You would be OK with your first proposed installation if your stack ran straight up from your sink tie-in point and your sink was within 6 feet or 42" of the stack, depending on which plumbing "theory" you are under. But your main line runs horizontally under your floor joists for some distance, so as Twowaxhack indicated, the sink will need to be dry vented. You do not show the length of the horizontal main drain run between your vertical tie-in point and where the upper-level(s) vertical run drops into that horizontal line. Perhaps an easier way to vent this sink and maintain a code installation without having to install a new vent through the roof, would be to run a 1 1/4" vent pipe from the top of your sink drain line into your attic. If this is a single-story house, that may be relatively easy to do. All you need to find is an open space between a set of studs relatively close to the vent stack. If this isn't a single-story house, this approach could be exceeding difficult, however. After you are in the attic, all you need to do is tie this 1 1/4" vent into the existing vent stack. I've been able to do this twice and it was really easy in a single-story house; once for a new basement sink, and once for a new basement washing machine installation. Just note that any horizontal runs need to be sloped 1/4" per foot minimum but can be sloped steeper as this is strictly a vent, not a drain line.