4 research outputs found
Small Cuts and Connectivity Certificates: A Fault Tolerant Approach
We revisit classical connectivity problems in the {CONGEST} model of distributed computing. By using techniques from fault tolerant network design, we show improved constructions, some of which are even "local" (i.e., with O~(1) rounds) for problems that are closely related to hard global problems (i.e., with a lower bound of Omega(Diam+sqrt{n}) rounds).
Distributed Minimum Cut: Nanongkai and Su presented a randomized algorithm for computing a (1+epsilon)-approximation of the minimum cut using O~(D +sqrt{n}) rounds where D is the diameter of the graph. For a sufficiently large minimum cut lambda=Omega(sqrt{n}), this is tight due to Das Sarma et al. [FOCS \u2711], Ghaffari and Kuhn [DISC \u2713].
- Small Cuts: A special setting that remains open is where the graph connectivity lambda is small (i.e., constant). The only lower bound for this case is Omega(D), with a matching bound known only for lambda <= 2 due to Pritchard and Thurimella [TALG \u2711]. Recently, Daga, Henzinger, Nanongkai and Saranurak [STOC \u2719] raised the open problem of computing the minimum cut in poly(D) rounds for any lambda=O(1). In this paper, we resolve this problem by presenting a surprisingly simple algorithm, that takes a completely different approach than the existing algorithms. Our algorithm has also the benefit that it computes all minimum cuts in the graph, and naturally extends to vertex cuts as well. At the heart of the algorithm is a graph sampling approach usually used in the context of fault tolerant (FT) design.
- Deterministic Algorithms: While the existing distributed minimum cut algorithms are randomized, our algorithm can be made deterministic within the same round complexity. To obtain this, we introduce a novel definition of universal sets along with their efficient computation. This allows us to derandomize the FT graph sampling technique, which might be of independent interest.
- Computation of all Edge Connectivities: We also consider the more general task of computing the edge connectivity of all the edges in the graph. In the output format, it is required that the endpoints u,v of every edge (u,v) learn the cardinality of the u-v cut in the graph. We provide the first sublinear algorithm for this problem for the case of constant connectivity values. Specifically, by using the recent notion of low-congestion cycle cover, combined with the sampling technique, we compute all edge connectivities in poly(D) * 2^{O(sqrt{log n log log n})} rounds.
Sparse Certificates: For an n-vertex graph G and an integer lambda, a lambda-sparse certificate H is a subgraph H subseteq G with O(lambda n) edges which is lambda-connected iff G is lambda-connected. For D-diameter graphs, constructions of sparse certificates for lambda in {2,3} have been provided by Thurimella [J. Alg. \u2797] and Dori [PODC \u2718] respectively using O~(D) number of rounds. The problem of devising such certificates with o(D+sqrt{n}) rounds was left open by Dori [PODC \u2718] for any lambda >= 4. Using connections to fault tolerant spanners, we considerably improve the round complexity for any lambda in [1,n] and epsilon in (0,1), by showing a construction of (1-epsilon)lambda-sparse certificates with O(lambda n) edges using only O(1/epsilon^2 * log^{2+o(1)} n) rounds
On Packing Low-Diameter Spanning Trees
Edge connectivity of a graph is one of the most fundamental graph-theoretic
concepts. The celebrated tree packing theorem of Tutte and Nash-Williams from
1961 states that every -edge connected graph contains a collection
of edge-disjoint spanning trees, that we refer
to as a tree packing; the diameter of the tree packing is the largest
diameter of any tree in . A desirable property of a tree packing, that
is both sufficient and necessary for leveraging the high connectivity of a
graph in distributed communication, is that its diameter is low. Yet, despite
extensive research in this area, it is still unclear how to compute a tree
packing, whose diameter is sublinear in , in a low-diameter graph ,
or alternatively how to show that such a packing does not exist.
In this paper we provide first non-trivial upper and lower bounds on the
diameter of tree packing. First, we show that, for every -edge connected
-vertex graph of diameter , there is a tree packing of size
, diameter , that causes edge-congestion at most
. Second, we show that for every -edge connected -vertex graph of
diameter , the diameter of is with high
probability, where is obtained by sampling each edge of
independently with probability . This provides a packing of
edge-disjoint trees of diameter at most
each. We then prove that these two results are nearly tight. Lastly, we show
that if every pair of vertices in a graph has edge-disjoint paths of length
at most connecting them, then there is a tree packing of size , diameter
, causing edge-congestion . We also provide several
applications of low-diameter tree packing in distributed computation
Topology Dependent Bounds For FAQs
In this paper, we prove topology dependent bounds on the number of rounds
needed to compute Functional Aggregate Queries (FAQs) studied by Abo Khamis et
al. [PODS 2016] in a synchronous distributed network under the model considered
by Chattopadhyay et al. [FOCS 2014, SODA 2017]. Unlike the recent work on
computing database queries in the Massively Parallel Computation model, in the
model of Chattopadhyay et al., nodes can communicate only via private
point-to-point channels and we are interested in bounds that work over an {\em
arbitrary} communication topology. This is the first work to consider more
practically motivated problems in this distributed model. For the sake of
exposition, we focus on two special problems in this paper: Boolean Conjunctive
Query (BCQ) and computing variable/factor marginals in Probabilistic Graphical
Models (PGMs). We obtain tight bounds on the number of rounds needed to compute
such queries as long as the underlying hypergraph of the query is
-degenerate and has -arity. In particular, the -degeneracy
condition covers most well-studied queries that are efficiently computable in
the centralized computation model like queries with constant treewidth. These
tight bounds depend on a new notion of `width' (namely internal-node-width) for
Generalized Hypertree Decompositions (GHDs) of acyclic hypergraphs, which
minimizes the number of internal nodes in a sub-class of GHDs. To the best of
our knowledge, this width has not been studied explicitly in the theoretical
database literature. Finally, we consider the problem of computing the product
of a vector with a chain of matrices and prove tight bounds on its round
complexity (over the finite field of two elements) using a novel min-entropy
based argument.Comment: A conference version was presented at PODS 201
Improved distributed algorithms for fundamental graph problems
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2017.Cataloged from PDF version of thesis.Includes bibliographical references (pages 237-255).Distributed graph algorithms provide efficient and theoretically sound methods for solving graph problems in distributed settings and more generally for performing distributed computation in networks. These algorithms are applicable in a wide variety of settings, ranging from computer networks to massively parallel computing and beyond. This thesis addresses a number of the central problems of distributed graph algorithms. These problems generally revolve around two of the principal challenges of the area, locality and congestion. The problems include computing maximal independent set, minimum spanning tree, minimum edge cut and minimum vertex cut, graph connectivity decompositions, network information dissemination, minimum-weight connected dominating set, and scheduling distributed protocols. We develop novel techniques, concepts, and tools for these problems, and present algorithms and impossibility results which improve considerably on the state of the art, in several cases resolving or advancing long-standing open problems.by Mohsen Ghaffari.Ph. D