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19
Distributed verification and hardness of distributed approximation
 CoRR
"... We study the verification problem in distributed networks, stated as follows. Let H be a subgraph of a network G where each vertex of G knows which edges incident on it are in H. We would like to verify whether H has some properties, e.g., if it is a tree or if it is connected (every node knows in t ..."
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Cited by 19 (6 self)
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We study the verification problem in distributed networks, stated as follows. Let H be a subgraph of a network G where each vertex of G knows which edges incident on it are in H. We would like to verify whether H has some properties, e.g., if it is a tree or if it is connected (every node knows in the end of the process whether H has the specified property or not). We would like to perform this verification in a decentralized fashion via a distributed algorithm. The time complexity of verification is measured as the number of rounds of distributed communication. In this paper we initiate a systematic study of distributed verification, and give almost tight lower bounds on the running time of distributed verification algorithms for many A full version of this paper is available as [5] at
Local MST Computation with Short Advice
 SPAA
, 2007
"... We use the recently introduced advising scheme framework for measuring the difficulty of locally distributively computing a Minimum Spanning Tree (MST). An (m, t)advising scheme for a distributed problem P is a way, for every possible input I of P, to provide an ”advice” (i.e., a bit string) about ..."
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Cited by 16 (9 self)
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We use the recently introduced advising scheme framework for measuring the difficulty of locally distributively computing a Minimum Spanning Tree (MST). An (m, t)advising scheme for a distributed problem P is a way, for every possible input I of P, to provide an ”advice” (i.e., a bit string) about I to each node so that: (1) the maximum size of the advices is at most m bits, and (2) the problem P can be solved distributively in at most t rounds using the advices as inputs. In case of MST, the output returned by each node of a weighted graph G is the edge leading to its parent in some rooted MST T of G. Clearly, there is a trivial (⌈log n⌉, 0)advising scheme for MST (each node is given the local port number of the edge leading to the root of some MST T), and it is known that any (0, t)advising scheme satisfies t ≥ ˜ Ω ( √ n). Our main result is the construction of an (O(1), O(log n))advising scheme for MST. That is, by only giving a constant number of bits of advice to each node, one can decrease exponentially the distributed computation time of MST in arbitrary graph, compared to algorithms dealing with the problem in absence of any a priori information. We also consider the average size of the advices. On the one hand, we show that any (m, 0)advising scheme for MST gives advices of average size Ω(log n). On the other hand we design an (m, 1)advising scheme for MST with advices of constant average size, that is one round is enough to decrease the average size of the advices from log n to constant.
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 ..."
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Cited by 9 (7 self)
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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
Labeling Schemes for Vertex Connectivity
"... This paper studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of labeling the nodes of any nnode graph is such a way that given the labels of two nodes u and v, one can decide whether u and v are kvertex connected in G, i.e., whether there exis ..."
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Cited by 8 (7 self)
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This paper studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of labeling the nodes of any nnode graph is such a way that given the labels of two nodes u and v, one can decide whether u and v are kvertex connected in G, i.e., whether there exist k vertex disjoint paths connecting u and v. The paper establishes an upper bound of k 2 log n on the number of bits used in a label. The best previous upper bound for the label size of such labeling scheme is 2 k log n.
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 ..."
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Cited by 7 (2 self)
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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).
Controller and Estimator for Dynamic Networks
, 2007
"... Awerbuch, Afek, Plotkin, and Saks identified an important fundamental problem inherent to distributed networks, which they called the Resource Controller problem. Consider, first, the problem in which one node (called the ‘root’) is required to estimate the number of events that occurred all over th ..."
