With undetectable nitrate and phosphate, water changes will do nothing to reduce algae. Keeping your alkalinity, calcium, and magnesium at steady levels may be all you need to keep your corals happy. Icp tests every now and then will confirm nothing toxic has built up in the water/ nothing is depleted. I would consider dosing all for reef. That will really keep your trace elements, calcium, alk, etc at acceptable levels with 0 water changes.
As for the algae, you could shorten the light period. A uv sterilizer is my go to with algae problems but I have not tried that with hair algae. Melevs reef claims flux rx will get rid of hair algae. I have never tried it, but if I trust anybody on the internet it would be him.