1 research outputs found
Fully Dynamic Spectral Vertex Sparsifiers and Applications
We study \emph{dynamic} algorithms for maintaining spectral vertex
sparsifiers of graphs with respect to a set of terminals of our choice.
Such objects preserve pairwise resistances, solutions to systems of linear
equations, and energy of electrical flows between the terminals in . We give
a data structure that supports insertions and deletions of edges, and terminal
additions, all in sublinear time. Our result is then applied to the following
problems.
(1) A data structure for maintaining solutions to Laplacian systems
, where is the Laplacian
matrix and is a demand vector. For a bounded degree, unweighted
graph, we support modifications to both and while
providing access to -approximations to the energy of routing an
electrical flow with demand , as well as query access to entries of
a vector such that in expected
amortized update and query time.
(2) A data structure for maintaining All-Pairs Effective Resistance. For an
intermixed sequence of edge insertions, deletions, and resistance queries, our
data structure returns -approximation to all the resistance
queries against an oblivious adversary with high probability. Its expected
amortized update and query times are on an unweighted graph, and
on weighted graphs.
These results represent the first data structures for maintaining key
primitives from the Laplacian paradigm for graph algorithms in sublinear time
without assumptions on the underlying graph topologies.Comment: STOC 2019. arXiv admin note: text overlap with arXiv:1804.0403