Half Dome cables usually go up by Memorial weekend. (but I like to go up before the cables are "up", or after they're "down". The cables are there, but not elevated on their posts. So you can just pick up the cable and climb. They get a little heavy where the slope curves over the hump about 3/4 way up. Yosemite N.P. always recommends that you NOT go up the cables when they are down. But there are no permits required then, either.
Whitney will have snow on the trail in June, but many people will be going up. You will want at least microspikes, and maybe crampons if the snow is still heavy. There will be a well-beaten path through whatever snow is still there. Watch the current conditions thread in June to see.
