Optimal step complexity under the space-optimal bound

Determine whether an atomic-from-atomic implementation of a multi-word single-writer register can achieve the optimal space complexity [?]Theta(r + log m/log b) together with read and write step complexity [?]Theta(log m/log b), where m is the number of simulated-register values, b is the number of values representable by each base register, and r is the number of readers.

Background

The paper develops a wait-free algorithm for implementing an atomic m-valued single-writer register from atomic b-valued single-writer base registers using Theta(r + log m/log b) space. This improves the prior visible-reader space bound of Theta(r log m/log b), but the algorithm requires Theta(log2 m/log2 b) read steps and Theta(r + log m/log b) write steps.

The unresolved issue is whether the asymptotically optimal space bound can coexist with the faster Theta(log m/log b) read and write step complexities achieved by the previous algorithm. The paper explicitly leaves this tradeoff unresolved.

References

Whether the optimal $\Theta(r+\frac{\log m}{\log b})$ space bound can be achieved together with $\Theta(\frac{\log m}{\log b})$ read and write step complexity remains open.

— Upper and Lower Bounds on the Space Complexity of Multi-word Single-Writer Registers  (2608.19167 - Wei et al., 19 Aug 2026) in Section 1, Introduction