Results 1  10
of
18
What can be approximated locally?  Case study: dominating sets in planar graphs
 SPAA'08
, 2008
"... Whether local algorithms can compute constant approximations of NPhard problems is of both practical and theoretical interest. So far, no algorithms achieving this goal are known, as either the approximation ratio or the running time exceed O(1), or the nodes are provided with nontrivial additiona ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
Whether local algorithms can compute constant approximations of NPhard problems is of both practical and theoretical interest. So far, no algorithms achieving this goal are known, as either the approximation ratio or the running time exceed O(1), or the nodes are provided with nontrivial additional information. In this paper, we present the first distributed algorithm approximating a minimum dominating set on a planar graph within a constant factor in constant time. Moreover, the nodes do not need any additional information.
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 9 (7 self)
 Add to MetaCart
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
Communication Algorithms with Advice
, 2009
"... We study the amount of knowledge about a communication network that must be given to its nodes in order to efficiently disseminate information. Our approach is quantitative: we investigate the minimum total number of bits of information (minimum size of advice) that has to be available to nodes, reg ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
We study the amount of knowledge about a communication network that must be given to its nodes in order to efficiently disseminate information. Our approach is quantitative: we investigate the minimum total number of bits of information (minimum size of advice) that has to be available to nodes, regardless of the type of information provided. We compare the size of advice needed to perform broadcast and wakeup (the latter is a broadcast in which nodes can transmit only after getting the source information), both using a linear number of messages (which is optimal). We show that the minimum size of advice permitting the wakeup with a linear number of messages in a nnode network, is Θ(nlog n), while the broadcast with a linear number of messages can be achieved with advice of size O(n). We also show that the latter size of advice is almost optimal: no advice of size o(n) can permit to broadcast with a linear number of messages. Thus an
Locally Checkable Proofs
"... This work studies decision problems from the perspective of nondeterministic distributed algorithms. For a yesinstance there must exist a proof that can be verified with a distributed algorithm: all nodes must accept a valid proof, and at least one node must reject an invalid proof. We focus on loc ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
This work studies decision problems from the perspective of nondeterministic distributed algorithms. For a yesinstance there must exist a proof that can be verified with a distributed algorithm: all nodes must accept a valid proof, and at least one node must reject an invalid proof. We focus on locally checkable proofs that can be verified with a constanttime distributed algorithm. For example, it is easy to prove that a graph is bipartite: the locally checkable proof gives a 2colouring of the graph, which only takes 1 bit per node. However, it is more difficult to prove that a graph is not bipartite—it turns out that any locally checkable proof requires Ω(log n) bits per node. In this work we classify graph problems according to their local proof complexity, i.e., how many bits per node are needed in a locally checkable proof. We establish tight or neartight results for classical graph properties such as the chromatic number. We show that the proof complexities form a natural hierarchy of complexity classes: for many classical graph problems, the proof complexity is either 0, Θ(1), Θ(log n), or poly(n) bits per node. Among the most difficult graph properties are symmetric graphs, which require Ω(n 2) bits per node, and non3colourable graphs, which require Ω(n 2 / log n) bits per node—any pure graph property admits a trivial proof of size O(n 2).
Distributed Computing with Advice
 Information Sensitivity of Graph Coloring, in "ICALP
"... Abstract. We consider a model for online computation in which the online algorithm receives, together with each request, some information regarding the future, referred to as advice. The advice provided to the online algorithm may allow an improvement in its performance, compared to the classical mo ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Abstract. We consider a model for online computation in which the online algorithm receives, together with each request, some information regarding the future, referred to as advice. The advice provided to the online algorithm may allow an improvement in its performance, compared to the classical model of complete lack of information regarding the future. We are interested in the impact of such advice on the competitive ratio, and in particular, in the relation between the size b of the advice, measured in terms of bits of information per request, and the (improved) competitive ratio. Since b = 0 corresponds to the classical online model, and b = ⌈log A⌉, where A is the algorithm’s action space, corresponds to the optimal (offline) one, our model spans a spectrum of settings ranging from classical online algorithms to offline ones. In this paper we propose the above model and illustrate its applicability by considering two of the most extensively studied online problems, namely, metrical task systems (MTS) and the kserver problem. For MTS we establish tight (up to constant factors) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms with advice for any choice of 1 ≤ b ≤ Θ(log n), where n is the number of states in the system: we prove that any randomized online algorithm for MTS has competitive ratio Ω(log(n)/b) and we present a deterministic online algorithm for MTS with competitive ratio O(log(n)/b). For the kserver problem we construct a deterministic online algorithm for general metric spaces with competitive ratio k O(1/b) for any choice of Θ(1) ≤ b ≤ log k. 1
Toward More Localized Local Algorithms: Removing Assumptions Concerning Global Knowledge
"... Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and ( ∆ + 1)coloring algorithms by Barenboim and Elkin [6], by Kuhn [22], and by Panconesi and Srinivasan [34], as well as the O(∆2)coloring algorithm by Linial [2 ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and ( ∆ + 1)coloring algorithms by Barenboim and Elkin [6], by Kuhn [22], and by Panconesi and Srinivasan [34], as well as the O(∆2)coloring algorithm by Linial [28]. Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are nonuniform, that is, they assume that all nodes know good estimations of one or more global parameters of the network, e.g., the maximum degree ∆ or the number of nodes n. This paper provides a rather general method for transforming a nonuniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original nonuniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all of the state of the art nonuniform algorithms regarding MIS and Maximal Matching, as well as to many results concerning the coloring problem. (In particular, it applies to all aforementioned algorithms.) To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest.
