Problem with understanding multi-party secure stable matching cirsuit

Hi,
I am reading the following paper:
MPCircuits: Optimized Circuit Generation for Secure Multi-Party Computation

Paper Link

I have following questions:
1. why Preference of group1 starts from L_0 where as preference list from group2 starts with L_(n/2+1)?
2. Any reason why they used the priority encoder?
3. What is the technique for the security of preference list?
4. A brief description how the circuit works?

Somebody please guide me.




Zulfi.