I wouldn't be concerned with any sagging. I would be concerned of the
liability risk of having a load bearing overhead structure who's sole
purpose is to have people under it.

Get a steel I-beam and box it with a thin facia to make a "fake" beam.
You can make this look slightly curved (the facia exterior) or make a
much larger beam than the steel inside. Many cosmetic possibilities. It
will never sag and that span in steel is trivial.

Or

Purchase an exterior grade glulam. (usually cedar)

Steel I-beams in my area are cheaper than the correct wood would cost. (a
redwood 2x10 should span 16' 8" (5.03m) for a rafter with minimal snow
load)
http://www.awc.org/calculators/span/...rcalcstyle.asp


Don't worry about sagging, worry about hurting a friend's child who's
over for a barbeque when the structure fails. Even though you may have
seen poorly built structures that survived a long time, remember that
much of the wood today is nowhere near the quality of the old growth wood
of the past.