The One-Out-of-k Retrieval Problem and Linear Network Coding

In this paper we show how linear network coding can reduce the number of queries needed to retrieve one specific message among k distinct ones replicated across a large number of randomly accessed nodes storing one message each. Without network coding, this would require k queries on average. After proving that no scheme can perform better than a straightforward lower bound of 0.5k average queries, we propose and asymptotically evaluate, using mean field arguments, a few example practical schemes, the best of which attains 0.82k queries on average. The paper opens two complementary challenges: a systematic analysis of practical schemes so as to identify the best performing ones and design guideline strategies, as well as the need to identify tighter, nontrivial, lower bounds.