not satisfied, we can always design a helper selection scheme that strictly outperforms BHS, the [*achievability*]{} part of the problem. For the achievability part, we propose a new low-complexity solution, termed the [*family helper selection (FHS) scheme*]{}, that is guaranteed to harvest the benefits of (careful) helper selection when compared to a BHS solution. We then characterize analytically the storage-bandwidth tradeoff of the FHS scheme and its extension, the family-plus helper selection scheme, and prove that they are *optimal* (as good as any helper selection one can envision) in some cases and [*weakly optimal*]{} in general, see Sections \[sec:achievability\_result\] and \[sec:optimality\_result\]). We also note that even though the purpose of introducing FHS and its extension is to prove the achievability part in theory, the FHS schemes have the same complexity as the existing BHS solution [@dimakis2010network] and demonstrate superior performance for practical system parameters.

In this Part I, we focus exclusively on the graph-based analysis of helper selection. In Part II [@part2], we provide an explicit construction of an exact-repair code that can achieve the minimum-bandwidth-regenerating (MBR) points of the family and family-plus helper selection schemes predicted by the graph-based analysis. The new MBR-point code in Part II is termed 