18 research outputs found
Fast Partial Distance Estimation and Applications
We study approximate distributed solutions to the weighted {\it
all-pairs-shortest-paths} (APSP) problem in the CONGEST model. We obtain the
following results.
A deterministic -approximation to APSP in
rounds. This improves over the best previously known algorithm, by both
derandomizing it and by reducing the running time by a factor.
In many cases, routing schemes involve relabeling, i.e., assigning new names
to nodes and require that these names are used in distance and routing queries.
It is known that relabeling is necessary to achieve running times of . In the relabeling model, we obtain the following results.
A randomized -approximation to APSP, for any integer ,
running in rounds, where is the hop diameter of
the network. This algorithm simplifies the best previously known result and
reduces its approximation ratio from to . Also, the new
algorithm uses uses labels of asymptotically optimal size, namely
bits.
A randomized -approximation to APSP, for any integer ,
running in time and producing {\it
compact routing tables} of size . The node lables consist
of bits. This improves on the approximation ratio of
for tables of that size achieved by the best previously known algorithm, which
terminates faster, in rounds
On Efficient Distributed Construction of Near Optimal Routing Schemes
Given a distributed network represented by a weighted undirected graph
on vertices, and a parameter , we devise a distributed
algorithm that computes a routing scheme in
rounds, where is the hop-diameter of the network. The running time matches
the lower bound of rounds (which holds for any
scheme with polynomial stretch), up to lower order terms. The routing tables
are of size , the labels are of size , and
every packet is routed on a path suffering stretch at most . Our
construction nearly matches the state-of-the-art for routing schemes built in a
centralized sequential manner. The previous best algorithms for building
routing tables in a distributed small messages model were by \cite[STOC
2013]{LP13} and \cite[PODC 2015]{LP15}. The former has similar properties but
suffers from substantially larger routing tables of size ,
while the latter has sub-optimal running time of
Parallel Metric Tree Embedding based on an Algebraic View on Moore-Bellman-Ford
A \emph{metric tree embedding} of expected \emph{stretch~}
maps a weighted -node graph to a weighted tree with such that, for all ,
and
. Such embeddings are highly useful for designing
fast approximation algorithms, as many hard problems are easy to solve on tree
instances. However, to date the best parallel -depth algorithm that achieves an asymptotically optimal expected stretch of
requires
work and a metric as input.
In this paper, we show how to achieve the same guarantees using
depth and
work, where and is an arbitrarily small constant.
Moreover, one may further reduce the work to at the expense of increasing the expected stretch to
.
Our main tool in deriving these parallel algorithms is an algebraic
characterization of a generalization of the classic Moore-Bellman-Ford
algorithm. We consider this framework, which subsumes a variety of previous
"Moore-Bellman-Ford-like" algorithms, to be of independent interest and discuss
it in depth. In our tree embedding algorithm, we leverage it for providing
efficient query access to an approximate metric that allows sampling the tree
using depth and work.
We illustrate the generality and versatility of our techniques by various
examples and a number of additional results