Lower and upper bounds for single-scanner snapshot implementations

Abstract

We present a collection of upper and lower bounds on the complexity of asynchronous, wait-free, linearizable, single-scanner snapshot implementations from read---write registers. We argue that at least m registers are needed to implement a single-scanner snapshot with m components and we prove that, in space-optimal implementations, SCANS execute $$\varOmega (m^2)$$Ω(m2) steps. We present an algorithm that runs in $$O(m^2)$$O(m2) steps and uses $$m+1$$m+1 registers. We also present three implementations (namely, T-Opt, RT and RT-Opt) that beat the $$\varOmega (m^2)$$Ω(m2) lower bound by using more registers. Specifically, T-Opt has step complexity O(1) for UPDATE and O(m) for SCAN. This step complexity is optimal, but the number of registers that T-Opt uses is unbounded. We then present interesting recycling techniques to bound the number and the size of registers used, resulting in RT and RT-Opt. Specifically, RT-Opt, which has optimal step complexity, uses O(mn) bounded-size registers, where n is the total number of processes. Our implementations are the first with step complexities that are (linear or quadratic) functions only of m (and not of n). Moreover, T-Opt and RT-Opt are the first implementations with optimal step complexity.