Omer Reingold's proof that $L=SL$ gives an algorithm for USTCON (In an Undirected graph with special vertices $s$ and $t$, are they Connected?) using only logspace. The basic idea is to build an expander graph from the original graph, and then do the walk in the expander graph. The expander graph is made by squaring the original graph logarithmically many times. In the expander graph, the diameter is only logarithmic, so a DFS search of logarithmic depth suffices.
Extending the result to $L=NL$ would imply the existence of a logspace algorithm for DSTCON -- the same, but for Directed graphs. (Sometimes just STCON.) My question, maybe slightly soft, is what are the primary obstructions to extending Reingold's proof to that?
It feels slightly like there should be a sort of "directed expander" graph. A similar sort of construction, where you add in edges corresponding to medium-length directed paths, and then some corresponding to long ones; and then you can traverse the graph with logarithmic depth by moving across short paths to get to a long one; then back to short paths at the end.
Is there a major flaw in this concept? Or is it that there aren't any good constructions of such expanders? Or does it somehow require more memory than the undirected version?
I unfortunately cannot find much at all on directed expander graphs. In fact essentially all I could find was https://math.stackexchange.com/questions/2628930/how-can-one-construct-a-directed-expander-graph-with-varying-degree-distribution (which is unanswered) and https://repository.upenn.edu/cgi/viewcontent.cgi?article=1202&context=cis_papers . Is there a different term I should be searching under?