigurationsproblem fĂŒr Nutzer dieser Software fĂŒhren. Das Ziel dieser Arbeit ist die Entwicklung einer generischen und automatisierten Auswahl- und Adaptionsmethode fĂŒr Simulatoren. DarĂŒber hinaus wird das Potential von spezifischen und approximativen Simulatoren anhand der Modellierungssprache ML-Rules untersucht, welche die EffektivitĂ€t des entwickelten Adaptionsmechanismus erhöhen können
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