80 research outputs found
On Constructing Spanners from Random Gaussian Projections
Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA\u2712) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area.
We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA\u2721), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work
The Sketching Complexity of Graph and Hypergraph Counting
Subgraph counting is a fundamental primitive in graph processing, with
applications in social network analysis (e.g., estimating the clustering
coefficient of a graph), database processing and other areas. The space
complexity of subgraph counting has been studied extensively in the literature,
but many natural settings are still not well understood. In this paper we
revisit the subgraph (and hypergraph) counting problem in the sketching model,
where the algorithm's state as it processes a stream of updates to the graph is
a linear function of the stream. This model has recently received a lot of
attention in the literature, and has become a standard model for solving
dynamic graph streaming problems.
In this paper we give a tight bound on the sketching complexity of counting
the number of occurrences of a small subgraph in a bounded degree graph
presented as a stream of edge updates. Specifically, we show that the space
complexity of the problem is governed by the fractional vertex cover number of
the graph . Our subgraph counting algorithm implements a natural vertex
sampling approach, with sampling probabilities governed by the vertex cover of
. Our main technical contribution lies in a new set of Fourier analytic
tools that we develop to analyze multiplayer communication protocols in the
simultaneous communication model, allowing us to prove a tight lower bound. We
believe that our techniques are likely to find applications in other settings.
Besides giving tight bounds for all graphs , both our algorithm and lower
bounds extend to the hypergraph setting, albeit with some loss in space
complexity
Expander Decomposition in Dynamic Streams
In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning ? of vertices V such that the subgraphs of G induced by the clusters C ? ? are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model.
Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (?, ?)-multiplicative/additive error with high probability. The power cut sparsifier uses O?(n/??) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant ?
Dynamic Graph Stream Algorithms in Space
In this paper we study graph problems in dynamic streaming model, where the
input is defined by a sequence of edge insertions and deletions. As many
natural problems require space, where is the number of
vertices, existing works mainly focused on designing space
algorithms. Although sublinear in the number of edges for dense graphs, it
could still be too large for many applications (e.g. is huge or the graph
is sparse). In this work, we give single-pass algorithms beating this space
barrier for two classes of problems.
We present space algorithms for estimating the number of connected
components with additive error and
-approximating the weight of minimum spanning tree, for any
small constant . The latter improves previous
space algorithm given by Ahn et al. (SODA 2012) for connected graphs with
bounded edge weights.
We initiate the study of approximate graph property testing in the dynamic
streaming model, where we want to distinguish graphs satisfying the property
from graphs that are -far from having the property. We consider
the problem of testing -edge connectivity, -vertex connectivity,
cycle-freeness and bipartiteness (of planar graphs), for which, we provide
algorithms using roughly space, which is
for any constant .
To complement our algorithms, we present space
lower bounds for these problems, which show that such a dependence on
is necessary.Comment: ICALP 201
Expander Decomposition in Dynamic Streams
In this paper we initiate the study of expander decompositions of a graph
in the streaming model of computation. The goal is to find a
partitioning of vertices such that the subgraphs of
induced by the clusters are good expanders, while the
number of intercluster edges is small. Expander decompositions are classically
constructed by a recursively applying balanced sparse cuts to the input graph.
In this paper we give the first implementation of such a recursive sparsest cut
process using small space in the dynamic streaming model.
Our main algorithmic tool is a new type of cut sparsifier that we refer to as
a power cut sparsifier - it preserves cuts in any given vertex induced subgraph
(or, any cluster in a fixed partition of ) to within a -multiplicative/additive error with high probability. The power cut
sparsifier uses space and edges, which we show is
asymptotically tight up to polylogarithmic factors in for constant
.Comment: 31 pages, 0 figures, to appear in ITCS 202
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
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