464 research outputs found
Improved Deterministic Network Decomposition
Network decomposition is a central tool in distributed graph algorithms. We
present two improvements on the state of the art for network decomposition,
which thus lead to improvements in the (deterministic and randomized)
complexity of several well-studied graph problems.
- We provide a deterministic distributed network decomposition algorithm with
round complexity, using -bit messages. This improves
on the -round algorithm of Rozho\v{n} and Ghaffari [STOC'20],
which used large messages, and their -round algorithm with -bit messages. This directly leads to similar improvements for a wide range
of deterministic and randomized distributed algorithms, whose solution relies
on network decomposition, including the general distributed derandomization of
Ghaffari, Kuhn, and Harris [FOCS'18].
- One drawback of the algorithm of Rozho\v{n} and Ghaffari, in the
model, was its dependence on the length of the identifiers.
Because of this, for instance, the algorithm could not be used in the
shattering framework in the model. Thus, the state of the
art randomized complexity of several problems in this model remained with an
additive term, which was a clear leftover of the
older network decomposition complexity [Panconesi and Srinivasan STOC'92]. We
present a modified version that remedies this, constructing a decomposition
whose quality does not depend on the identifiers, and thus improves the
randomized round complexity for various problems
The Complexity of Distributed Approximation of Packing and Covering Integer Linear Programs
In this paper, we present a low-diameter decomposition algorithm in the LOCAL
model of distributed computing that succeeds with probability .
Specifically, we show how to compute an low-diameter decomposition in
round
Further developing our techniques, we show new distributed algorithms for
approximating general packing and covering integer linear programs in the LOCAL
model. For packing problems, our algorithm finds an -approximate
solution in rounds
with probability . For covering problems, our algorithm finds an
-approximate solution in rounds with probability . These results improve upon the previous -round algorithm by Ghaffari, Kuhn, and Maus [STOC 2017]
which is based on network decompositions.
Our algorithms are near-optimal for many fundamental combinatorial graph
optimization problems in the LOCAL model, such as minimum vertex cover and
minimum dominating set, as their -approximate solutions
require rounds to compute.Comment: To appear in PODC 202
Faster Deterministic Distributed MIS and Approximate Matching
We present an
round deterministic distributed algorithm for the maximal independent set
problem. By known reductions, this round complexity extends also to maximal
matching, vertex coloring, and edge coloring. These four
problems are among the most central problems in distributed graph algorithms
and have been studied extensively for the past four decades. This improved
round complexity comes closer to the lower bound of
maximal independent set and maximal matching [Balliu et al. FOCS '19]. The
previous best known deterministic complexity for all of these problems was
. Via the shattering technique, the improvement permeates
also to the corresponding randomized complexities, e.g., the new randomized
complexity of vertex coloring is now
rounds.
Our approach is a novel combination of the previously known two methods for
developing deterministic algorithms for these problems, namely global
derandomization via network decomposition (see e.g., [Rozhon, Ghaffari STOC'20;
Ghaffari, Grunau, Rozhon SODA'21; Ghaffari et al. SODA'23]) and local rounding
of fractional solutions (see e.g., [Fischer DISC'17; Harris FOCS'19; Fischer,
Ghaffari, Kuhn FOCS'17; Ghaffari, Kuhn FOCS'21; Faour et al. SODA'23]). We
consider a relaxation of the classic network decomposition concept, where
instead of requiring the clusters in the same block to be non-adjacent, we
allow each node to have a small number of neighboring clusters. We also show a
deterministic algorithm that computes this relaxed decomposition faster than
standard decompositions. We then use this relaxed decomposition to
significantly improve the integrality of certain fractional solutions, before
handing them to the local rounding procedure that now has to do fewer rounding
steps
Undirected -Shortest Paths via Minor-Aggregates: Near-Optimal Deterministic Parallel & Distributed Algorithms
This paper presents near-optimal deterministic parallel and distributed
algorithms for computing -approximate single-source shortest
paths in any undirected weighted graph.
On a high level, we deterministically reduce this and other shortest-path
problems to Minor-Aggregations. A Minor-Aggregation computes an
aggregate (e.g., max or sum) of node-values for every connected component of
some subgraph.
Our reduction immediately implies:
Optimal deterministic parallel (PRAM) algorithms with depth
and near-linear work.
Universally-optimal deterministic distributed (CONGEST) algorithms, whenever
deterministic Minor-Aggregate algorithms exist. For example, an optimal
-round deterministic CONGEST algorithm for
excluded-minor networks.
Several novel tools developed for the above results are interesting in their
own right:
A local iterative approach for reducing shortest path computations "up to
distance " to computing low-diameter decompositions "up to distance
". Compared to the recursive vertex-reduction approach of [Li20],
our approach is simpler, suitable for distributed algorithms, and eliminates
many derandomization barriers.
A simple graph-based -competitive -oblivious routing
based on low-diameter decompositions that can be evaluated in near-linear work.
The previous such routing [ZGY+20] was -competitive and required
more work.
A deterministic algorithm to round any fractional single-source transshipment
flow into an integral tree solution.
The first distributed algorithms for computing Eulerian orientations
Fast Coloring Despite Congested Relays
We provide a -round randomized algorithm for distance-2
coloring in CONGEST with colors. For
, this improves exponentially on the
algorithm of [Halld\'orsson,
Kuhn, Maus, Nolin, DISC'20].
Our study is motivated by the ubiquity and hardness of local reductions in
CONGEST. For instance, algorithms for the Local Lov\'asz Lemma [Moser, Tardos,
JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume
communication on the conflict graph, which can be simulated in LOCAL with only
constant overhead, while this may be prohibitively expensive in CONGEST. We
hope our techniques help tackle in CONGEST other coloring problems defined by
local relations.Comment: 37 pages. To appear in proceedings of DISC 202
Robust network computation
Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (p. 91-98).In this thesis, we present various models of distributed computation and algorithms for these models. The underlying theme is to come up with fast algorithms that can tolerate faults in the underlying network. We begin with the classical message-passing model of computation, surveying many known results. We give a new, universally optimal, edge-biconnectivity algorithm for the classical model. We also give a near-optimal sub-linear algorithm for identifying bridges, when all nodes are activated simultaneously. After discussing some ways in which the classical model is unrealistic, we survey known techniques for adapting the classical model to the real world. We describe a new balancing model of computation. The intent is that algorithms in this model should be automatically fault-tolerant. Existing algorithms that can be expressed in this model are discussed, including ones for clustering, maximum flow, and synchronization. We discuss the use of agents in our model, and give new agent-based algorithms for census and biconnectivity. Inspired by the balancing model, we look at two problems in more depth.(cont.) First, we give matching upper and lower bounds on the time complexity of the census algorithm, and we show how the census algorithm can be used to name nodes uniquely in a faulty network. Second, we consider using discrete harmonic functions as a computational tool. These functions are a natural exemplar of the balancing model. We prove new results concerning the stability and convergence of discrete harmonic functions, and describe a method which we call Eulerization for speeding up convergence.by David Pritchard.M.Eng
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