- Wojciech Golab
- Xiaozhou (Steve) Li
- Alejandro López-Ortiz
- Naomi Nishimura

## Abstract

The k-atomicity property can be used to describe the consistency of data operations in large distributed storage systems. The weak consistency guarantees offered by such systems are seen as a necessary compromise in view of Brewer's CAP principle. The k-atomicity property requires that every read operation obtains a value that is at most k updates (writes) old, and becomes a useful way to quantify weak consistency if k is treated as a variable that can be computed from a history of operations. Specifically, the value of k quantifies how far the history deviates from Lamport's atomicity property for read/write registers. We address the problem of computing k indirectly by solving the k-atomicity verification problem (k-AV): given a history of read/write operations and a positive integer k, decide whether the history is k-atomic. Gibbons and Korach showed that in general this problem is NP-complete when k = 1, and hence not solvable in polynomial time unless P = NP. In this paper we present two algorithms that solve the k-AV problem for any k >= 2 in special cases. Similarly to known solutions for k = 1 and k = 2, both algorithms assume that all the values written to a given object are distinct. The first algorithm places an additional restriction on the structure of the input history and solves k-AV in O(n^2 + n (k log k) time. The second algorithm does not place any additional restrictions on the input but is efficient only when k is small and when concurrency among write operations is limited. Its time complexity is O(n^2) if both k and our particular measure of write concurrency are bounded by constants.

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