A tight time lower bound for space-optimal implementations of multi-writer snapshots

A snapshot object consists of a collection of m > 1 components, each capable of storing a value, shared by n processes in an asynchronous shared-memory distributed system. It supports two operations: a process can UPDATE any individual component or atomically SCAN the entire collection to obtain the values of all the components. It is possible to implement a snapshot object using m registers so that each operation takes O(mn) time.In a previous paper, we proved that m registers are necessary to implement a snapshot object with m < n-1 components. Here we prove that, for any such space-optimal implementation, Ω(mn) steps are required to perform a SCAN operation in the worst case, matching the upper bound. We also extend our space and time lower bounds to implementations that use single-writer registers in addition to the multi-writer registers. Specifically, we prove that at least m multi-writer registers are still needed, provided the SCANS do not read a large fraction of the single-writer registers. We also prove that any implementation that uses single-writer registers in addition to $m$ multi-writer registers uses Ω(√mn) steps in the worst case. Our proof yields insight into the structure of any implementation that uses only m multi-writer registers, showing that processes must access the multi-writer registers in a very constrained way.