Results 1 
6 of
6
Local Distributed Decision
 In FOCS 2011
"... A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired ..."
Abstract

Cited by 17 (11 self)
 Add to MetaCart
(Show Context)
A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard LOCAL model of computation and define LD(t) (for local decision) as the class of decision problems that can be solved in t communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD(t, p, q), containing all languages for which there exists a randomized algorithm that runs in t rounds, accepts correct instances with probability at least p and rejects incorrect ones with probability at least q. We show that p 2 +q = 1 is a threshold for the containment of LD(t) in BPLD(t, p, q). More precisely, we show that there exists a language that does not belong to LD(t) for any t = o(n) but does belong to BPLD(0, p, q) for any p, q ∈ (0, 1] such that p 2 +q ≤ 1. On the other hand, we show that, restricted to
Distributed Algorithms for Ultrasparse Spanners and Linear Size Skeletons
"... We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an O(2 log ∗ n log n)spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size O(n(log log n) φ) where φ = 1+ √ 5 2 is the golden ratio. As the distance increases the Fibonacci spanner’s multiplicative distortion passes through four discrete stages, moving from logarithmic to loglogarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion.
Multipath Spanners
"... This paper concerns graph spanners that approximate multipaths between pair of vertices of an undirected graphs with n vertices. Classically, a spanner H of stretch s for a graph G is a spanning subgraph such that the distance in H between any two vertices is at most s times the distance in G. We s ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
This paper concerns graph spanners that approximate multipaths between pair of vertices of an undirected graphs with n vertices. Classically, a spanner H of stretch s for a graph G is a spanning subgraph such that the distance in H between any two vertices is at most s times the distance in G. We study in this paper spanners that approximate short cycles, and more generally p edgedisjoint paths with p> 1, between any pair of vertices. For every unweighted graph G, we construct a 2multipath 3spanner of O(n 3/2) edges. In other words, for any two vertices u, v of G, the length of the shortest cycle (with no edge replication) traversing u, v in the spanner is at most thrice the length of the shortest one in G. This construction is shown to be optimal in term of stretch and of size. In a second construction, we produce a 2multipath (2, 8)spanner of O(n 3/2) edges, i.e., the length of the shortest cycle traversing any two vertices have length at most twice the shortest length in G plus eight. For arbitrary p, we observe that, for each integer k � 1, every weighted graph has a pmultipath p(2k−1)spanner with O(pn 1+1/k) edges, leaving open the question whether, with similar size, the stretch of the spanner can be reduced to 2k − 1 for all p> 1.
Towards a Complexity Theory for Local Distributed Computing
, 2013
"... A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Yet despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Insp ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Yet despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard LOCAL model of computation and define LD(t) (for local decision) as the class of decision problems that can be solved in t communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD(t, p, q), containing all languages for which there exists a randomized algorithm that runs in t rounds, accepts correct instances with probability at least p, and rejects incorrect ones with probability at least q. We
3.3. Theoretical Validation 7
"... c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1 ..."
Abstract
 Add to MetaCart
c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1
3.1.2. Small World Phenomenon 2 3.1.3. Doubling Metrics 2
"... c t i v it y e p o r t 2009 Table of contents ..."
(Show Context)