5 research outputs found
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
Routing Schemes and Distance Oracles in the Hybrid Model
The model was introduced as a means for theoretical study
of distributed networks that use various communication modes. Conceptually, it
is a synchronous message passing model with a local communication mode, where
in each round each node can send large messages to all its neighbors in a local
network (a graph), and a global communication mode, where each node is allotted
limited (polylogarithmic) bandwidth per round which it can use to communicate
with any node in the network.
Prior work has often focused on shortest paths problems in the local network,
as their global nature makes these an interesting case study how combining
communication modes in the model can overcome the individual
lower bounds of either mode. In this work we consider a similar problem, namely
computation of distance oracles and routing schemes. In the former, all nodes
have to compute local tables, which allows them to look up the distance
(estimates) to any target node in the local network when provided with the
label of the target. In the latter, it suffices that nodes give the next node
on an (approximately) shortest path to the target.
Our goal is to compute these local tables as fast as possible with labels as
small as possible. We show that this can be done exactly in communication rounds and labels of size bits. For
constant stretch approximations we achieve labels of size in the
same time. Further, as our main technical contribution, we provide
computational lower bounds for a variety of problem parameters. For instance,
we show that computing solutions with stretch below a certain constant takes
rounds even for labels of size
Round- and Message-Optimal Distributed Graph Algorithms
Distributed graph algorithms that separately optimize for either the number
of rounds used or the total number of messages sent have been studied
extensively. However, algorithms simultaneously efficient with respect to both
measures have been elusive. For example, only very recently was it shown that
for Minimum Spanning Tree (MST), an optimal message and round complexity is
achievable (up to polylog terms) by a single algorithm in the CONGEST model of
communication.
In this paper we provide algorithms that are simultaneously round- and
message-optimal for a number of well-studied distributed optimization problems.
Our main result is such a distributed algorithm for the fundamental primitive
of computing simple functions over each part of a graph partition. From this
algorithm we derive round- and message-optimal algorithms for multiple
problems, including MST, Approximate Min-Cut and Approximate Single Source
Shortest Paths, among others. On general graphs all of our algorithms achieve
worst-case optimal round complexity and
message complexity. Furthermore, our algorithms require an optimal
rounds and messages on planar, genus-bounded,
treewidth-bounded and pathwidth-bounded graphs.Comment: To appear in PODC 201
Minor Excluded Network Families Admit Fast Distributed Algorithms
Distributed network optimization algorithms, such as minimum spanning tree,
minimum cut, and shortest path, are an active research area in distributed
computing. This paper presents a fast distributed algorithm for such problems
in the CONGEST model, on networks that exclude a fixed minor.
On general graphs, many optimization problems, including the ones mentioned
above, require rounds of communication in the CONGEST
model, even if the network graph has a much smaller diameter. Naturally, the
next step in algorithm design is to design efficient algorithms which bypass
this lower bound on a restricted class of graphs. Currently, the only known
method of doing so uses the low-congestion shortcut framework of Ghaffari and
Haeupler [SODA'16]. Building off of their work, this paper proves that excluded
minor graphs admit high-quality shortcuts, leading to an round
algorithm for the aforementioned problems, where is the diameter of the
network graph. To work with excluded minor graph families, we utilize the Graph
Structure Theorem of Robertson and Seymour. To the best of our knowledge, this
is the first time the Graph Structure Theorem has been used for an algorithmic
result in the distributed setting.
Even though the proof is involved, merely showing the existence of good
shortcuts is sufficient to obtain simple, efficient distributed algorithms. In
particular, the shortcut framework can efficiently construct near-optimal
shortcuts and then use them to solve the optimization problems. This, combined
with the very general family of excluded minor graphs, which includes most
other important graph classes, makes this result of significant interest