186 research outputs found
Probabilistic Spectral Sparsification In Sublinear Time
In this paper, we introduce a variant of spectral sparsification, called
probabilistic -spectral sparsification. Roughly speaking,
it preserves the cut value of any cut with an
multiplicative error and a additive error. We show how
to produce a probabilistic -spectral sparsifier with
edges in time
time for unweighted undirected graph. This gives fastest known sub-linear time
algorithms for different cut problems on unweighted undirected graph such as
- An time -approximation
algorithm for the sparsest cut problem and the balanced separator problem.
- A time approximation minimum s-t cut algorithm
with an additive error
Densest Subgraph in Dynamic Graph Streams
In this paper, we consider the problem of approximating the densest subgraph
in the dynamic graph stream model. In this model of computation, the input
graph is defined by an arbitrary sequence of edge insertions and deletions and
the goal is to analyze properties of the resulting graph given memory that is
sub-linear in the size of the stream. We present a single-pass algorithm that
returns a approximation of the maximum density with high
probability; the algorithm uses O(\epsilon^{-2} n \polylog n) space,
processes each stream update in \polylog (n) time, and uses \poly(n)
post-processing time where is the number of nodes. The space used by our
algorithm matches the lower bound of Bahmani et al.~(PVLDB 2012) up to a
poly-logarithmic factor for constant . The best existing results for
this problem were established recently by Bhattacharya et al.~(STOC 2015). They
presented a approximation algorithm using similar space and
another algorithm that both processed each update and maintained a
approximation of the current maximum density in \polylog (n)
time per-update.Comment: To appear in MFCS 201
Effective Resistances in Non-Expander Graphs
Effective resistances are ubiquitous in graph algorithms and network
analysis. In this work, we study sublinear time algorithms to approximate the
effective resistance of an adjacent pair and . We consider the classical
adjacency list model for local algorithms. While recent works have provided
sublinear time algorithms for expander graphs, we prove several lower bounds
for general graphs of vertices and edges:
1.It needs queries to obtain -approximations of the
effective resistance of an adjacent pair and , even for graphs of degree
at most 3 except and .
2.For graphs of degree at most and any parameter , it needs
queries to obtain -approximations
where is a universal constant.
Moreover, we supplement the first lower bound by providing a sublinear time
-approximation algorithm for graphs of degree 2 except the pair
and .
One of our technical ingredients is to bound the expansion of a graph in
terms of the smallest non-trivial eigenvalue of its Laplacian matrix after
removing edges. We discover a new lower bound on the eigenvalues of perturbed
graphs (resp. perturbed matrices) by incorporating the effective resistance of
the removed edge (resp. the leverage scores of the removed rows), which may be
of independent interest
On Solving Linear Systems in Sublinear Time
We study sublinear algorithms that solve linear systems locally. In the classical version of this problem the input is a matrix S in R^{n x n} and a vector b in R^n in the range of S, and the goal is to output x in R^n satisfying Sx=b. For the case when the matrix S is symmetric diagonally dominant (SDD), the breakthrough algorithm of Spielman and Teng [STOC 2004] approximately solves this problem in near-linear time (in the input size which is the number of non-zeros in S), and subsequent papers have further simplified, improved, and generalized the algorithms for this setting.
Here we focus on computing one (or a few) coordinates of x, which potentially allows for sublinear algorithms. Formally, given an index u in [n] together with S and b as above, the goal is to output an approximation x^_u for x^*_u, where x^* is a fixed solution to Sx=b.
Our results show that there is a qualitative gap between SDD matrices and the more general class of positive semidefinite (PSD) matrices. For SDD matrices, we develop an algorithm that approximates a single coordinate x_{u} in time that is polylogarithmic in n, provided that S is sparse and has a small condition number (e.g., Laplacian of an expander graph). The approximation guarantee is additive | x^_u-x^*_u | 0. We further prove that the condition-number assumption is necessary and tight.
In contrast to the SDD matrices, we prove that for certain PSD matrices S, the running time must be at least polynomial in n (for the same additive approximation), even if S has bounded sparsity and condition number
Quantum Speedup for Graph Sparsification, Cut Approximation and Laplacian Solving
Graph sparsification underlies a large number of algorithms, ranging from
approximation algorithms for cut problems to solvers for linear systems in the
graph Laplacian. In its strongest form, "spectral sparsification" reduces the
number of edges to near-linear in the number of nodes, while approximately
preserving the cut and spectral structure of the graph. In this work we
demonstrate a polynomial quantum speedup for spectral sparsification and many
of its applications. In particular, we give a quantum algorithm that, given a
weighted graph with nodes and edges, outputs a classical description of
an -spectral sparsifier in sublinear time
. This contrasts with the optimal classical
complexity . We also prove that our quantum algorithm is optimal
up to polylog-factors. The algorithm builds on a string of existing results on
sparsification, graph spanners, quantum algorithms for shortest paths, and
efficient constructions for -wise independent random strings. Our algorithm
implies a quantum speedup for solving Laplacian systems and for approximating a
range of cut problems such as min cut and sparsest cut.Comment: v2: several small improvements to the text. An extended abstract will
appear in FOCS'20; v3: corrected a minor mistake in Appendix
Multi-Scale Matrix Sampling and Sublinear-Time PageRank Computation
A fundamental problem arising in many applications in Web science and social
network analysis is, given an arbitrary approximation factor , to output a
set of nodes that with high probability contains all nodes of PageRank at
least , and no node of PageRank smaller than . We call this
problem {\sc SignificantPageRanks}. We develop a nearly optimal, local
algorithm for the problem with runtime complexity on
networks with nodes. We show that any algorithm for solving this problem
must have runtime of , rendering our algorithm optimal up
to logarithmic factors.
Our algorithm comes with two main technical contributions. The first is a
multi-scale sampling scheme for a basic matrix problem that could be of
interest on its own. In the abstract matrix problem it is assumed that one can
access an unknown {\em right-stochastic matrix} by querying its rows, where the
cost of a query and the accuracy of the answers depend on a precision parameter
. At a cost propositional to , the query will return a
list of entries and their indices that provide an
-precision approximation of the row. Our task is to find a set that
contains all columns whose sum is at least , and omits any column whose
sum is less than . Our multi-scale sampling scheme solves this
problem with cost , while traditional sampling algorithms
would take time .
Our second main technical contribution is a new local algorithm for
approximating personalized PageRank, which is more robust than the earlier ones
developed in \cite{JehW03,AndersenCL06} and is highly efficient particularly
for networks with large in-degrees or out-degrees. Together with our multiscale
sampling scheme we are able to optimally solve the {\sc SignificantPageRanks}
problem.Comment: Accepted to Internet Mathematics journal for publication. An extended
abstract of this paper appeared in WAW 2012 under the title "A Sublinear Time
Algorithm for PageRank Computations
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