The algae is too thin and tenacious for washing to get it off - you need to kill it, preferably first so it gets removed with the moss.
In the long run, the zinc strip at the ridge will kill it, and copper strip actually does a better job for algae and lichen than the zinc, which works best on moss. I guess you either have to decide which is more important, or maybe use the zinc powder (MossOut - I mistakenly said MossOff, which is an organic product that works but not as well) and copper strip, though purchase cost for that is about $100 for 50 feet versus $35-50 for zinc. Or go the other way if the moss is the major problem (the algae does not significantly damage roofs as moss does), and use the cheaper zinc strip and a copper-based granule or spray to kill the algae right off.
NOTE - zinc is corrosive to copper, so if you use both zinc and copper strips the copper should be the top one on the roof, and don't sprinkle zinc granules uphill of it. Also, bleach will neutralize the zinc or copper granules and react violently with the strip, so they should not be applied till after the bleach has been used and then washed off with the cleaning. If you don't like the bleach use, there are alternative products like Roof Cleaner QSE and Roof Shingle CLeaner OX available.
If you want immediate kill, then you can use chlorine bleach (50% mix with water) sprayed on the roof to kill both the algae and the moss, let them sit a week or so to die, then the moss cleaning will go easier and also take most of the algae with it. Then coat the roof with the zinc MossOut powder as residual kill measure for the bits left behind, backup up long-term by the metal strip.
Before using the chlorine bleach wet plants and grass nearby, then wash them off right after. If really valuable plants, cover them and use oxygenated rather than chlorine based - still wet first and rinse after, but has minimal effect on plants in small quantities.
One note on the moss removal - be sure the contract includes picking the mess up off the ground and disposing of it, and washing out your gutters AND downspouts to get the moss clumps out or they will plug it up.