106 research outputs found
Stable Leader Election in Population Protocols Requires Linear Time
A population protocol *stably elects a leader* if, for all , starting from
an initial configuration with agents each in an identical state, with
probability 1 it reaches a configuration that is correct (exactly
one agent is in a special leader state ) and stable (every configuration
reachable from also has a single agent in state ). We show
that any population protocol that stably elects a leader requires
expected "parallel time" --- expected total pairwise interactions
--- to reach such a stable configuration. Our result also informs the
understanding of the time complexity of chemical self-organization by showing
an essential difficulty in generating exact quantities of molecular species
quickly.Comment: accepted to Distributed Computing special issue of invited papers
from DISC 2015; significantly revised proof structure and intuitive
explanation
Succinct Population Protocols for Presburger Arithmetic
International audienceIn [5], Angluin et al. proved that population protocols compute exactly the predicates definable in Presburger arithmetic (PA), the first-order theory of addition. As part of this result, they presented a procedure that translates any formula of quantifier-free PA with remainder predicates (which has the same expressive power as full PA) into a population protocol with states that computes . More precisely, the number of states of the protocol is exponential in both the bit length of the largest coefficient in the formula, and the number of nodes of its syntax tree. In this paper, we prove that every formula of quantifier-free PA with remainder predicates is computable by a leaderless population protocol with states. Our proof is based on several new constructions, which may be of independent interest. Given a formula of quantifier-free PA with remainder predicates, a first construction produces a succinct protocol (with leaders) that computes Ï; this completes the work initiated in [8], where we constructed such protocols for a fragment of PA. For large enough inputs, we can get rid of these leaders. If the input is not large enough, then it is small, and we design another construction producing a succinct protocol with one leader that computes . Our last construction gets rid of this leader for small inputs
Uniform Partition in Population Protocol Model Under Weak Fairness
We focus on a uniform partition problem in a population protocol model. The uniform partition problem aims to divide a population into k groups of the same size, where k is a given positive integer. In the case of k=2 (called uniform bipartition), a previous work clarified space complexity under various assumptions: 1) an initialized base station (BS) or no BS, 2) weak or global fairness, 3) designated or arbitrary initial states of agents, and 4) symmetric or asymmetric protocols, except for the setting that agents execute a protocol from arbitrary initial states under weak fairness in the model with an initialized base station. In this paper, we clarify the space complexity for this remaining setting. In this setting, we prove that P states are necessary and sufficient to realize asymmetric protocols, and that P+1 states are necessary and sufficient to realize symmetric protocols, where P is the known upper bound of the number of agents. From these results and the previous work, we have clarified the solvability of the uniform bipartition for each combination of assumptions. Additionally, we newly consider an assumption on a model of a non-initialized BS and clarify solvability and space complexity in the assumption. Moreover, the results in this paper can be applied to the case that k is an arbitrary integer (called uniform k-partition)
Global Versus Local Computations: Fast Computing with Identifiers
This paper studies what can be computed by using probabilistic local
interactions with agents with a very restricted power in polylogarithmic
parallel time. It is known that if agents are only finite state (corresponding
to the Population Protocol model by Angluin et al.), then only semilinear
predicates over the global input can be computed. In fact, if the population
starts with a unique leader, these predicates can even be computed in a
polylogarithmic parallel time. If identifiers are added (corresponding to the
Community Protocol model by Guerraoui and Ruppert), then more global predicates
over the input multiset can be computed. Local predicates over the input sorted
according to the identifiers can also be computed, as long as the identifiers
are ordered. The time of some of those predicates might require exponential
parallel time. In this paper, we consider what can be computed with Community
Protocol in a polylogarithmic number of parallel interactions. We introduce the
class CPPL corresponding to protocols that use , for some k,
expected interactions to compute their predicates, or equivalently a
polylogarithmic number of parallel expected interactions. We provide some
computable protocols, some boundaries of the class, using the fact that the
population can compute its size. We also prove two impossibility results
providing some arguments showing that local computations are no longer easy:
the population does not have the time to compare a linear number of consecutive
identifiers. The Linearly Local languages, such that the rational language
, are not computable.Comment: Long version of SSS 2016 publication, appendixed version of SIROCCO
201
Nerio: Leader Election and Edict Ordering
Coordination in a distributed system is facilitated if there is a unique
process, the leader, to manage the other processes. The leader creates edicts
and sends them to other processes for execution or forwarding to other
processes. The leader may fail, and when this occurs a leader election protocol
selects a replacement. This paper describes Nerio, a class of such leader
election protocols
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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