319 research outputs found
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
Communication-Optimal Distributed Dynamic Graph Clustering
We consider the problem of clustering graph nodes over large-scale dynamic
graphs, such as citation networks, images and web networks, when graph updates
such as node/edge insertions/deletions are observed distributively. We propose
communication-efficient algorithms for two well-established communication
models namely the message passing and the blackboard models. Given a graph with
nodes that is observed at remote sites over time , the two
proposed algorithms have communication costs and
( hides a polylogarithmic factor), almost matching
their lower bounds, and , respectively, in the
message passing and the blackboard models. More importantly, we prove that at
each time point in our algorithms generate clustering quality nearly as
good as that of centralizing all updates up to that time and then applying a
standard centralized clustering algorithm. We conducted extensive experiments
on both synthetic and real-life datasets which confirmed the communication
efficiency of our approach over baseline algorithms while achieving comparable
clustering results.Comment: Accepted and to appear in AAAI'1
Sublinear Estimation of Weighted Matchings in Dynamic Data Streams
This paper presents an algorithm for estimating the weight of a maximum
weighted matching by augmenting any estimation routine for the size of an
unweighted matching. The algorithm is implementable in any streaming model
including dynamic graph streams. We also give the first constant estimation for
the maximum matching size in a dynamic graph stream for planar graphs (or any
graph with bounded arboricity) using space which also
extends to weighted matching. Using previous results by Kapralov, Khanna, and
Sudan (2014) we obtain a approximation for general graphs
using space in random order streams, respectively. In
addition, we give a space lower bound of for any
randomized algorithm estimating the size of a maximum matching up to a
factor for adversarial streams
Maximum Matching in Turnstile Streams
We consider the unweighted bipartite maximum matching problem in the one-pass
turnstile streaming model where the input stream consists of edge insertions
and deletions. In the insertion-only model, a one-pass -approximation
streaming algorithm can be easily obtained with space , where
denotes the number of vertices of the input graph. We show that no such result
is possible if edge deletions are allowed, even if space is
granted, for every . Specifically, for every , we show that in the one-pass turnstile streaming model, in order to compute
a -approximation, space is
required for constant error randomized algorithms, and, up to logarithmic
factors, space is sufficient. Our lower bound result is
proved in the simultaneous message model of communication and may be of
independent interest
Approximate F_2-Sketching of Valuation Functions
We study the problem of constructing a linear sketch of minimum dimension that allows approximation of a given real-valued function f : F_2^n - > R with small expected squared error. We develop a general theory of linear sketching for such functions through which we analyze their dimension for most commonly studied types of valuation functions: additive, budget-additive, coverage, alpha-Lipschitz submodular and matroid rank functions. This gives a characterization of how many bits of information have to be stored about the input x so that one can compute f under additive updates to its coordinates.
Our results are tight in most cases and we also give extensions to the distributional version of the problem where the input x in F_2^n is generated uniformly at random. Using known connections with dynamic streaming algorithms, both upper and lower bounds on dimension obtained in our work extend to the space complexity of algorithms evaluating f(x) under long sequences of additive updates to the input x presented as a stream. Similar results hold for simultaneous communication in a distributed setting
Graph Sketches: Sparsification, Spanners, and Subgraphs
When processing massive data sets, a core task is to construct synopses of the data. To be useful, a synopsis data structure should be easy to construct while also yielding good approximations of the relevant properties of the data set. A particularly useful class of synopses are sketches, i.e., those based on linear projections of the data. These are applicable in many models including various parallel, stream, and compressed sensing settings. A rich body of analytic and empirical work exists for sketching numerical data such as the frequencies of a set of entities. Our work investigates graph sketching where the graphs of interest encode the relationships between these entities. The main challenge is to capture this richer structure and build the necessary synopses with only linear measurements.
In this paper we consider properties of graphs including the size of the cuts, the distances between nodes, and the prevalence of dense sub-graphs. Our main result is a sketch-based sparsifier construction: we show that O̅(nε-2) random linear projections of a graph on n nodes suffice to (1 + ε) approximate all cut values. Similarly, we show that O(ε-2) linear projections suffice for (additively) approximating the fraction of induced sub-graphs that match a given pattern such as a small clique. Finally, for distance estimation we present sketch-based spanner constructions. In this last result the sketches are adaptive, i.e., the linear projections are performed in a small number of batches where each projection may be chosen dependent on the outcome of earlier sketches. All of the above results immediately give rise to data stream algorithms that also apply to dynamic graph streams where edges are both inserted and deleted. The non-adaptive sketches, such as those for sparsification and subgraphs, give us single-pass algorithms for distributed data streams with insertion and deletions. The adaptive sketches can be used to analyze MapReduce algorithms that use a small number of rounds
Online Row Sampling
Finding a small spectral approximation for a tall matrix is
a fundamental numerical primitive. For a number of reasons, one often seeks an
approximation whose rows are sampled from those of . Row sampling improves
interpretability, saves space when is sparse, and preserves row structure,
which is especially important, for example, when represents a graph.
However, correctly sampling rows from can be costly when the matrix is
large and cannot be stored and processed in memory. Hence, a number of recent
publications focus on row sampling in the streaming setting, using little more
space than what is required to store the outputted approximation [KL13,
KLM+14].
Inspired by a growing body of work on online algorithms for machine learning
and data analysis, we extend this work to a more restrictive online setting: we
read rows of one by one and immediately decide whether each row should be
kept in the spectral approximation or discarded, without ever retracting these
decisions. We present an extremely simple algorithm that approximates up to
multiplicative error and additive error using online samples, with memory overhead
proportional to the cost of storing the spectral approximation. We also present
an algorithm that uses ) memory but only requires
samples, which we show is
optimal.
Our methods are clean and intuitive, allow for lower memory usage than prior
work, and expose new theoretical properties of leverage score based matrix
approximation
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