Tradeoffs between the size of advice and broadcasting time in trees
 IN SPAA
, 2008
"... We study the problem of the amount of information required to perform fast broadcasting in tree networks. The source located at the root of a tree has to disseminate a message to all nodes. In each round each informed node can transmit to one child. Nodes do not know the topology of the tree but an ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
We study the problem of the amount of information required to perform fast broadcasting in tree networks. The source located at the root of a tree has to disseminate a message to all nodes. In each round each informed node can transmit to one child. Nodes do not know the topology of the tree but an oracle knowing it can give a string of bits of advice to the source which can then pass it down the tree with the source message. The quality of a broadcasting algorithm with advice is measured by its competitive ratio: the worst case ratio, taken over nnode trees, between the time of this algorithm and the optimal broadcasting time in the given tree. Our goal is to find a tradeoff between the size of advice and the best competitive ratio of a broadcasting algorithm for nnode trees. We establish such a tradeoff with an approximation factor of O(n ɛ), for an arbitrarily small
LOCAL MULTICOLORING ALGORITHMS: COMPUTING A NEARLYOPTIMAL TDMA SCHEDULE IN CONSTANT TIME
, 2009
"... We are given a set V of autonomous agents (e.g. the computers of a distributed system) that are connected to each other by a graph G = (V, E) (e.g. by a communication network connecting the agents). Assume that all agents have a unique ID between 1 and N for a parameter N ≥ V  and that each agent ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We are given a set V of autonomous agents (e.g. the computers of a distributed system) that are connected to each other by a graph G = (V, E) (e.g. by a communication network connecting the agents). Assume that all agents have a unique ID between 1 and N for a parameter N ≥ V  and that each agent knows its ID as well as the IDs of its neighbors in G. Based on this limited information, every agent v must autonomously compute a set of colors Sv ⊆ C such that the color sets Su and Sv of adjacent agents u and v are disjoint. We prove that there is a deterministic algorithm that uses a total of C  = O( ∆ 2 log(N)/ε 2) colors such that for every node v of G (i.e., for every agent), we have Sv  ≥ C·(1−ε)/(δv+1), where δv is the degree of v and where ∆ is the maximum degree of G. For N = Ω( ∆ 2 log ∆), Ω( ∆ 2 + log log N) colors are necessary even to assign at least one color to every node (i.e., to compute a standard vertex coloring). Using randomization, it is possible to assign an (1 − ε)/(δ + 1)fraction of all colors to every node of degree δ using only O( ∆ log V /ε 2) colors w.h.p. We show that this is asymptotically almost optimal. For graphs with maximum degree ∆ = Ω(log V ), Ω( ∆ log V  / log log V ) colors are needed in expectation, even to compute a valid coloring. The described multicoloring problem has direct applications in the context of wireless ad hoc and
Fast radio broadcasting with advice
 Theor. Comput. Sci
, 2010
"... Abstract. We study deterministic broadcasting in radio networks in the recently introduced framework of network algorithms with advice. We concentrate on the problem of tradeoffs between the number of bits of information (size of advice) available to nodes and the time in which broadcasting can be ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Abstract. We study deterministic broadcasting in radio networks in the recently introduced framework of network algorithms with advice. We concentrate on the problem of tradeoffs between the number of bits of information (size of advice) available to nodes and the time in which broadcasting can be accomplished. In particular, we ask what is the minimum number of bits of information that must be available to nodes of the network, in order to broadcast very fast. For networks in which constant time broadcast is possible under complete knowledge of the network we give a tight answer to the above question: O(n) bits of advice are sufficient but o(n) bits are not, in order to achieve constant broadcasting time in all these networks. This is in sharp contrast with geometric radio networks of constant broadcasting time: we show that in these networks a constant number of bits suffices to broadcast in constant time. For arbitrary radio networks we present a broadcasting algorithm whose time is inverseproportional to the size of advice.
What Can be Observed Locally? Roundbased Models for Quantum Distributed
 Computing, in "23rd International Symposium on Distributed Computing (DISC) DISC, Espagne Elche/Elx
"... It is a wellknown fact that, by resorting to quantum processing in addition to manipulating classical information, it is possible to reduce the time complexity of some centralized algorithms, and also to decrease the bit size of messages exchanged in tasks requiring communication among several agen ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
It is a wellknown fact that, by resorting to quantum processing in addition to manipulating classical information, it is possible to reduce the time complexity of some centralized algorithms, and also to decrease the bit size of messages exchanged in tasks requiring communication among several agents. Recently, several claims have been made that certain fundamental problems of distributed computing, including Leader Election and Distributed Consensus, begin to admit feasible and efficient solutions when the model of distributed computation is extended so as to apply quantum processing. This has been achieved in one of two distinct ways: (1) by initializing the system in a quantum entangled state, and/or (2) by applying quantum communication channels. In this paper, we explain why some of these prior claims are misleading, in the sense that they rely on changes to the model unrelated to quantum processing. On the positive side, we consider the aforementioned quantum extensions when applied to Linial’s wellestablished LOCAL model of distributed computing. For both types of extensions, we put forward valid proofofconcept examples of distributed problems whose round complexity