13,417 research outputs found
Time vs. Information Tradeoffs for Leader Election in Anonymous Trees
The leader election task calls for all nodes of a network to agree on a
single node. If the nodes of the network are anonymous, the task of leader
election is formulated as follows: every node of the network must output a
simple path, coded as a sequence of port numbers, such that all these paths end
at a common node, the leader. In this paper, we study deterministic leader
election in anonymous trees.
Our aim is to establish tradeoffs between the allocated time and the
amount of information that has to be given to the nodes to
enable leader election in time in all trees for which leader election in
this time is at all possible. Following the framework of , this information (a single binary string) is provided to all
nodes at the start by an oracle knowing the entire tree. The length of this
string is called the . For an allocated time ,
we give upper and lower bounds on the minimum size of advice sufficient to
perform leader election in time .
We consider -node trees of diameter . While leader election
in time can be performed without any advice, for time we give
tight upper and lower bounds of . For time we give
tight upper and lower bounds of for even values of ,
and tight upper and lower bounds of for odd values of .
For the time interval for constant ,
we prove an upper bound of and a lower bound of
, the latter being valid whenever is odd or when
the time is at most . Finally, for time for any
constant (except for the case of very small diameters), we give
tight upper and lower bounds of
Leader Election in Anonymous Rings: Franklin Goes Probabilistic
We present a probabilistic leader election algorithm for anonymous, bidirectional, asynchronous rings. It is based on an algorithm from Franklin, augmented with random identity selection, hop counters to detect identity clashes, and round numbers modulo 2. As a result, the algorithm is finite-state, so that various model checking techniques can be employed to verify its correctness, that is, eventually a unique leader is elected with probability one. We also sketch a formal correctness proof of the algorithm for rings with arbitrary size
Breaking Symmetries
A well-known result by Palamidessi tells us that {\pi}mix (the {\pi}-calculus
with mixed choice) is more expressive than {\pi}sep (its subset with only
separate choice). The proof of this result argues with their different
expressive power concerning leader election in symmetric networks. Later on,
Gorla of- fered an arguably simpler proof that, instead of leader election in
symmetric networks, employed the reducibility of "incestual" processes (mixed
choices that include both enabled senders and receivers for the same channel)
when running two copies in parallel. In both proofs, the role of breaking (ini-
tial) symmetries is more or less apparent. In this paper, we shed more light on
this role by re-proving the above result-based on a proper formalization of
what it means to break symmetries-without referring to another layer of the
distinguishing problem domain of leader election.
Both Palamidessi and Gorla rephrased their results by stating that there is
no uniform and reason- able encoding from {\pi}mix into {\pi}sep . We indicate
how the respective proofs can be adapted and exhibit the consequences of
varying notions of uniformity and reasonableness. In each case, the ability to
break initial symmetries turns out to be essential
Computing in Additive Networks with Bounded-Information Codes
This paper studies the theory of the additive wireless network model, in
which the received signal is abstracted as an addition of the transmitted
signals. Our central observation is that the crucial challenge for computing in
this model is not high contention, as assumed previously, but rather
guaranteeing a bounded amount of \emph{information} in each neighborhood per
round, a property that we show is achievable using a new random coding
technique.
Technically, we provide efficient algorithms for fundamental distributed
tasks in additive networks, such as solving various symmetry breaking problems,
approximating network parameters, and solving an \emph{asymmetry revealing}
problem such as computing a maximal input.
The key method used is a novel random coding technique that allows a node to
successfully decode the received information, as long as it does not contain
too many distinct values. We then design our algorithms to produce a limited
amount of information in each neighborhood in order to leverage our enriched
toolbox for computing in additive networks
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
How Many Cooks Spoil the Soup?
In this work, we study the following basic question: "How much parallelism
does a distributed task permit?" Our definition of parallelism (or symmetry)
here is not in terms of speed, but in terms of identical roles that processes
have at the same time in the execution. We initiate this study in population
protocols, a very simple model that not only allows for a straightforward
definition of what a role is, but also encloses the challenge of isolating the
properties that are due to the protocol from those that are due to the
adversary scheduler, who controls the interactions between the processes. We
(i) give a partial characterization of the set of predicates on input
assignments that can be stably computed with maximum symmetry, i.e.,
, where is the minimum multiplicity of a state in
the initial configuration, and (ii) we turn our attention to the remaining
predicates and prove a strong impossibility result for the parity predicate:
the inherent symmetry of any protocol that stably computes it is upper bounded
by a constant that depends on the size of the protocol.Comment: 19 page
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