2 research outputs found
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
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)