Delimiting the power of bounded size synchronization objects

Theoretically, various shared synchronization objects, such as compare&swap and arbitrary read-modify-write registers, are universal [10, 20]. That is, any sequentially specified task can be solved in a concurrent system that supports these objects and and a large enough number of shared read/write registers. Are these objects indeed almighty? Or, are there other considerations that have to be kept in mind when analyzing their computation power. In this paper we show that progressively larger objects of these types are more powerful (larger in the number of different values they can hold). This provides a refinement of Herlihy's hierarchy. We consider a shared memory system with unbounded read/write memory and a size κ compare&swap register. Let nk be the maximum number of processes that can elect a leader in such a system (in a wait-free manner). In [1] we present an election algorithm for O{k!) processes in such a system, i.e. showing that nk, is at least O(k!). However, on the lower bound side only n₁,n₂, and n3 were shown to be bounded [1, 10, 18], while for k > 3 it was not known whether such a bound exists. Here we prove that for any k, nk is bounded by O(κ(k²+3)) that is, at most O(κ(k²+3) processes can elect a leader in such a system¹. Hence, the more values a strong shared memory object can hold the stronger it is! The proof of the lower bound (lower bound on space, which is an upper bound on number of processes) combines several techniques that were recently developed with novel new techniques, which are interesting on their own.