Group-Solvability

Recent advances in Network Attached Storage (NAS) devices has given rise to research on tasks in which the number of potentially participating processors is not known or even bounded in advance. In many of such situations the output of processors depends upon the group the processor belongs to, rather than upon the individual. Case in point: the renaming task in which processors dynamically acquire unique individual slots. In the group version of the renaming task, processors from the same group are allowed to share a slot. Sharing slots by processors for the same group may be applicable when processors are to post information and processors of the same group possess the same information. The difficulty in reducing the group version to the individual version arises from the fact that in an asynchronous READ-WRITE wait-free model of computation, a group cannot elect a leader to acquire slot on the group’s behalf and post the information. This paper generalizes the notion of a standard task solvability to solvability by groups. It is mainly concerned with solvability by groups of infinite size. It shows that the notion of group solvability by infinite size groups is proper restriction of standard solvability by proving that the Immediate Snapshots task on three processors is not group solvable. The paper’s main technical contribution is in reducing a question about infinite size groups to finite size. It characterizes group solvability of a task Tn over n+1 processors via solvability by groups of size n. Finally, it poses a challenging lower-bound conjecture on a proposed group-solvable version of the renaming task.