Given a group of order $n$, what are the least number of elements to specify on the Cayley table to specify a group?
Example
Consider this $Z_4$ group table for example:
Due to abelian-ness I can hide these elements and still lose no information:
Furthermore, I just need one slot filled to know the action of identity on everything else. So, I can shade these terms out as well:
Further more I can shade darken out one element on the second row:
This is because I know that purply shaded element must be '1' only as identity acts on it, and I also know that each row must have one identity.
Taking assumptions on the nature of Group will quickly reduce the upper bound of elements needed to be specified.
Is there any algorithm to strike out maximum number of slots in the cayley table?
If the group is abelian, as shown, we can quickly just strike elements above or below main diagonal.
Regarding your example. I argue that you can only keep three elements in the Cayley table - this $$ T(0,0),T(1,1), T(1,2). $$ The remaining elements of this table can be reconstructed by knowing the size of the table and it's three values.
For example, $T(2,2)\neq3$, $T(2,2)\neq2$, and $T(2,2)\neq1$. The first two are obvious. If $T(2,2)=1$, then $1+1+1+1=1$, i.e. $1+1+1=0$ and $T(1,2)=0$. Contradiction.