Let $A\in\mathbb R^{n\times n}$ be a symmetric positive-definite matrix and $A^{-1}$ is already known. Now I want to compute the matrix $(A+\sum_{i=1}^n B_i)^{-1}$ where each $B_i$ is a sparse symmetric matrix with only four non-zero elements that are placed on the rectangle, which has the following form: $$B_i=(e_j-e_k)(e_j-e_k)^T=\left[\begin{array}{ccccc} & & & & \\ & 1 & -1 & & \\ & & & & \\ & -1 & 1 & \\ & & & & \end{array}\right]$$ where $e_j$ is the $j$-th unit vector.
Such a problem has many practical values (such as in graph theory). I wonder if there is some algorithm to compute $(A+\sum_{i=1}^n B_i)^{-1}$ in $\Theta(n^2)$ time based on the known matrix $A^{-1}$. I have tried to use the Sherman-Morrison-Woodbury formula: $$(A+vv^T)^{-1}=A^{-1}-\frac {A^{-1}vv^T A^{-1}}{1+v^T A^{-1} v},$$ but it seems that each iteration has to update the full matrix and the overall complexity remains $\Theta(n^3)$ (which is the same as a direct matrix inversion). Since the summation $\sum_{i=1}^n B_i$ is still highly sparse and the structure of the matrix has a special form, I wonder whether it is possible to calculate the inversion efficiently in $\Theta(n^2)$ time. Thank you!