237 research outputs found
A Faster Distributed Single-Source Shortest Paths Algorithm
We devise new algorithms for the single-source shortest paths (SSSP) problem
with non-negative edge weights in the CONGEST model of distributed computing.
While close-to-optimal solutions, in terms of the number of rounds spent by the
algorithm, have recently been developed for computing SSSP approximately, the
fastest known exact algorithms are still far away from matching the lower bound
of rounds by Peleg and Rubinovich [SIAM
Journal on Computing 2000], where is the number of nodes in the network
and is its diameter. The state of the art is Elkin's randomized algorithm
[STOC 2017] that performs rounds. We
significantly improve upon this upper bound with our two new randomized
algorithms for polynomially bounded integer edge weights, the first performing
rounds and the second performing rounds. Our bounds also compare favorably to the
independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a
-approximation -round algorithm for directed SSSP and a new work/depth trade-off for exact
SSSP on directed graphs in the PRAM model.Comment: Presented at the the 59th Annual IEEE Symposium on Foundations of
Computer Science (FOCS 2018
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
Reachability and Shortest Paths in the Broadcast CONGEST Model
In this paper we study the time complexity of the single-source reachability problem and the single-source shortest path problem for directed unweighted graphs in the Broadcast CONGEST model. We focus on the case where the diameter D of the underlying network is constant.
We show that for the case where D = 1 there is, quite surprisingly, a very simple algorithm that solves the reachability problem in 1(!) round. In contrast, for networks with D = 2, we show that any distributed algorithm (possibly randomized) for this problem requires Omega(sqrt{n/ log{n}}) rounds. Our results therefore completely resolve (up to a small polylog factor) the complexity of the single-source reachability problem for a wide range of diameters.
Furthermore, we show that when D = 1, it is even possible to get an almost 3 - approximation for the all-pairs shortest path problem (for directed unweighted graphs) in just 2 rounds. We also prove a stronger lower bound of Omega(sqrt{n}) for the single-source shortest path problem for unweighted directed graphs that holds even when the diameter of the underlying network is 2. As far as we know this is the first lower bound that achieves Omega(sqrt{n}) for this problem
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