16,656 research outputs found
Space-Optimal Majority in Population Protocols
Population protocols are a model of distributed computing, in which
agents with limited local state interact randomly, and cooperate to
collectively compute global predicates. An extensive series of papers, across
different communities, has examined the computability and complexity
characteristics of this model. Majority, or consensus, is a central task, in
which agents need to collectively reach a decision as to which one of two
states or had a higher initial count. Two complexity metrics are
important: the time that a protocol requires to stabilize to an output
decision, and the state space size that each agent requires.
It is known that majority requires states per agent to
allow for poly-logarithmic time stabilization, and that states
are sufficient. Thus, there is an exponential gap between the upper and lower
bounds.
We address this question. We provide a new lower bound of
states for any protocol which stabilizes in time, for any constant. This result is conditional on basic monotonicity and output
assumptions, satisfied by all known protocols. Technically, it represents a
significant departure from previous lower bounds. Instead of relying on dense
configurations, we introduce a new surgery technique to construct executions
which contradict the correctness of algorithms that stabilize too fast.
Subsequently, our lower bound applies to general initial configurations.
We give an algorithm for majority which uses states, and
stabilizes in time. Central to the algorithm is a new leaderless
phase clock, which allows nodes to synchronize in phases of
consecutive interactions using states per node. We also employ our
phase clock to build a leader election algorithm with states,
which stabilizes in time
Efficient size estimation and impossibility of termination in uniform dense population protocols
We study uniform population protocols: networks of anonymous agents whose
pairwise interactions are chosen at random, where each agent uses an identical
transition algorithm that does not depend on the population size . Many
existing polylog time protocols for leader election and majority
computation are nonuniform: to operate correctly, they require all agents to be
initialized with an approximate estimate of (specifically, the exact value
). Our first main result is a uniform protocol for
calculating with high probability in time and
states ( bits of memory). The protocol is
converging but not terminating: it does not signal when the estimate is close
to the true value of . If it could be made terminating, this would
allow composition with protocols, such as those for leader election or
majority, that require a size estimate initially, to make them uniform (though
with a small probability of failure). We do show how our main protocol can be
indirectly composed with others in a simple and elegant way, based on the
leaderless phase clock, demonstrating that those protocols can in fact be made
uniform. However, our second main result implies that the protocol cannot be
made terminating, a consequence of a much stronger result: a uniform protocol
for any task requiring more than constant time cannot be terminating even with
probability bounded above 0, if infinitely many initial configurations are
dense: any state present initially occupies agents. (In particular,
no leader is allowed.) Crucially, the result holds no matter the memory or time
permitted. Finally, we show that with an initial leader, our size-estimation
protocol can be made terminating with high probability, with the same
asymptotic time and space bounds.Comment: Using leaderless phase cloc
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)
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
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
Brief Announcement: Fast Graphical Population Protocols
Let G be a graph on n nodes. In the stochastic population protocol model, a collection of n indistinguishable, resource-limited nodes collectively solve tasks via pairwise interactions. In each interaction, two randomly chosen neighbors first read each other’s states, and then update their local states. A rich line of research has established tight upper and lower bounds on the complexity of fundamental tasks, such as majority and leader election, in this model, when G is a clique. Specifically, in the clique, these tasks can be solved fast, i.e., in n polylog n pairwise interactions, with high probability, using at most polylog n states per node. In this work, we consider the more general setting where G is an arbitrary graph, and present a technique for simulating protocols designed for fully-connected networks in any connected regular graph. Our main result is a simulation that is efficient on many interesting graph families: roughly, the simulation overhead is polylogarithmic in the number of nodes, and quadratic in the conductance of the graph. As an example, this implies that, in any regular graph with conductance φ, both leader election and exact majority can be solved in φ^{-2} ⋅ n polylog n pairwise interactions, with high probability, using at most φ^{-2} ⋅ polylog n states per node. This shows that there are fast and space-efficient population protocols for leader election and exact majority on graphs with good expansion properties
Fast Graphical Population Protocols
Let be a graph on nodes. In the stochastic population protocol model,
a collection of indistinguishable, resource-limited nodes collectively
solve tasks via pairwise interactions. In each interaction, two randomly chosen
neighbors first read each other's states, and then update their local states. A
rich line of research has established tight upper and lower bounds on the
complexity of fundamental tasks, such as majority and leader election, in this
model, when is a clique. Specifically, in the clique, these tasks can be
solved fast, i.e., in pairwise interactions, with
high probability, using at most states per node.
In this work, we consider the more general setting where is an arbitrary
graph, and present a technique for simulating protocols designed for
fully-connected networks in any connected regular graph. Our main result is a
simulation that is efficient on many interesting graph families: roughly, the
simulation overhead is polylogarithmic in the number of nodes, and quadratic in
the conductance of the graph. As a sample application, we show that, in any
regular graph with conductance , both leader election and exact majority
can be solved in pairwise
interactions, with high probability, using at most states per node. This shows that there are fast and
space-efficient population protocols for leader election and exact majority on
graphs with good expansion properties. We believe our results will prove
generally useful, as they allow efficient technology transfer between the
well-mixed (clique) case, and the under-explored spatial setting.Comment: 47 pages, 5 figure
Message Complexity of Population Protocols
The standard population protocol model assumes that when two agents interact,
each observes the entire state of the other agent. We initiate the study of
for population protocols, where the state of an
agent is divided into an externally-visible and an internal
component, where only the message can be observed by the other agent in an
interaction.
We consider the case of message complexity. When time is unrestricted,
we obtain an exact characterization of the stably computable predicates based
on the number of internal states : If then the protocol
computes semilinear predicates (unlike the original model, which can compute
non-semilinear predicates with ), and otherwise it computes a
predicate decidable by a nondeterministic -space-bounded Turing
machine. We then introduce novel expected time
protocols for junta/leader election and general purpose broadcast correct with
high probability, and approximate and exact population size counting correct
with probability 1. Finally, we show that the main constraint on the power of
bounded-message-size protocols is the size of the internal states: with
unbounded internal states, any computable function can be computed with
probability 1 in the limit by a protocol that uses only
messages
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