our FHS scheme, along with its extension, are sufficient to prove the achievability part of Proposition \[prop:comparison\], we do not have to design a DHS scheme for that purpose. More specifically, we have proved that whenever there exists a DHS scheme that strictly outperforms BHS, there always exists another SHS scheme that strictly outperforms BHS. As a result, at least when considering only single node failure, there is no clear advantage of DHS over SHS. However, for the multiple failures scenario, we have shown in a separate work [@arxiv_multiple] that it is possible to have DHS$\succ$SHS$=$BHS. Specifically, under some scenarios, only DHS can strictly outperform BHS while the best SHS design is no better than the simple BHS solution.

The Information-Flow Graph and the Corresponding Graph-Based Analysis {#subsec:ifg_existing}
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As in [@dimakis2010network], the performance of a distributed storage system can be characterized by the concept of information flow graphs (IFGs). IFGs depict the storage in the network and the communication that takes place during repair. For readers who are not familiar with IFGs, we provide its detailed description in Appendix \[app:ifg\].

Intuitively, each IFG reflects one unique history of the failure patterns and the helper selection choices from time $1$ 