306 research outputs found
Robustness of Randomized Rumour Spreading
In this work we consider three well-studied broadcast protocols: Push, Pull
and Push&Pull. A key property of all these models, which is also an important
reason for their popularity, is that they are presumed to be very robust, since
they are simple, randomized, and, crucially, do not utilize explicitly the
global structure of the underlying graph. While sporadic results exist, there
has been no systematic theoretical treatment quantifying the robustness of
these models. Here we investigate this question with respect to two orthogonal
aspects: (adversarial) modifications of the underlying graph and message
transmission failures.
We explore in particular the following notion of Local Resilience: beginning
with a graph, we investigate up to which fraction of the edges an adversary has
to be allowed to delete at each vertex, so that the protocols need
significantly more rounds to broadcast the information. Our main findings
establish a separation among the three models. It turns out that Pull is robust
with respect to all parameters that we consider. On the other hand, Push may
slow down significantly, even if the adversary is allowed to modify the degrees
of the vertices by an arbitrarily small positive fraction only. Finally,
Push&Pull is robust when no message transmission failures are considered,
otherwise it may be slowed down.
On the technical side, we develop two novel methods for the analysis of
randomized rumour spreading protocols. First, we exploit the notion of
self-bounding functions to facilitate significantly the round-based analysis:
we show that for any graph the variance of the growth of informed vertices is
bounded by its expectation, so that concentration results follow immediately.
Second, in order to control adversarial modifications of the graph we make use
of a powerful tool from extremal graph theory, namely Szemer\`edi's Regularity
Lemma.Comment: version 2: more thorough literature revie
Gossip vs. Markov Chains, and Randomness-Efficient Rumor Spreading
We study gossip algorithms for the rumor spreading problem which asks one
node to deliver a rumor to all nodes in an unknown network. We present the
first protocol for any expander graph with nodes such that, the
protocol informs every node in rounds with high probability, and
uses random bits in total. The runtime of our protocol is
tight, and the randomness requirement of random bits almost
matches the lower bound of random bits for dense graphs. We
further show that, for many graph families, polylogarithmic number of random
bits in total suffice to spread the rumor in rounds.
These results together give us an almost complete understanding of the
randomness requirement of this fundamental gossip process.
Our analysis relies on unexpectedly tight connections among gossip processes,
Markov chains, and branching programs. First, we establish a connection between
rumor spreading processes and Markov chains, which is used to approximate the
rumor spreading time by the mixing time of Markov chains. Second, we show a
reduction from rumor spreading processes to branching programs, and this
reduction provides a general framework to derandomize gossip processes. In
addition to designing rumor spreading protocols, these novel techniques may
have applications in studying parallel and multiple random walks, and
randomness complexity of distributed algorithms.Comment: 41 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:1304.135
On the push&pull protocol for rumour spreading
The asynchronous push&pull protocol, a randomized distributed algorithm for
spreading a rumour in a graph , works as follows. Independent Poisson clocks
of rate 1 are associated with the vertices of . Initially, one vertex of
knows the rumour. Whenever the clock of a vertex rings, it calls a random
neighbour : if knows the rumour and does not, then tells the
rumour (a push operation), and if does not know the rumour and knows
it, tells the rumour (a pull operation). The average spread time of
is the expected time it takes for all vertices to know the rumour, and the
guaranteed spread time of is the smallest time such that with
probability at least , after time all vertices know the rumour. The
synchronous variant of this protocol, in which each clock rings precisely at
times , has been studied extensively. We prove the following results
for any -vertex graph: In either version, the average spread time is at most
linear even if only the pull operation is used, and the guaranteed spread time
is within a logarithmic factor of the average spread time, so it is . In the asynchronous version, both the average and guaranteed spread times
are . We give examples of graphs illustrating that these bounds
are best possible up to constant factors. We also prove theoretical
relationships between the guaranteed spread times in the two versions. Firstly,
in all graphs the guaranteed spread time in the asynchronous version is within
an factor of that in the synchronous version, and this is tight.
