Determinant of symmetric tridiagonal matrices

If you're asking if either of these determinants can be solved/calculated by a sub-linear time algorithm, the answer is obviously no because the $a_i$'s and $d_i$'s can all have different values and thus you need to read each of them at least once (and use the value read in some operation), so that gives $2n$ (minus some small constant, which I can't be bothered to calculate) a lower bound for the symmetric one and $3n$ for the general tridiagonal matrix determinant. Asymptotically, both are $\Omega(n)$ of course.

Cache-locality tweaks, like representing the [sparse] matrix by a [non-sparse] array would improve performance in practice for some sizes of problems, but I think such a code tuning discussion is rather off-topic on math.SE.

However if the values are known to be all the same, then you can have a constant-time, closed-form formula, e.g. the answer is the Fibonacci number if $a_i=-1$ and $c_i=d_i=1$. There are several papers about such determinants; here's an example one. A constant-time time algorithm for such closed formulas assumes the result is small enough to fit in a register, otherwise if you need arbitrary-precision arithmetic, then it's no longer proper to consider them constant time.