Analysis of Omni-Do in Asynchronous Partitionable Networks

In the settings where network partitions may interfere with the progress of computation, the challenge is to maintain efficiency in performing the tasks and learning the results of the tasks, despite the dynamically changing group connectivity. However, no amount of algorithmic sophistication can compensate for the possibility of groups of processors or even individual pro- cessors becoming disconnected during the computation. In general, an ad- versary that is able to partition the network into g components will cause any task-performing algorithm to have work Ω(n · g) even if each group of processors performs no more than the optimal number of Ө(n) tasks. In the extreme case where all processors are isolated from the beginning, the work of any algorithm is Ω(n · p). When the network can partition into disconnected components, it is not always sufficient to learn that all tasks are complete (e.g., to solve the Do-All problem). It may also be necessary for the processors in each network component to learn the results of the task completion. Thus here we pursue solutions to the Omni-Do problem (Definition 2.3): Given a set of n tasks and p message-passing processors, each processor must learn the results of all tasks. Even given the pessimistic lower bound of Ω(n · p) on work for partitionable networks, it is desirable to design and analyze efficient algorithmic approaches that can be shown to be better than the oblivious approach where each processor or each group performs all tasks. In particular, it is important to develop complexity bounds that are failure-sensitive, namely that capture the dependence of work complexity on the nature of network partitions. In this chapter we present an asynchronous Omni-Do algorithm, called AX, and we show that it is optimal in terms of worst case task-oriented work, under network fragmentations and merges. The algorithm uses a group communication service to provide membership and communication services.


Inductive Hypothesis Directed Acyclic Graph History Variable Message Complexity Liveness Property 
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© Springer Science+Business Media, LLC 2008

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