Sorting is an operation that has received a lot of attention. Writing your own sort isn't advisable if you are interested in high performance. I would consider something like thrust, back40computing, moderngpu, or CUB for sorting on the GPU.
Most of the above will be handling an array at a time, using the full GPU to sort an array. There are techniques within thrust to do a vectorized sort which can handle multiple arrays "at once", and CUB may also be an option for doing a "per-thread" sort (let's say, "per thread block").
Generally I would say the same thing about CPU sorting code. Don't write your own.
EDIT: I guess one more comment. I would lean heavily towards the first approach you mention (i.e. not doing a sort per thread.) There are two related reasons for this:
- Most of the fast sorting work has been done along the lines of your first method, not the second.
- The GPU is generally better at being fast when the work is well adapted for SIMD or SIMT. This means we generally want each thread to be doing the same thing and minimizing branching and warp divergence. This is harder to achieve (I think) in the second case, where each thread appears to be following the same sequence but in fact data dependencies are causing "algorithm divergence". On the surface of it, you might wonder if the same criticism might be levelled at the first approach, but since these libraries I mention arer written by experts, they are aware of how best to utilize the SIMT architecture. The thrust "vectorized sort" and CUB approaches will allow multiple sorts to be done per operation, while still taking advantage of SIMT architecture.