5 research outputs found
Density Independent Algorithms for Sparsifying k-Step Random Walks
We give faster algorithms for producing sparse approximations of the transition matrices of k-step random walks on undirected and weighted graphs. These transition matrices also form graphs, and arise as intermediate objects in a variety of graph algorithms. Our improvements are based on a better understanding of processes that sample such walks, as well as tighter bounds on key weights underlying these sampling processes. On a graph with n vertices and m edges, our algorithm produces a graph with about nlog(n) edges that approximates the k-step random walk graph in about m + k^2 nlog^4(n) time. In order to obtain this runtime bound, we also revisit "density independent" algorithms for sparsifying graphs whose runtime overhead is expressed only in terms of the number of vertices
Fully Dynamic Effective Resistances
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 among any pair of query vertices under an intermixed sequence of edge
insertions and deletions in . 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.
We give a data-structure that maintains -approximations to
all-pair effective resistances of a fully-dynamic unweighted, undirected
multi-graph with 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