Space bounds for reliable storage: Fundamental limits of coding

We study the inherent space requirements of reliable storage algorithms in asynchronous distributed systems. A number of recent works have used codes in order to achieve a bet- ter storage cost than the well-known replication approach. However, a closer look reveals that they incur extra costs in certain scenarios. Specifically, if multiple clients access the storage concurrently, then existing asynchronous code- based algorithms may store a number of copies of the data that grows linearly with the number of concurrent clients. We prove here that this is inherent. Given three parameters, (1) the data size - D bits, (2) the concurrency level - c, and (3) the number of storage node failures that need to be tol- erated - f, we show a lower bound of ω (min(f, c) · D) bits on the space complexity of asynchronous distributed stor- age algorithms. Intuitively, this implies that the asymptotic storage cost is either as high as with replication, namely O(fD), or as high under concurrency as with the aforemen- tioned code-based algorithms, i.e., O(cD). We further present a technique for combining erasure codes with replication so as to obtain the best of both. We present an adaptive f - tolerant storage algorithm whose storage cost is O(min(f; c) · D). Together, our results show that the space complexity of providing reliable storage in asyn- chronous distributed systems is θ(min(f; c) · D).