Complexity tradeoffs for read and update operations

Recent work established that some restricted-use objects, such as max registers, counters and atomic snapshots, admit polylogarithmic step-complexity wait-free implementations using only reads and writes: when only polynomially-many updates are allowed, reading the object (by performing a ReadMax, CounterRead or Scan operation, depending on the object's type) incurs O (log N) steps (where N is the number of processes), which was shown to be optimal. But what about the step-complexity of update operations? With these implementations, updating the object's state (by performing a WriteMax, CounterIncrement or Update operation, depending on the object's type) requires ω(log N) steps. The question that we address in this work is the following: are there read-optimal implementations of these restricted-use objects for which the asymptotic step-complexity of update operations is sub-logarithmic? We present tradeoffs between the step-complexity of read and update operations on these objects, establishing that updating a read-optimal counter or snapshot incurs ω(log N) steps. These tradeoffs hold also if compare-and-swap (CAS) operations may be used, in addition to reads and writes. We also derive a tradeoff between the step-complexities of read and update operations of M-bounded max registers: if the step-complexity of the ReadMax operation is 0(f(min(N, M))), then the step-complexity of the Write-Max operation is ω(log "Equation Presented ") . It follows from this tradeoff that the step-complexity of WriteMax in any readoptimal implementation of a max register from read, write and CAS is ω(log log/min(N, M)). On the positive side, we present a wait-free implementation of an M-bounded max register from read, write and CAS for which the step complexities of ReadMax and WriteMax operations are 0(1) and O ( log min (N, M)), respectively. Partially supported by the Israel Science Foundation (grant number 1227/10) and by the Lynne and William Frankel Center for Computing Science at Ben-Gurion University.