630 research outputs found
Distributed Symmetry Breaking in Hypergraphs
Fundamental local symmetry breaking problems such as Maximal Independent Set
(MIS) and coloring have been recognized as important by the community, and
studied extensively in (standard) graphs. In particular, fast (i.e.,
logarithmic run time) randomized algorithms are well-established for MIS and
-coloring in both the LOCAL and CONGEST distributed computing
models. On the other hand, comparatively much less is known on the complexity
of distributed symmetry breaking in {\em hypergraphs}. In particular, a key
question is whether a fast (randomized) algorithm for MIS exists for
hypergraphs.
In this paper, we study the distributed complexity of symmetry breaking in
hypergraphs by presenting distributed randomized algorithms for a variety of
fundamental problems under a natural distributed computing model for
hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can
be solved in rounds ( is the number of nodes of the
hypergraph) in the LOCAL model. We then present a key result of this paper ---
an -round hypergraph MIS algorithm in
the CONGEST model where is the maximum node degree of the hypergraph
and is any arbitrarily small constant.
To demonstrate the usefulness of hypergraph MIS, we present applications of
our hypergraph algorithm to solving problems in (standard) graphs. In
particular, the hypergraph MIS yields fast distributed algorithms for the {\em
balanced minimal dominating set} problem (left open in Harris et al. [ICALP
2013]) and the {\em minimal connected dominating set problem}. We also present
distributed algorithms for coloring, maximal matching, and maximal clique in
hypergraphs.Comment: Changes from the previous version: More references adde
Distributed local approximation algorithms for maximum matching in graphs and hypergraphs
We describe approximation algorithms in Linial's classic LOCAL model of
distributed computing to find maximum-weight matchings in a hypergraph of rank
. Our main result is a deterministic algorithm to generate a matching which
is an -approximation to the maximum weight matching, running in rounds. (Here, the
notations hides and factors).
This is based on a number of new derandomization techniques extending methods
of Ghaffari, Harris & Kuhn (2017).
As a main application, we obtain nearly-optimal algorithms for the
long-studied problem of maximum-weight graph matching. Specifically, we get a
approximation algorithm using randomized time and deterministic time.
The second application is a faster algorithm for hypergraph maximal matching,
a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of
local graph algorithms. This gives an algorithm for -edge-list
coloring in rounds deterministically or
rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most for a graph of
arboricity ; for fixed this runs in
rounds deterministically or rounds randomly
On Derandomizing Local Distributed Algorithms
The gap between the known randomized and deterministic local distributed
algorithms underlies arguably the most fundamental and central open question in
distributed graph algorithms. In this paper, we develop a generic and clean
recipe for derandomizing LOCAL algorithms. We also exhibit how this simple
recipe leads to significant improvements on a number of problem. Two main
results are:
- An improved distributed hypergraph maximal matching algorithm, improving on
Fischer, Ghaffari, and Kuhn [FOCS'17], and giving improved algorithms for
edge-coloring, maximum matching approximation, and low out-degree edge
orientation. The first gives an improved algorithm for Open Problem 11.4 of the
book of Barenboim and Elkin, and the last gives the first positive resolution
of their Open Problem 11.10.
- An improved distributed algorithm for the Lov\'{a}sz Local Lemma, which
gets closer to a conjecture of Chang and Pettie [FOCS'17], and moreover leads
to improved distributed algorithms for problems such as defective coloring and
-SAT.Comment: 37 page
Fast Local Computation Algorithms
For input , let denote the set of outputs that are the "legal"
answers for a computational problem . Suppose and members of are
so large that there is not time to read them in their entirety. We propose a
model of {\em local computation algorithms} which for a given input ,
support queries by a user to values of specified locations in a legal
output . When more than one legal output exists for a given
, the local computation algorithm should output in a way that is consistent
with at least one such . Local computation algorithms are intended to
distill the common features of several concepts that have appeared in various
algorithmic subfields, including local distributed computation, local
algorithms, locally decodable codes, and local reconstruction.
We develop a technique, based on known constructions of small sample spaces
of -wise independent random variables and Beck's analysis in his algorithmic
approach to the Lov{\'{a}}sz Local Lemma, which under certain conditions can be
applied to construct local computation algorithms that run in {\em
polylogarithmic} time and space. We apply this technique to maximal independent
set computations, scheduling radio network broadcasts, hypergraph coloring and
satisfying -SAT formulas.Comment: A preliminary version of this paper appeared in ICS 2011, pp. 223-23
Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching
We present a deterministic distributed algorithm that computes a
-edge-coloring, or even list-edge-coloring, in any -node graph
with maximum degree , in rounds. This answers
one of the long-standing open questions of \emph{distributed graph algorithms}
from the late 1980s, which asked for a polylogarithmic-time algorithm. See,
e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and
Elkin. The previous best round complexities were by
Panconesi and Srinivasan [STOC'92] and
by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our
deterministic list-edge-coloring also improves the randomized complexity of
-edge-coloring to poly rounds.
The key technical ingredient is a deterministic distributed algorithm for
\emph{hypergraph maximal matching}, which we believe will be of interest beyond
this result. In any hypergraph of rank --- where each hyperedge has at most
vertices --- with nodes and maximum degree , this algorithm
computes a maximal matching in rounds.
This hypergraph matching algorithm and its extensions lead to a number of
other results. In particular, a polylogarithmic-time deterministic distributed
maximal independent set algorithm for graphs with bounded neighborhood
independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a
-round deterministic
algorithm for -approximation of maximum matching, and a
quasi-polylogarithmic-time deterministic distributed algorithm for orienting
-arboricity graphs with out-degree at most ,
for any constant , hence partially answering Open Problem 10 of
Barenboim and Elkin's book
On Computing Maximal Independent Sets of Hypergraphs in Parallel
Whether or not the problem of finding maximal independent sets (MIS) in
hypergraphs is in (R)NC is one of the fundamental problems in the theory of
parallel computing. Unlike the well-understood case of MIS in graphs, for the
hypergraph problem, our knowledge is quite limited despite considerable work.
It is known that the problem is in \emph{RNC} when the edges of the hypergraph
have constant size. For general hypergraphs with vertices and edges,
the fastest previously known algorithm works in time with
processors. In this paper we give an EREW PRAM algorithm
that works in time with processors on general
hypergraphs satisfying , where
and . Our algorithm is
based on a sampling idea that reduces the dimension of the hypergraph and
employs the algorithm for constant dimension hypergraphs as a subroutine
On distributed scheduling in wireless networks exploiting broadcast and network coding
In this paper, we consider cross-layer optimization in wireless networks with wireless broadcast advantage, focusing on the problem of distributed scheduling of broadcast links. The wireless broadcast advantage is most useful in multicast scenarios. As such, we include network coding in our design to exploit the throughput gain brought in by network coding for multicasting. We derive a subgradient algorithm for joint rate control, network coding and scheduling, which however requires centralized link scheduling. Under the primary interference model, link scheduling problem is equivalent to a maximum weighted hypergraph matching problem that is NP-complete. To solve the scheduling problem distributedly, locally greedy and randomized approximation algorithms are proposed and shown to have bounded worst-case performance. With random network coding, we obtain a fully distributed cross-layer design. Numerical results show promising throughput gain using the proposed algorithms, and surprisingly, in some cases even with less complexity than cross-layer design without broadcast advantage
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