Time Bounds for Shared Objects in Partially Synchronous Systems

• Main Author: Wang, Jiaqi
• Other Authors: Welch, Jennifer
• Format: Others
• Language: en_US
• Published: 2012
• Subjects: Shared objects; linearizability; partially synchronous system; bounds on time complexity
• Online Access: http://hdl.handle.net/1969.1/ETD-TAMU-2011-12-10214

Shared objects are a key component in today's large distributed systems. Linearizability is a popular consistency condition for such shared objects which gives the illusion of sequential execution of operations. The time bound of an operation is the worst-case time complexity from the operation invocation to its response. Some time bounds have been proved for certain operations on linearizable shared objects in partially synchronous systems but there are some gaps between time upper bound and lower bound for each operation. In this work, the goal is to narrow or eliminate the gaps and find optimally fast implementations. To reach this goal, we prove larger lower bounds and show smaller upper bounds (compared to 2d for all operations in previous folklore implementations) by proposing an implementation for a shared object with an arbitrary data type in distributed systems of n processes in which every message delay is bounded within [d-u, d] and the maximum skew between processes' clocks is epsilon. Considering any operation for which there exist two instances such that individually, each instance is legal but in sequence they are not, we prove a lower bound of d + min{epsilon, u, d/3}, improving from d, and show this bound is tight when epsilon < d/3 and epsilon < u. Considering any operation for which there exist k instances such that each instance separately is legal and any sequence of them is legal, but the state of the object is different after different sequences, we prove a lower bound of (1-1/k)u, improving from u/2, and show this bound is tight when k = n. A pure mutator only modifies the object but does not return anything about the object. A pure accessor does not modify the object. For a pure mutator OP1 and a pure accessor OP2, if given a set of instances of OP1, the state of the object reflects the order in which the instances occur and an instance of OP2 can detect whether an instance of OP1 occurs, we prove the sum of the time bound for OP1 and OP2 is at least d + min{epsilon, u, d/3}, improving from d. The upper bound is d + 2*epsilon from our implementation.