Competitive Analysis of Omni-Do in Partitionable Networks

The efficiency of an algorithm solving the Omni-Do problem can only be partially understood through its worst case work analysis, such as we did for algorithm AX in the previous chapter. This is because the worst case upper and lower bounds might depend on unusual or extreme patterns of regroup- ings. In such cases, worst case work may not be the best way to compare the efficiency of algorithms. Hence, in this chapter, in order to understand better the practical implications of performing work in partitionable settings, we treat the Omni-Do problem as an on-line problem and we pursue compet- itive analysis, that is we compare the efficiency of a given algorithm to the efficiency of an “off-line” algorithm that has full knowledge of future changes in the communication medium. We consider asynchronous processors under arbitrary patterns of regroupings (including, but not limited to, fragmenta- tion and merges). A processor crash is modeled as the creation of a singleton group (containing the crashed processor) that remains disconnected for the entire computation; the processors in such groups are charged for complet- ing all remaining tasks, in other words, the analysis assumes the worst case situation where a crashed processor becomes disconnected, but manages to complete all tasks before the crash. In this chapter we view algorithms as a rule that, given a group of proces- sors and a set of tasks known by this group to be completed, determines a task for the group to complete next. We assume that task executions are atomic with respect to regroupings (a task considered for execution by a group is either executed or not prior a subsequent regrouping). Processors in the same group can share their knowledge of completed tasks and, while they remain connected, avoid doing redundant work. The challenge is to avoid redundant work “globally”, in the sense that processors should be performing tasks with anticipation of future changes in the network topology. An optimal algorithm, with full knowledge of the future regroupings, can schedule the execution of the tasks in each group in such a way that the overall task-oriented work is the smallest possible, given the particular sequence of regroupings.