Proof of value discovery in complete graphs
If I understand correctly, a rough approach could be as follows.

Each processor $p_i$ can read $s_{i+1\ {\sf mod}\ n}$ and write its content to $r_i$.

Then each processor $p_i$ can read the wanted value by reading $r_{i-1\ {\sf mod}\ n}$.

The main issue of this naive algorithm is that some synchronization is needed between the two steps: $p_i$ must not read $r_{i-1\ {\sf mod}\ n}$ too soon, before it is written by $p_{i-1\ {\sf mod}\ n}$.