2,832 research outputs found
Deterministic Population Protocols for Exact Majority and Plurality
In this paper we study space-efficient deterministic population protocols for several variants of the majority problem including plurality consensus. We focus on space efficient majority protocols in populations with an arbitrary number of colours C represented by k-bit labels, where k = ceiling (log C). In particular, we present asymptotically space-optimal (with respect to the adopted k-bit representation of colours) protocols for (1) the absolute majority problem, i.e., a protocol which decides whether a single colour dominates all other colours considered together, and (2) the relative majority problem, also known in the literature as plurality consensus, in which colours declare their volume superiority versus other individual colours. The new population protocols proposed in this paper rely on a dynamic formulation of the majority problem in which the colours originally present in the population can be changed by an external force during the communication process. The considered dynamic formulation is based on the concepts studied by D. Angluin et al. and O. Michail et al. about stabilizing inputs and composition of population protocols. Also, the protocols presented in this paper use a composition of some known protocols for static and dynamic majority
On the Necessary Memory to Compute the Plurality in Multi-Agent Systems
We consider the Relative-Majority Problem (also known as Plurality), in
which, given a multi-agent system where each agent is initially provided an
input value out of a set of possible ones, each agent is required to
eventually compute the input value with the highest frequency in the initial
configuration. We consider the problem in the general Population Protocols
model in which, given an underlying undirected connected graph whose nodes
represent the agents, edges are selected by a globally fair scheduler.
The state complexity that is required for solving the Plurality Problem
(i.e., the minimum number of memory states that each agent needs to have in
order to solve the problem), has been a long-standing open problem. The best
protocol so far for the general multi-valued case requires polynomial memory:
Salehkaleybar et al. (2015) devised a protocol that solves the problem by
employing states per agent, and they conjectured their upper bound
to be optimal. On the other hand, under the strong assumption that agents
initially agree on a total ordering of the initial input values, Gasieniec et
al. (2017), provided an elegant logarithmic-memory plurality protocol.
In this work, we refute Salehkaleybar et al.'s conjecture, by providing a
plurality protocol which employs states per agent. Central to our
result is an ordering protocol which allows to leverage on the plurality
protocol by Gasieniec et al., of independent interest. We also provide a
-state lower bound on the necessary memory to solve the problem,
proving that the Plurality Problem cannot be solved within the mere memory
necessary to encode the output.Comment: 14 pages, accepted at CIAC 201
Noisy Rumor Spreading and Plurality Consensus
Error-correcting codes are efficient methods for handling \emph{noisy}
communication channels in the context of technological networks. However, such
elaborate methods differ a lot from the unsophisticated way biological entities
are supposed to communicate. Yet, it has been recently shown by Feinerman,
Haeupler, and Korman {[}PODC 2014{]} that complex coordination tasks such as
\emph{rumor spreading} and \emph{majority consensus} can plausibly be achieved
in biological systems subject to noisy communication channels, where every
message transferred through a channel remains intact with small probability
, without using coding techniques. This result is a
considerable step towards a better understanding of the way biological entities
may cooperate. It has been nevertheless be established only in the case of
2-valued \emph{opinions}: rumor spreading aims at broadcasting a single-bit
opinion to all nodes, and majority consensus aims at leading all nodes to adopt
the single-bit opinion that was initially present in the system with (relative)
majority. In this paper, we extend this previous work to -valued opinions,
for any .
Our extension requires to address a series of important issues, some
conceptual, others technical. We had to entirely revisit the notion of noise,
for handling channels carrying -\emph{valued} messages. In fact, we
precisely characterize the type of noise patterns for which plurality consensus
is solvable. Also, a key result employed in the bivalued case by Feinerman et
al. is an estimate of the probability of observing the most frequent opinion
from observing the mode of a small sample. We generalize this result to the
multivalued case by providing a new analytical proof for the bivalued case that
is amenable to be extended, by induction, and that is of independent interest.Comment: Minor revisio
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)
Communication cost of consensus for nodes with limited memory
Motivated by applications in blockchains and sensor networks, we consider a
model of nodes trying to reach consensus on their majority bit. Each node
is assigned a bit at time zero, and is a finite automaton with bits of
memory (i.e., states) and a Poisson clock. When the clock of rings,
can choose to communicate, and is then matched to a uniformly chosen node
. The nodes and may update their states based on the state of the
other node. Previous work has focused on minimizing the time to consensus and
the probability of error, while our goal is minimizing the number of
communications. We show that when , consensus can be
reached at linear communication cost, but this is impossible if
. We also study a synchronous variant of the model, where
our upper and lower bounds on for achieving linear communication cost are
and , respectively. A key step is to
distinguish when nodes can become aware of knowing the majority bit and stop
communicating. We show that this is impossible if their memory is too low.Comment: 62 pages, 5 figure
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
Population Protocols for Exact Plurality Consensus -- How a small chance of failure helps to eliminate insignificant opinions
We consider the \emph{exact plurality consensus} problem for \emph{population
protocols}. Here, anonymous agents start each with one of opinions.
Their goal is to agree on the initially most frequent opinion (the
\emph{plurality opinion}) via random, pairwise interactions. The case of opinions is known as the \emph{majority problem}. Recent breakthroughs led
to an always correct, exact majority population protocol that is both time- and
space-optimal, needing states per agent and, with high probability,
time~[Doty, Eftekhari, Gasieniec, Severson, Stachowiak, and
Uznanski; 2021]. We know that any always correct protocol requires
states, while the currently best protocol needs
states~[Natale and Ramezani; 2019]. For ordered opinions, this can be improved
to ~[Gasieniec, Hamilton, Martin, Spirakis, and Stachowiak; 2016]. We
design protocols for plurality consensus that beat the quadratic lower bound by
allowing a negligible failure probability. While our protocols might fail, they
identify the plurality opinion with high probability even if the bias is .
Our first protocol achieves this via tournaments in time using states. While it assumes an ordering on the opinions,
we remove this restriction in our second protocol, at the cost of a slightly
increased time . By efficiently pruning
insignificant opinions, our final protocol reduces the number of tournaments at
the cost of a slightly increased state complexity . This improves the time to , where
is the initial size of the plurality. Note that is at
most and can be much smaller (e.g., in case of a large bias or if there are
many small opinions)
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