10,123 research outputs found
Distributed Edge Connectivity in Sublinear Time
We present the first sublinear-time algorithm for a distributed
message-passing network sto compute its edge connectivity exactly in
the CONGEST model, as long as there are no parallel edges. Our algorithm takes
time to compute and a
cut of cardinality with high probability, where and are the
number of nodes and the diameter of the network, respectively, and
hides polylogarithmic factors. This running time is sublinear in (i.e.
) whenever is. Previous sublinear-time
distributed algorithms can solve this problem either (i) exactly only when
[Thurimella PODC'95; Pritchard, Thurimella, ACM
Trans. Algorithms'11; Nanongkai, Su, DISC'14] or (ii) approximately [Ghaffari,
Kuhn, DISC'13; Nanongkai, Su, DISC'14].
To achieve this we develop and combine several new techniques. First, we
design the first distributed algorithm that can compute a -edge connectivity
certificate for any in time .
Second, we show that by combining the recent distributed expander decomposition
technique of [Chang, Pettie, Zhang, SODA'19] with techniques from the
sequential deterministic edge connectivity algorithm of [Kawarabayashi, Thorup,
STOC'15], we can decompose the network into a sublinear number of clusters with
small average diameter and without any mincut separating a cluster (except the
`trivial' ones). Finally, by extending the tree packing technique from [Karger
STOC'96], we can find the minimum cut in time proportional to the number of
components. As a byproduct of this technique, we obtain an -time
algorithm for computing exact minimum cut for weighted graphs.Comment: Accepted at 51st ACM Symposium on Theory of Computing (STOC 2019
Near-linear Time Algorithm for Approximate Minimum Degree Spanning Trees
Given a graph , we wish to compute a spanning tree whose maximum
vertex degree, i.e. tree degree, is as small as possible. Computing the exact
optimal solution is known to be NP-hard, since it generalizes the Hamiltonian
path problem. For the approximation version of this problem, a
time algorithm that computes a spanning tree of degree at most is
previously known [F\"urer \& Raghavachari 1994]; here denotes the
minimum tree degree of all the spanning trees. In this paper we give the first
near-linear time approximation algorithm for this problem. Specifically
speaking, we propose an time algorithm that
computes a spanning tree with tree degree for any constant .
Thus, when , we can achieve approximate solutions with
constant approximate ratio arbitrarily close to 1 in near-linear time.Comment: 17 page
Distributed Computing on Core-Periphery Networks: Axiom-based Design
Inspired by social networks and complex systems, we propose a core-periphery
network architecture that supports fast computation for many distributed
algorithms and is robust and efficient in number of links. Rather than
providing a concrete network model, we take an axiom-based design approach. We
provide three intuitive (and independent) algorithmic axioms and prove that any
network that satisfies all axioms enjoys an efficient algorithm for a range of
tasks (e.g., MST, sparse matrix multiplication, etc.). We also show the
minimality of our axiom set: for networks that satisfy any subset of the
axioms, the same efficiency cannot be guaranteed for any deterministic
algorithm
Content Distribution by Multiple Multicast Trees and Intersession Cooperation: Optimal Algorithms and Approximations
In traditional massive content distribution with multiple sessions, the
sessions form separate overlay networks and operate independently, where some
sessions may suffer from insufficient resources even though other sessions have
excessive resources. To cope with this problem, we consider the universal
swarming approach, which allows multiple sessions to cooperate with each other.
We formulate the problem of finding the optimal resource allocation to maximize
the sum of the session utilities and present a subgradient algorithm which
converges to the optimal solution in the time-average sense. The solution
involves an NP-hard subproblem of finding a minimum-cost Steiner tree. We cope
with this difficulty by using a column generation method, which reduces the
number of Steiner-tree computations. Furthermore, we allow the use of
approximate solutions to the Steiner-tree subproblem. We show that the
approximation ratio to the overall problem turns out to be no less than the
reciprocal of the approximation ratio to the Steiner-tree subproblem.
Simulation results demonstrate that universal swarming improves the performance
of resource-poor sessions with negligible impact to resource-rich sessions. The
proposed approach and algorithm are expected to be useful for
infrastructure-based content distribution networks with long-lasting sessions
and relatively stable network environment
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