418 research outputs found
On the Distributed Construction of Stable Networks in Polylogarithmic Parallel Time
We study the class of networks, which can be created in polylogarithmic parallel time by network constructors: groups of anonymous agents that interact randomly under a uniform random scheduler with the ability to form connections between each other. Starting from an empty network, the goal is to construct a stable network that belongs to a given family. We prove that the class of trees where each node has any k≥2 children can be constructed in O(logn) parallel time with high probability. We show that constructing networks that are k-regular is Ω(n) time, but a minimal relaxation to (l,k)-regular networks, where l=k−1, can be constructed in polylogarithmic parallel time for any fixed k, where k>2. We further demonstrate that when the finite-state assumption is relaxed and k is allowed to grow with n, then k=loglogn acts as a threshold above which network construction is, again, polynomial time. We use this to provide a partial characterisation of the class of polylogarithmic time network constructors.</jats:p
Active Self-Assembly of Algorithmic Shapes and Patterns in Polylogarithmic Time
We describe a computational model for studying the complexity of
self-assembled structures with active molecular components. Our model captures
notions of growth and movement ubiquitous in biological systems. The model is
inspired by biology's fantastic ability to assemble biomolecules that form
systems with complicated structure and dynamics, from molecular motors that
walk on rigid tracks and proteins that dynamically alter the structure of the
cell during mitosis, to embryonic development where large-scale complicated
organisms efficiently grow from a single cell. Using this active self-assembly
model, we show how to efficiently self-assemble shapes and patterns from simple
monomers. For example, we show how to grow a line of monomers in time and
number of monomer states that is merely logarithmic in the length of the line.
Our main results show how to grow arbitrary connected two-dimensional
geometric shapes and patterns in expected time that is polylogarithmic in the
size of the shape, plus roughly the time required to run a Turing machine
deciding whether or not a given pixel is in the shape. We do this while keeping
the number of monomer types logarithmic in shape size, plus those monomers
required by the Kolmogorov complexity of the shape or pattern. This work thus
highlights the efficiency advantages of active self-assembly over passive
self-assembly and motivates experimental effort to construct general-purpose
active molecular self-assembly systems
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
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)
Fast Routing Table Construction Using Small Messages
We describe a distributed randomized algorithm computing approximate
distances and routes that approximate shortest paths. Let n denote the number
of nodes in the graph, and let HD denote the hop diameter of the graph, i.e.,
the diameter of the graph when all edges are considered to have unit weight.
Given 0 < eps <= 1/2, our algorithm runs in weak-O(n^(1/2 + eps) + HD)
communication rounds using messages of O(log n) bits and guarantees a stretch
of O(eps^(-1) log eps^(-1)) with high probability. This is the first
distributed algorithm approximating weighted shortest paths that uses small
messages and runs in weak-o(n) time (in graphs where HD in weak-o(n)). The time
complexity nearly matches the lower bounds of weak-Omega(sqrt(n) + HD) in the
small-messages model that hold for stateless routing (where routing decisions
do not depend on the traversed path) as well as approximation of the weigthed
diameter. Our scheme replaces the original identifiers of the nodes by labels
of size O(log eps^(-1) log n). We show that no algorithm that keeps the
original identifiers and runs for weak-o(n) rounds can achieve a
polylogarithmic approximation ratio.
Variations of our techniques yield a number of fast distributed approximation
algorithms solving related problems using small messages. Specifically, we
present algorithms that run in weak-O(n^(1/2 + eps) + HD) rounds for a given 0
< eps <= 1/2, and solve, with high probability, the following problems:
- O(eps^(-1))-approximation for the Generalized Steiner Forest (the running
time in this case has an additive weak-O(t^(1 + 2eps)) term, where t is the
number of terminals);
- O(eps^(-2))-approximation of weighted distances, using node labels of size
O(eps^(-1) log n) and weak-O(n^(eps)) bits of memory per node;
- O(eps^(-1))-approximation of the weighted diameter;
- O(eps^(-3))-approximate shortest paths using the labels 1,...,n.Comment: 40 pages, 2 figures, extended abstract submitted to STOC'1
Restricted Mobility Improves Delay-Throughput Trade-offs in Mobile Ad-Hoc Networks
In this paper we revisit two classes of mobility models which are widely used to repre-sent users ’ mobility in wireless networks: Random Waypoint (RWP) and Random Direction (RD). For both models we obtain systems of partial differential equations which describe the evolution of the users ’ distribution. For the RD model, we show how the equations can be solved analytically both in the stationary and transient regime adopting standard mathematical techniques. Our main contributions are i) simple expressions which relate the transient dura-tion to the model parameters; ii) the definition of a generalized random direction model whose stationary distribution of mobiles in the physical space corresponds to an assigned distribution
Fast Byzantine Leader Election in Dynamic Networks
International audienceWe study the fundamental Byzantine leader election problem in dynamic networks where the topology can change from round to round and nodes can also experience heavy churn (i.e., nodes can join and leave the network continuously over time). We assume the full information model where the Byzantine nodes have complete knowledge about the entire state of the network at every round (including random choices made by all the nodes), have unbounded computational power and can deviate arbitrarily from the protocol. The churn is controlled by an adversary that has complete knowledge and control over which nodes join and leave and at what times and also may rewire the topology in every round and has unlimited computational power, but is oblivious to the random choices made by the algorithm.Our main contribution is an O(log^3 n) round algorithm that achieves Byzantine leader election under the presence of up to O(n^(1/2)−ε) Byzantinenodes (for a small constant ε > 0) and a churn of up to O( √n/ polylog(n)) nodes per round (where n is the stable network size). The algorithm elects a leader with probability at least 1 − n^(−Ω(1)) and guarantees that it is an honest node with probability at least 1 − n^(−Ω(1)); assuming the algorithm succeeds, the leader’s identity will be known to a 1 − o(1) fraction of the honest nodes. Our algorithm is fully-distributed, lightweight, and is simple to implement. It is also scalable, as it runs in polylogarithmic (in n) time and requires nodes to send and receive messages of only polylogarithmic size per round. To the best of our knowledge, our algorithm is the first scalable solution for Byzantine leader election in a dynamic network with a high rate of churn; our protocol can also be used to solve Byzantine agreement in a straightforward way. We also show how to implement an (almost-everywhere) public coin with constant bias in a dynamic network with Byzantine nodes and provide a mechanism for enabling honest nodes to store information reliably in the network, which might be of independent interest
Avatar: A Time- and Space-Efficient Self-Stabilizing Overlay Network
Overlay networks present an interesting challenge for fault-tolerant
computing. Many overlay networks operate in dynamic environments (e.g. the
Internet), where faults are frequent and widespread, and the number of
processes in a system may be quite large. Recently, self-stabilizing overlay
networks have been presented as a method for managing this complexity.
\emph{Self-stabilizing overlay networks} promise that, starting from any
weakly-connected configuration, a correct overlay network will eventually be
built. To date, this guarantee has come at a cost: nodes may either have high
degree during the algorithm's execution, or the algorithm may take a long time
to reach a legal configuration. In this paper, we present the first
self-stabilizing overlay network algorithm that does not incur this penalty.
Specifically, we (i) present a new locally-checkable overlay network based upon
a binary search tree, and (ii) provide a randomized algorithm for
self-stabilization that terminates in an expected polylogarithmic number of
rounds \emph{and} increases a node's degree by only a polylogarithmic factor in
expectation
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