Lower bounds for restricted-use objects

Concurrent objects play a key role in the design of applications for multi-core architectures, making it imperative to precisely understand their complexity requirements. For some objects, it is known that implementations can be significantly more efficient when their usage is restricted. However, apart from the specific restriction of one-shot implementations, where each process may apply only a single operation to the object, very little is known about the complexities of objects under general restrictions. This paper draws a more complete picture by defining a large class of objects for which an operation applied to the object can be “perturbed ” L consecutive times, and by proving lower bounds on their space complexity and on the time complexity of deterministic implementations of such objects. This class includes bounded-value max registers, limited-use approximate and exact counters, and limited-use collect and compare-and-swap objects; L depends on the number of times the object can be accessed or the maximum value it can support. For n-process implementations that use only historyless primitives, we prove Ω(min(L, n)) space complexity lower bounds, which hold for both deterministic and randomized implemen-tations. For deterministic implementations, we prove lower bounds of Ω(min(logL, n)) on the worst-case step complexity of an operation. When arbitrary primitives can be used, we prove that either some operation incurs Ω(min(logL, n)) memory stalls or some operation performs Ω(min(logL, n)) steps. In addition to our deterministic time lower bounds, the paper establishes lower bounds on the expected step complexity of restricted-use randomized versions of many of these objects in a weak oblivious adversary model.