589 research outputs found
Solving Directed Laplacian Systems in Nearly-Linear Time through Sparse LU Factorizations
We show how to solve directed Laplacian systems in nearly-linear time. Given
a linear system in an Eulerian directed Laplacian with nonzero
entries, we show how to compute an -approximate solution in time . Through reductions from [Cohen et al.
FOCS'16] , this gives the first nearly-linear time algorithms for computing
-approximate solutions to row or column diagonally dominant linear
systems (including arbitrary directed Laplacians) and computing
-approximations to various properties of random walks on directed
graphs, including stationary distributions, personalized PageRank vectors,
hitting times, and escape probabilities. These bounds improve upon the recent
almost-linear algorithms of [Cohen et al. STOC'17], which gave an algorithm to
solve Eulerian Laplacian systems in time .
To achieve our results, we provide a structural result that we believe is of
independent interest. We show that Laplacians of all strongly connected
directed graphs have sparse approximate LU-factorizations. That is, for every
such directed Laplacian , there is a lower triangular matrix
and an upper triangular matrix
, each with at most
nonzero entries, such that their product spectrally approximates
in an appropriate norm. This claim can be viewed as an analogue of recent work
on sparse Cholesky factorizations of Laplacians of undirected graphs. We show
how to construct such factorizations in nearly-linear time and prove that, once
constructed, they yield nearly-linear time algorithms for solving directed
Laplacian systems.Comment: Appeared in FOCS 201
Fourth-order time-stepping for stiff PDEs on the sphere
We present in this paper algorithms for solving stiff PDEs on the unit sphere
with spectral accuracy in space and fourth-order accuracy in time. These are
based on a variant of the double Fourier sphere method in coefficient space
with multiplication matrices that differ from the usual ones, and
implicit-explicit time-stepping schemes. Operating in coefficient space with
these new matrices allows one to use a sparse direct solver, avoids the
coordinate singularity and maintains smoothness at the poles, while
implicit-explicit schemes circumvent severe restrictions on the time-steps due
to stiffness. A comparison is made against exponential integrators and it is
found that implicit-explicit schemes perform best. Implementations in MATLAB
and Chebfun make it possible to compute the solution of many PDEs to high
accuracy in a very convenient fashion
Kirchhoff Index As a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms
Most previous work of centralities focuses on metrics of vertex importance
and methods for identifying powerful vertices, while related work for edges is
much lesser, especially for weighted networks, due to the computational
challenge. In this paper, we propose to use the well-known Kirchhoff index as
the measure of edge centrality in weighted networks, called -Kirchhoff
edge centrality. The Kirchhoff index of a network is defined as the sum of
effective resistances over all vertex pairs. The centrality of an edge is
reflected in the increase of Kirchhoff index of the network when the edge
is partially deactivated, characterized by a parameter . We define two
equivalent measures for -Kirchhoff edge centrality. Both are global
metrics and have a better discriminating power than commonly used measures,
based on local or partial structural information of networks, e.g. edge
betweenness and spanning edge centrality.
Despite the strong advantages of Kirchhoff index as a centrality measure and
its wide applications, computing the exact value of Kirchhoff edge centrality
for each edge in a graph is computationally demanding. To solve this problem,
for each of the -Kirchhoff edge centrality metrics, we present an
efficient algorithm to compute its -approximation for all the
edges in nearly linear time in . The proposed -Kirchhoff edge
centrality is the first global metric of edge importance that can be provably
approximated in nearly-linear time. Moreover, according to the
-Kirchhoff edge centrality, we present a -Kirchhoff vertex
centrality measure, as well as a fast algorithm that can compute
-approximate Kirchhoff vertex centrality for all the vertices in
nearly linear time in
- …