A simple approach for adapting continuous load balancing processes to discrete settings

Abstract

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 T rounds, we convert the algorithm into a discrete version. This new algorithm is deterministic and balances the load in T rounds so that the difference between the average and the maximum load is at most 2d • w max , where d is the maximum degree of the network and w max 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 O(√ d log n) provided that the initial load on every node is large enough.

A small world based overlay network for improving dynamic load-balancing

“…The objective of the user is to allocate his/her task to a processor with minimum load, where the load of a processor is defined as the weight of its tasks divided by its speed. Neighborhood load balancing algorithms (Akbari et al, 2012) are diffusion algorithm that have the advantage that they are very simple and that the vertices do not need any global information to base their balancing decisions on.…”

Improved Analysis of Deterministic Load-Balancing Schemes

“…The only other result in the literature known to the authors which achieves a discrepancy of O(d) in O(T ) steps in the diffusive model is the one of [4] (see [18] for O(d) discrepancy in the dimension exchange model where nodes balance with only one neighbor per round). Their algorithms simulate and mimic (with the discrete tokens) the continuous flow.…”

“…In [4], the authors propose an algorithm that achieves discrepancy of 2d after T steps for any graph. For every edge e and step t, their algorithm calculates the number of tokens that should be sent over e in t such that the total number of tokens forwarded over e (over the first t steps) stays as close as possible to the amount of load that is sent by the continuous algorithm over e during the first t steps.…”

“…Within the class of schemes considered in [17], the bound of O(d log n/µ) on discrepancy cannot be improved for many important graph classes, such as constant-degree expanders. Since the work [17], many different refinements and variants of this approach have been proposed [4,9,10,18], as well as extensions to other models, including systems with non-uniform tokens [4] and non-uniform machines [2].…”

Ergodic Effects in Token Circulation

“…In [33], a process that sends x(u)/(d + 1) or x(u)/(d + 1) load over each edge is shown to achieve vertex discrepancy of O(d log n/µ) after O(log(Kn)/µ) steps for d-regular graphs, where K is the initial load discrepancy and µ is the spectral gap of the transition matrix of the underlying Markov chain. Since [33], many variants of discretization of diffusive process have been proposed, see [4,8,21,22,35].…”