185 research outputs found
Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models
We present a method for solving the transshipment problem - also known as
uncapacitated minimum cost flow - up to a multiplicative error of in undirected graphs with non-negative edge weights using a
tailored gradient descent algorithm. Using to hide
polylogarithmic factors in (the number of nodes in the graph), our gradient
descent algorithm takes iterations, and in each
iteration it solves an instance of the transshipment problem up to a
multiplicative error of . In particular, this allows
us to perform a single iteration by computing a solution on a sparse spanner of
logarithmic stretch. Using a randomized rounding scheme, we can further extend
the method to finding approximate solutions for the single-source shortest
paths (SSSP) problem. As a consequence, we improve upon prior work by obtaining
the following results: (1) Broadcast CONGEST model: -approximate SSSP using rounds, where is the (hop) diameter of the network.
(2) Broadcast congested clique model: -approximate
transshipment and SSSP using rounds. (3)
Multipass streaming model: -approximate transshipment and
SSSP using space and passes. The
previously fastest SSSP algorithms for these models leverage sparse hop sets.
We bypass the hop set construction; computing a spanner is sufficient with our
method. The above bounds assume non-negative edge weights that are polynomially
bounded in ; for general non-negative weights, running times scale with the
logarithm of the maximum ratio between non-zero weights.Comment: Accepted to SIAM Journal on Computing. Preliminary version in DISC
2017. Abstract shortened to fit arXiv's limitation to 1920 character
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
Spanners and Sparsifiers in Dynamic Streams
Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements. The underlying idea of taking randomized linear measurements of input data has been extremely successful in providing space-efficient algorithms for classical problems such as frequency moment estimation and computing heavy hitters, and was very recently shown to be a powerful technique for solving graph problems in dynamic streams [AGM’12]. Ideally, one would like to obtain algorithms that use one or a small constant number of passes over the data and a small amount of space (i.e. sketching dimension) to preserve some useful properties of the input graph presented as a sequence of edge insertions and edge deletions. In this paper, we concentrate on the problem of constructing linear sketches of graphs that (approximately) preserve the spectral information of the graph in a few passes over the stream. We do so by giving the first sketch-based algorithm for constructing multiplicative graph spanners in only two passes over the stream. Our spanners use Õ(n1+1/k) bits of space and have stretch 2 k. While this stretch is larger than the conjectured optimal 2k − 1 for this amount of space, we show for an appropriate k that it implies the first 2-pass spectral sparsifier with n 1+o(1) bits of space. Previous constructions of spectral sparsifiers in this model with a constant number of passes would require n 1+c bits of space for a constant c> 0. We also give an algorithm for constructing spanners that provides an additive approximation to the shortest path metric using a single pass over the data stream, also achieving an essentially best possible space/approximation tradeoff. 1
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
Massively Parallel Algorithms for Distance Approximation and Spanners
Over the past decade, there has been increasing interest in
distributed/parallel algorithms for processing large-scale graphs. By now, we
have quite fast algorithms -- usually sublogarithmic-time and often
-time, or even faster -- for a number of fundamental graph
problems in the massively parallel computation (MPC) model. This model is a
widely-adopted theoretical abstraction of MapReduce style settings, where a
number of machines communicate in an all-to-all manner to process large-scale
data. Contributing to this line of work on MPC graph algorithms, we present
round MPC algorithms for computing
-spanners in the strongly sublinear regime of local memory. To
the best of our knowledge, these are the first sublogarithmic-time MPC
algorithms for spanner construction. As primary applications of our spanners,
we get two important implications, as follows:
-For the MPC setting, we get an -round algorithm for
approximation of all pairs shortest paths (APSP) in the
near-linear regime of local memory. To the best of our knowledge, this is the
first sublogarithmic-time MPC algorithm for distance approximations.
-Our result above also extends to the Congested Clique model of distributed
computing, with the same round complexity and approximation guarantee. This
gives the first sub-logarithmic algorithm for approximating APSP in weighted
graphs in the Congested Clique model
(1 + )-Approximate shortest paths in dynamic streams
Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2−1 versus n1+1/, for an integer parameter . (In fact, existing solutions also incur an extra factor of 1 + in the stretch for weighted graphs, and an additional factor of logO(1) n in the space.) The only existing solution of the second type uses n1/2−O(1/) passes over the stream (for space O(n1+1/)), and applies only to unweighted graphs. In this paper we show that (1+)-approximate single-source shortest paths can be computed with ˜O (n1+1/) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs. Moreover, the same result applies for multi-source shortest paths, as long as the number of sources is O(n1/). We achieve these results by devising efficient dynamic streaming constructions of (1 + , )-spanners and hopsets. On our way to these results, we also devise a new dynamic streaming algorithm for the 1-sparse recovery problem. Even though our algorithm for this task is slightly inferior to the existing algorithms of [26, 11], we believe that it is of independent interest. 2012 ACM Subject Classification Theory of computation ! Streaming models; Theory of computation ! Streaming, sublinear and near linear time algorithms; Theory of computation ! Shortest paths; Theory of computation ! Sparsification and spanner
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