We consider the problem of designing optimal and effcient algorithms for solving tridiagonal linear systems on a mesh interconnection network. We derive precise upper and lower bounds for these solvers using odd-even cyclic reduction. We present various important lower bounds on execution time for solving these systems including general lower bounds which are independent of initial data assignment, lower bounds based on classifications of initial data assignments which classify assignments via the proportion of initial data assigned amongst processors, and lower bounds for commonly-used data layouts for tridiagonal solvers. Finally, algorithms are provided which have running times not only within a small constant factor of the lower bounds provided but which are within a small constant additive term of the lower bounds.