Work-Competitive Scheduling for Cooperative Computing with Dynamic Groups

• Published in: SIAM journal on computing Vol. 34; no. 4; pp. 848 - 862
• Main Authors: Georgiou, Chryssis, Russell, Alexander, Shvartsman, Alexander A.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2005
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Communication; Competition; Computer science; Computer science; control theory; systems; Computer systems performance. Reliability; Distributed processing; Exact sciences and technology; Knowledge; Scheduling; Software; Theoretical computing; 68W20; Lower bound; 68Q25; 68W40; Task complexity; partitionable networks; independent tasks; Competitiveness; on-line algorithms; distributed computation; Scheduling; Independent task; Computing; Distributed computing; Competitive algorithms; Upper bound; Partitioning; Algorithm performance; Multiplicity; randomized algorithms; work complexity 68W15; Counting; 68Q85; competitive analysis
• ISSN: 0097-5397; 1095-7111
• DOI: 10.1137/S0097539704440442

The problem of cooperatively performing a set of t tasks in a decentralized computing environment subject to failures is one of the fundamental problems in distributed computing. The setting with partitionable networks is especially challenging, as algorithmic solutions must accommodate the possibility that groups of processors become disconnected (and, perhaps, reconnected) during the computation. The efficiency of task-performing algorithms is often assessed in terms of work: the total number of tasks, counting multiplicities, performed by all of the processors during the computation. In general, the scenario where the processors are partitioned into g disconnected components causes any task-performing algorithm to have work $\Omega(t\cdot g)$ even if each group of processors performs no more than the optimal number of $\Theta(t)$ tasks. Given that such pessimistic lower bounds apply to any scheduling algorithm, we pursue a competitive analysis. Specifically, this paper studies a simple randomized scheduling algorithm for p asynchronous processors, connected by a dynamically changing communication medium, to complete t known tasks. The performance of this algorithm is compared against that of an omniscient off-line algorithm with full knowledge of the future changes in the communication medium. The paper describes a notion of computation width, which associates a natural number with a history of changes in the communication medium, and shows both upper and lower bounds on work-competitiveness in terms of this quantity. Specifically, it is shown that the simple randomized algorithm obtains the competitive ratio $(1+\mathbf{cw}/e)$, where $\mathbf{cw}$ is the computation width and $e$ is the base of the natural logarithm ($e=2.7182\ldots$); this competitive ratio is then shown to be tight.