Fully Dynamic Effective Resistances.
arXiv: Data Structures and Algorithms(2018)
摘要
In this paper we consider the emph{fully-dynamic} All-Pairs Effective Resistance problem, where the goal is to maintain effective resistances on a graph $G$ among any pair of query vertices under an intermixed sequence of edge insertions and deletions in $G$. The effective resistance between a pair of vertices is a physics-motivated quantity that encapsulates both the congestion and the dilation of a flow. It is directly related to random walks, and it has been instrumental in the recent works for designing fast algorithms for combinatorial optimization problems, graph sparsification, and network science. give a data-structure that maintains $(1+epsilon)$-approximations to all-pair effective resistances of a fully-dynamic unweighted, undirected multi-graph $G$ with $tilde{O}(m^{4/5}epsilon^{-4})$ expected amortized update and query time, against an oblivious adversary. Key to our result is the maintenance of a dynamic emph{Schur complement}~(also known as vertex resistance sparsifier) onto a set of terminal vertices of our choice. This maintenance is obtained (1) by interpreting the Schur complement as a sum of random walks and (2) by randomly picking the vertex subset into which the sparsifier is constructed. We can then show that each update in the graph affects a small number of such walks, which in turn leads to our sub-linear update time. We believe that this local representation of vertex sparsifiers may be of independent interest.
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