I tried to use Floyd to find a shortest path, but I don't know how to
output the path! I know a recursion way to output the path, But it
wastes time and memory. And it cannot be used in such ring problem .

IF YOU CAN'T - YOU MUSTN'T DO!
If you can't use Floyd, you can use Dijkstra with references to
previous vertices - it is easier and doesn't cause the TIME LIMIT.