Linear equation solving with special sparse coefficient matrix
There are iterative methods in which each iteration takes time $O(nnz)$, where $nnz$ is the number of non-zero entries. See for example Chapter 4 of Saad's lecture notes. For the state-of-the-art, consult Section 5 of Lee and Sidford (2013). For special types of matrices, see Lee, Peng and Spielman (2016).