7,955 research outputs found
Silent MST approximation for tiny memory
In network distributed computing, minimum spanning tree (MST) is one of the
key problems, and silent self-stabilization one of the most demanding
fault-tolerance properties. For this problem and this model, a polynomial-time
algorithm with memory is known for the state model. This is
memory optimal for weights in the classic range (where
is the size of the network). In this paper, we go below this
memory, using approximation and parametrized complexity.
More specifically, our contributions are two-fold. We introduce a second
parameter~, which is the space needed to encode a weight, and we design a
silent polynomial-time self-stabilizing algorithm, with space . In turn, this allows us to get an approximation algorithm for the problem,
with a trade-off between the approximation ratio of the solution and the space
used. For polynomial weights, this trade-off goes smoothly from memory for an -approximation, to memory for exact solutions,
with for example memory for a 2-approximation
Trade-Offs in Distributed Interactive Proofs
The study of interactive proofs in the context of distributed network computing is a novel topic, recently introduced by Kol, Oshman, and Saxena [PODC 2018]. In the spirit of sequential interactive proofs theory, we study the power of distributed interactive proofs. This is achieved via a series of results establishing trade-offs between various parameters impacting the power of interactive proofs, including the number of interactions, the certificate size, the communication complexity, and the form of randomness used. Our results also connect distributed interactive proofs with the established field of distributed verification. In general, our results contribute to providing structure to the landscape of distributed interactive proofs
Polynomial-Time Space-Optimal Silent Self-Stabilizing Minimum-Degree Spanning Tree Construction
Motivated by applications to sensor networks, as well as to many other areas,
this paper studies the construction of minimum-degree spanning trees. We
consider the classical node-register state model, with a weakly fair scheduler,
and we present a space-optimal \emph{silent} self-stabilizing construction of
minimum-degree spanning trees in this model. Computing a spanning tree with
minimum degree is NP-hard. Therefore, we actually focus on constructing a
spanning tree whose degree is within one from the optimal. Our algorithm uses
registers on bits, converges in a polynomial number of rounds, and
performs polynomial-time computation at each node. Specifically, the algorithm
constructs and stabilizes on a special class of spanning trees, with degree at
most . Indeed, we prove that, unless NP coNP, there are no
proof-labeling schemes involving polynomial-time computation at each node for
the whole family of spanning trees with degree at most . Up to our
knowledge, this is the first example of the design of a compact silent
self-stabilizing algorithm constructing, and stabilizing on a subset of optimal
solutions to a natural problem for which there are no time-efficient
proof-labeling schemes. On our way to design our algorithm, we establish a set
of independent results that may have interest on their own. In particular, we
describe a new space-optimal silent self-stabilizing spanning tree
construction, stabilizing on \emph{any} spanning tree, in rounds, and
using just \emph{one} additional bit compared to the size of the labels used to
certify trees. We also design a silent loop-free self-stabilizing algorithm for
transforming a tree into another tree. Last but not least, we provide a silent
self-stabilizing algorithm for computing and certifying the labels of a
NCA-labeling scheme
Introduction to local certification
A distributed graph algorithm is basically an algorithm where every node of a
graph can look at its neighborhood at some distance in the graph and chose its
output. As distributed environment are subject to faults, an important issue is
to be able to check that the output is correct, or in general that the network
is in proper configuration with respect to some predicate. One would like this
checking to be very local, to avoid using too much resources. Unfortunately
most predicates cannot be checked this way, and that is where certification
comes into play. Local certification (also known as proof-labeling schemes,
locally checkable proofs or distributed verification) consists in assigning
labels to the nodes, that certify that the configuration is correct. There are
several point of view on this topic: it can be seen as a part of
self-stabilizing algorithms, as labeling problem, or as a non-deterministic
distributed decision.
This paper is an introduction to the domain of local certification, giving an
overview of the history, the techniques and the current research directions.Comment: Last update: minor editin
Distributed Certification for Classes of Dense Graphs
A proof-labeling scheme (PLS) for a boolean predicate on labeled graphs
is a mechanism used for certifying the legality with respect to of global
network states in a distributed manner. In a PLS, a certificate is assigned to
each processing node of the network, and the nodes are in charge of checking
that the collection of certificates forms a global proof that the system is in
a correct state, by exchanging the certificates once, between neighbors only.
The main measure of complexity is the size of the certificates. Many PLSs have
been designed for certifying specific predicates, including cycle-freeness,
minimum-weight spanning tree, planarity, etc.
In 2021, a breakthrough has been obtained, as a meta-theorem stating that a
large set of properties have compact PLSs in a large class of networks. Namely,
for every property on labeled graphs, there exists a PLS
for with -bit certificates for all graphs of bounded
tree-depth. This result has been extended to the larger class of graphs with
bounded {tree-width}, using certificates on bits.
We extend this result even further, to the larger class of graphs with
bounded clique-width, which, as opposed to the other two aforementioned
classes, includes dense graphs. We show that, for every
property on labeled graphs, there exists a PLS for with bit certificates for all graphs of bounded clique-width
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