88 research outputs found
Gossip routing, percolation, and restart in wireless multi-hop networks
Route and service discovery in wireless multi-hop networks applies flooding or
gossip routing to disseminate and gather information. Since packets may get
lost, retransmissions of lost packets are required. In many protocols the
retransmission timeout is fixed in the protocol specification. In this
technical report we demonstrate that optimization of the timeout is required
in order to ensure proper functioning of flooding schemes. Based on an
experimental study, we apply percolation theory and derive analytical models
for computing the optimal restart timeout. To the best of our knowledge, this
is the first comprehensive study of gossip routing, percolation, and restart
in this context
Sharp Thresholds in Random Simple Temporal Graphs
A graph whose edges only appear at certain points in time is called a temporal graph (among other names). Such a graph is temporally connected if each ordered pair of vertices is connected by a path which traverses edges in chronological order (i.e., a temporal path). In this paper, we consider a simple model of random temporal graph, obtained from an Erd\H{o}s-R\'enyi random graph by considering a random permutation of the edges and interpreting the ranks in as presence times. Temporal reachability in this model exhibits a surprisingly regular sequence of thresholds. In particular, we show that at any fixed pair of vertices can a.a.s. reach each other; at at least one vertex (and in fact, any fixed vertex) can a.a.s. reach all others; and at all the vertices can a.a.s. reach each other, i.e., the graph is temporally connected. Furthermore, the graph admits a temporal spanner of size as soon as it becomes temporally connected, which is nearly optimal as is a lower bound. This result is significant because temporal graphs do not admit spanners of size in general (Kempe et al, STOC 2000). In fact, they do not even admit spanners of size (Axiotis et al, ICALP 2016). Thus, our result implies that the obstructions found in these works, and more generally, all non-negligible obstructions, must be statistically insignificant: nearly optimal spanners always exist in random temporal graphs. All the above thresholds are sharp. Carrying the study of temporal spanners further, we show that pivotal spanners -- i.e., spanners of size made of two spanning trees glued at a single vertex (one descending in time, the other ascending subsequently) -- exist a.a.s. at , this threshold being also sharp. Finally, we show that optimal spanners (of size ) also exist a.a.s. at
Random Geometric Graphs and the Initialization Problem for Wireless Networks
32 pages. Tutorial invitéInternational audienceThe initialization problem, also known as naming, assigns one unique identifier (ranging from 1 to ) to a set of n indistinguishable nodes (stations or processors) in a given wireless network . is composed of nodes randomly deployed within a square (or a cube) . We assume the time to be slotted and to be synchronous; two nodes are able to communicate if they are within a distance at most of each other ( is the transmitting/receiving range). Moreover, if two or more neighbors of a processor transmit concurrently at the same round, does not receive either messages. After the analysis of various critical transmitting/sensing ranges for connectivity and coverage of randomly deployed sensor networks, we design sub-linear randomized initialization and gossip algorithms achieving and O(n^3/4 \log (n)^1/4) rounds
Reachability of Five Gossip Protocols
Gossip protocols use point-to-point communication to spread information within a network until every agent knows everything. Each agent starts with her own piece of information (‘secret’) and in each call two agents will exchange all secrets they currently know. Depending on the protocol, this leads to different distributions of secrets among the agents during its execution. We investigate which distributions of secrets are reachable when using several distributed epistemic gossip protocols from the literature. Surprisingly, a protocol may reach the distribution where all agents know all secrets, but not all other distributions. The five protocols we consider are called 햠햭햸, 햫햭햲, 햢햮, 햳햮햪, and 햲햯햨. We find that 햳햮햪 and 햠햭햸 reach the same distributions but all other protocols reach different sets of distributions, with some inclusions. Additionally, we show that all distributions are subreachable with all five protocols: any distribution can be reached, if there are enough additional agents
On Primitivity of Sets of Matrices
A nonnegative matrix is called primitive if is positive for some
integer . A generalization of this concept to finite sets of matrices is
as follows: a set of matrices is
primitive if is positive for some indices
. The concept of primitive sets of matrices comes up in a
number of problems within the study of discrete-time switched systems. In this
paper, we analyze the computational complexity of deciding if a given set of
matrices is primitive and we derive bounds on the length of the shortest
positive product.
We show that while primitivity is algorithmically decidable, unless it
is not possible to decide primitivity of a matrix set in polynomial time.
Moreover, we show that the length of the shortest positive sequence can be
superpolynomial in the dimension of the matrices. On the other hand, defining
to be the set of matrices with no zero rows or columns, we give
a simple combinatorial proof of a previously-known characterization of
primitivity for matrices in which can be tested in polynomial
time. This latter observation is related to the well-known 1964 conjecture of
Cerny on synchronizing automata; in fact, any bound on the minimal length of a
synchronizing word for synchronizing automata immediately translates into a
bound on the length of the shortest positive product of a primitive set of
matrices in . In particular, any primitive set of
matrices in has a positive product of length
Extrema propagation: fast distributed estimation of sums and network sizes
Aggregation of data values plays an important role on distributed computations, in particular, over peer-to-peer and sensor networks, as it can provide a summary of some global system property and direct the actions of self-adaptive distributed algorithms. Examples include using estimates of the network size to dimension distributed hash tables or estimates of the average system load to direct load balancing. Distributed aggregation using nonidempotent functions, like sums, is not trivial as it is not easy to prevent a given value from being accounted for multiple times; this is especially the case if no centralized algorithms or global identifiers can be used. This paper introduces Extrema Propagation, a probabilistic technique for distributed estimation of the sum of positive real numbers. The technique relies on the exchange of duplicate insensitive messages and can be applied in flood and/or epidemic settings, where multipath routing occurs; it is tolerant of message loss; it is fast, as the number of message exchange steps can be made just slightly above the theoretical minimum; and it is fully distributed, with no single point of failure and the result produced at every node
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