Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs

We consider the problem of dynamically maintaining (approximate) all-pairs effective resistances in separable graphs, which are those that contain small balanced separators. We give a fully dynamic algorithm that maintains (1 + ε)-approximations of the all-pairs effective resistances of an n-vertex graph G undergoing edge insertions and deletions with Õ( √ n/ε) worst-case update time and Õ( √ n/ε) worst-case query time, if G is guaranteed to be O( √ n)separable (i.e., admit a balanced separator of size O( √ n)) and its separator can be computed in Õ(n) time. Our algorithm is built upon a dynamic algorithm for maintaining approximate Schur complement that approximately preserves pairwise effective resistances among a set of terminals for separable graphs, which might be of independent interest. We also show that our algorithm is close to optimal by proving that for any two fixed vertices s and t, there is no incremental or decremental (and thus, also no fully dynamic) algorithm that (1+O( 1 n ))-approximates the s− t effective resistance for O( √ n)-separable graphs with worstcase update time O(n) and query time O(n) for any δ > 0, unless the Online Matrix Vector Multiplication (OMv) conjecture is false. We further show that for general graphs, no incremental or decremental algorithm can maintain a (1 + O( 1 n ))-approximation of the s− t effective resistance with worst-case update time O(n) and query-time O(n) for any δ > 0, unless the OMv conjecture is false.