Lower bounds for adaptive collect and related objects

An <i>adaptive</i> algorithm, whose step complexity adjusts to the number of active processes, is attractive for situations in which the number of participating processes is highly variable. This paper studies the number and type of multi-writer registers that are needed for adaptive algorithms. We prove that if a collect algorithm is <i>f</i> -adaptive to total contention, namely, its step complexity is <i>f(k)</i>, where <i>k</i> is the number of processes that ever took a step, then it uses Ω(<i>f</i>^-1(<i>n</i>) multi-writer registers, where <i>n</i> is the total number of processes in the system.Furthermore, we show that competition for the underlying registers is inherent for adaptive collect algorithms. We consider <i>c-write</i> registers, to which at most <i>c</i> processes can be concurrently about to write. Special attention is given to <i>exclusive-write</i> registers, the case <i>c</i>=1 where no competition is allowed, and <i>concurrent-write</i> registers, the case <i>c</i>=<i>n</i> where any amount of competition is allowed. A collect algorithm is <i>f</i>-adaptive to point contention, if its step complexity is <i>f</i>(<i>k</i>), where <i>k</i> is the maximum number of simultaneously active processes. Such an algorithm is shown to require Ω(<i>f</i>^-1 (n<over>c)) concurrent-write registers, even if an unlimited number of <i>c</i>-write registers are available. A smaller lower bound is also obtained in this situation for collect algorithms that are <i>f</i>-adaptive to total contention.The lower bounds also hold for nondeterministic implementations of <i>sensitive</i> objects from historyless objects.Finally, we present lower bounds on the step complexity in solo executions (i.e., without any contention), when only <i>c</i>-write registers are used: For weak test&set objects, we present an ΩΩ(log <i>n</i><over>log <i>c</i> +log log <i>n</i>) lower bound. Our lower bound for collect and sensitive objects is Ω(<i>n</i>-1<over><i>c</i>).