163 research outputs found
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
With Great Speed Come Small Buffers: Space-Bandwidth Tradeoffs for Routing
We consider the Adversarial Queuing Theory (AQT) model, where packet arrivals
are subject to a maximum average rate and burstiness
. In this model, we analyze the size of buffers required to avoid
overflows in the basic case of a path. Our main results characterize the space
required by the average rate and the number of distinct destinations: we show
that space suffice, where is the number of distinct
destinations and ; and we show that space is necessary. For directed trees, we describe an algorithm
whose buffer space requirement is at most where is the
maximum number of destinations on any root-leaf path
Guaranteeing the diversity of number generators
A major problem in using iterative number generators of the form
x_i=f(x_{i-1}) is that they can enter unexpectedly short cycles. This is hard
to analyze when the generator is designed, hard to detect in real time when the
generator is used, and can have devastating cryptanalytic implications. In this
paper we define a measure of security, called_sequence_diversity_, which
generalizes the notion of cycle-length for non-iterative generators. We then
introduce the class of counter assisted generators, and show how to turn any
iterative generator (even a bad one designed or seeded by an adversary) into a
counter assisted generator with a provably high diversity, without reducing the
quality of generators which are already cryptographically strong.Comment: Small update
A Note on Distributed Stable Matching
We consider the distributed complexity of the stable mar-riage problem. In this problem, the communication graph is undirected and bipartite, and each node ranks its neigh-bors. Given a matching of the nodes, a pair of unmatched nodes is called blocking if they prefer each other to their assigned match. A matching is called stable if it does not induce any blocking pair. In the distributed model, nodes exchange messages in each round over the communication links, until they find a stable matching. We show that if mes-sages may contain at most B bits each, then any distributed algorithm that solves the stable marriage problem requires Ω( n/B log n) communication rounds in the worst case, even for graphs of diameter O(log n), where n is the num-ber of nodes in the graph. Furthermore, the lower bound holds even if we allow the output to contain O( n) block-ing pairs. We also consider ε-stability, where a pair is called ε-blocking if they can improve the quality of their match by more than an ε fraction, for some 0 ≤ ε ≤ 1. Our lower bound extends to ε-stability where ε is arbitrarily close to 1/2. We also present a simple distributed algorithm for ε-stability whose time complexity is O(n/ε).
Length-based cryptanalysis: The case of Thompson's Group
The length-based approach is a heuristic for solving randomly generated
equations in groups which possess a reasonably behaved length function. We
describe several improvements of the previously suggested length-based
algorithms, that make them applicable to Thompson's group with significant
success rates. In particular, this shows that the Shpilrain-Ushakov public key
cryptosystem based on Thompson's group is insecure, and suggests that no
practical public key cryptosystem based on this group can be secure.Comment: Final version, to appear in JM
Randomized Proof-Labeling Schemes
International audienceA proof-labeling scheme, introduced by Korman, Kutten and Peleg [PODC 2005], is a mechanism enabling to certify the legality of a network configuration with respect to a boolean predicate. Such a mechanism finds applications in many frameworks, including the design of fault-tolerant distributed algorithms. In a proof-labeling scheme, the verification phase consists of exchanging labels between neighbors. The size of these labels depends on the network predicate to be checked. There are predicates requiring large labels, of poly-logarithmic size (e.g., MST), or even polynomial size (e.g., Symmetry). In this paper, we introduce the notion of randomized proof-labeling schemes. By reduction from deterministic schemes, we show that randomization enables the amount of communication to be exponentially reduced. As a consequence, we show that checking any network predicate can be done with probability of correctness as close to one as desired by exchanging just a logarithmic number of bits between neighbors. Moreover, we design a novel space lower bound technique that applies to both deterministic and randomized proof-labeling schemes. Using this technique, we establish several tight bounds on the verification complexity of classical distributed computing problems, such as MST construction, and of classical predicates such as acyclicity, connectivity, and cycle length
Stable Secretaries
We define and study a new variant of the secretary problem. Whereas in the
classic setting multiple secretaries compete for a single position, we study
the case where the secretaries arrive one at a time and are assigned, in an
on-line fashion, to one of multiple positions. Secretaries are ranked according
to talent, as in the original formulation, and in addition positions are ranked
according to attractiveness. To evaluate an online matching mechanism, we use
the notion of blocking pairs from stable matching theory: our goal is to
maximize the number of positions (or secretaries) that do not take part in a
blocking pair. This is compared with a stable matching in which no blocking
pair exists. We consider the case where secretaries arrive randomly, as well as
that of an adversarial arrival order, and provide corresponding upper and lower
bounds.Comment: Accepted for presentation at the 18th ACM conference on Economics and
Computation (EC 2017
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