I have a small budget for some directory submissions; I was wondering if my site is listed on a couple of prominent directories could it speed up my insertion into DMOZ?
It will have no influence at all.
DMOZ decides to list websites based on their content, and only their content.
Ofcourse it might happen that an editor finds a website through another directory and decides that it is listable. This might happen without looking at the suggestion you made to DMOZ.
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