  We are concerned with the fastest possible direct numerical solution
algorithm for a thin-banded or tridiagonal linear system of dimension $N$ on a
distributed computing network of $N$ nodes that is connected in a binary
communication tree. Our research is driven by the need for faster ways of
numerically solving discretized systems of coupled one-dimensional black-box
boundary-value problems.
  Our paper presents two major results: First, we provide an algorithm that
achieves the optimal parallel time complexity for solving a tridiagonal linear
system and thin-banded linear systems. Second, we prove that it is impossible
to improve the time complexity of this method by any polynomial degree.
  To solve a system of dimension $m\cdot N$ and bandwidth $m \in
\Omega(N^{1/6})$ on $2 \cdot N-1$ computing nodes, our method needs time
complexity $\mathcal{O}(\log(N)^2 \cdot m^3)$.
