Discrete distributed load balancing

The neighbourhood load balancing problem considers a network along with a distribution of tasks over its nodes. The aim is to minimize discrepancy which is the difference between the maximum and the minimum load. This is done by a distributed process that exchanges load between neighbouring nodes. One distinguishes between the continuous model - where tasks can be split arbitrarily - and the more practical discrete model in which tasks are atomic. The former has been comprehensively studied in early results [20, 57, 29]. In the latter model, various algorithms have been developed and analyzed [10, 9, 27, 28, 41, 39, 44, 57, 62, 64]. Nevertheless, several aspects are yet to be explored, such as devising new discrete algorithms, and generalizing or tightening the existing analyses. We study the problem of discrete neighbourhood load balancing. We propose a new approach that can transform continuous load balancing algorithms into deterministic or randomized discrete versions. Despite its simplicity, the proposed approach works in quite general settings (arbitrary network topology, weighted tasks and heterogeneous processors) and in many cases achieves improved discrepancy bounds. We also study the usage of rotor-router walks - deterministic analogue of random walks - for discrete load balancing, and in particular, derandomization of existing randomized approaches. After that, we obtain discrepancy bounds for deterministic and randomized discrete second-order processes [57], where the amount of load transferred in each round depends both on the current load distribution and the amount of load transferred in the previous round. In second-order processes a node may attempt to send out more load than its available load. To address this issue, we provide bounds on the minimum load of any node which is sufficient to prevent such conditions.