Next, we find examples of graphs whose asynchronous spread times are
logarithmic, but the synchronous versions are polynomially large. Finally, we
show for any graph that the ratio of the synchronous spread time to the
asynchronous spread time is .Comment: 25 page
Global Computation in a Poorly Connected World: Fast Rumor Spreading with No Dependence on Conductance
In this paper, we study the question of how efficiently a collection of
interconnected nodes can perform a global computation in the widely studied
GOSSIP model of communication. In this model, nodes do not know the global
topology of the network, and they may only initiate contact with a single
neighbor in each round. This model contrasts with the much less restrictive
LOCAL model, where a node may simultaneously communicate with all of its
neighbors in a single round. A basic question in this setting is how many
rounds of communication are required for the information dissemination problem,
in which each node has some piece of information and is required to collect all
others. In this paper, we give an algorithm that solves the information
dissemination problem in at most rounds in a network
of diameter , withno dependence on the conductance. This is at most an
additive polylogarithmic factor from the trivial lower bound of , which
applies even in the LOCAL model. In fact, we prove that something stronger is
true: any algorithm that requires rounds in the LOCAL model can be
simulated in rounds in the GOSSIP model. We thus
prove that these two models of distributed computation are essentially
equivalent
Quasi-Random Rumor Spreading: Reducing Randomness Can Be Costly
We give a time-randomness tradeoff for the quasi-random rumor spreading
protocol proposed by Doerr, Friedrich and Sauerwald [SODA 2008] on complete
graphs. In this protocol, the goal is to spread a piece of information
originating from one vertex throughout the network. Each vertex is assumed to
have a (cyclic) list of its neighbors. Once a vertex is informed by one of its
neighbors, it chooses a position in its list uniformly at random and then
informs its neighbors starting from that position and proceeding in order of
the list. Angelopoulos, Doerr, Huber and Panagiotou [Electron.~J.~Combin.~2009]
showed that after rounds, the rumor will have been
broadcasted to all nodes with probability .
We study the broadcast time when the amount of randomness available at each
node is reduced in natural way. In particular, we prove that if each node can
only make its initial random selection from every -th node on its list,
then there exists lists such that steps are needed to inform every vertex with
probability at least . This shows that a further reduction of the amount of
randomness used in a simple quasi-random protocol comes at a loss of
efficiency
Strong Robustness of Randomized Rumor Spreading Protocols
Randomized rumor spreading is a classical protocol to disseminate information
across a network. At SODA 2008, a quasirandom version of this protocol was
proposed and competitive bounds for its run-time were proven. This prompts the
question: to what extent does the quasirandom protocol inherit the second
principal advantage of randomized rumor spreading, namely robustness against
transmission failures?
In this paper, we present a result precise up to factors. We
limit ourselves to the network in which every two vertices are connected by a
direct link. Run-times accurate to their leading constants are unknown for all
other non-trivial networks.
We show that if each transmission reaches its destination with a probability
of , after (1+\e)(\frac{1}{\log_2(1+p)}\log_2n+\frac{1}{p}\ln n)
rounds the quasirandom protocol has informed all nodes in the network with
probability at least 1-n^{-p\e/40}. Note that this is faster than the
intuitively natural factor increase over the run-time of approximately
for the non-corrupted case.
We also provide a corresponding lower bound for the classical model. This
demonstrates that the quasirandom model is at least as robust as the fully
random model despite the greatly reduced degree of independent randomness.Comment: Accepted for publication in "Discrete Applied Mathematics". A short
version appeared in the proceedings of the 20th International Symposium on
Algorithms and Computation (ISAAC 2009). Minor typos fixed in the second
version. Proofs of Lemma 11 and Theorem 12 fixed in the third version. Proof
of Lemma 8 fixed in the fourth versio
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