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Cited by 6 (1 self)
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Awerbuch, Afek, Plotkin, and Saks identified an important fundamental problem inherent to distributed networks, which they called the Resource Controller problem. Consider, first, the problem in which one node (called the ‘root’) is required to estimate the number of events that occurred all over the network. The counting can be viewed as a useful variant of the heavily studied and used task of topology update (that deals with collecting all remote information). The Resource Controller problem generalizes the counting problem further: such remote events are considered as requests, and the counting node, i.e., the ‘root’, also issues permits for the requests. That way, the number of request granted can be controlled (bounded). In the paper by Awerbuch et al., it was assumed that the network is spanned by a tree that may only grow, and only by allowing leaves to join the tree (after receiving a permit). In contrast, the Resource Controller presented here can operate under a more general dynamic model. Specifically, the dynamic model considered in this paper allows both controlled insertions and deletions of leaves as well as controlled insertions and deletions of internal nodes. Despite the more dynamic network model we allow, the message complexity of our controller is always at most the message complexity of the
Constructing Labeling schemes through Universal Matrices
 In Proc. 17th Int. Symp. on Algorithms and Computation
, 2006
"... Abstract. Let f be a function on pairs of vertices. An flabeling scheme for a family of graphs F labels the vertices of all graphs in F such that for every graph G ∈ F and every two vertices u, v ∈ G, f(u, v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme ..."
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Cited by 6 (5 self)
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Abstract. Let f be a function on pairs of vertices. An flabeling scheme for a family of graphs F labels the vertices of all graphs in F such that for every graph G ∈ F and every two vertices u, v ∈ G, f(u, v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in F. This paper illustrates that the notion of universal matrices can be used to efficiently construct flabeling schemes. Let F(n) be a family of connected graphs of size at most n and let C(F, n) denote the collection of graphs of size at most n, such that each graph in C(F, n) is composed of a disjoint union of some graphs in F(n). We first investigate methods for translating flabeling schemes for F(n) to flabeling schemes for C(F, n). In particular, we show that in many cases, given an flabeling scheme of size g(n) for a graph family F(n), one can construct an flabeling scheme of size g(n) + log log n + O(1) for C(F, n). We also show that in several cases, the above mentioned extra additive term of log log n + O(1) is necessary. In addition, we show that the family of nnode graphs which are unions of disjoint circles enjoys an adjacency labeling scheme of size log n + O(1). This illustrates a nontrivial example showing that the above mentioned extra additive term is sometimes not necessary. We then turn to investigate distance labeling schemes on the class of circles of at most n vertices and show an upper bound of 1.5 log n + O(1) and a lower bound of 4/3 log n − O(1) for the size of any such labeling scheme.
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 ..."
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Cited by 6 (1 self)
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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 ..."
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Cited by 5 (5 self)
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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.
Compact Ancestry Labeling Schemes for XML Trees
 In Proc. 21st ACMSIAM Symp. on Discrete Algorithms (SODA
, 2010
"... An ancestry labeling scheme labels the nodes of any tree in such a way that ancestry queries between any two nodes can be answered just by looking at their corresponding labels. The common measure to evaluate the quality of an ancestry scheme is by its label size, that is the maximum number of bits ..."
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Cited by 4 (2 self)
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An ancestry labeling scheme labels the nodes of any tree in such a way that ancestry queries between any two nodes can be answered just by looking at their corresponding labels. The common measure to evaluate the quality of an ancestry scheme is by its label size, that is the maximum number of bits stored in a label, taken over all nnode trees. The design of ancestry labeling schemes finds applications in XML search engines. In these contexts, even small improvements in the label size are important. As a result, following the proposal of a simple interval based ancestry scheme with label size 2 log n bits (Kannan et al., STOC 88), a considerable amount of work was devoted to improve the bound on the label size. The current state of the art upper bound is log n + O ( √ log n) bits (Abiteboul et al., SICOMP 06) which is still far from the known log n + Ω(log log n) lower bound (Alstrup et al., SODA 03). Motivated by the fact that typical XML trees have extremely small depth, this paper parameterizes the quality measure of an ancestry scheme not only by the number of nodes in the given tree but also by its depth. Our main result is the construction of an ancestry scheme that labels nnode trees of depth d with labels of size log n + 2 log d + O(1). In addition to our main result, we prove a result that may be of independent interest concerning the existence of a small universal graph for the family of trees with bounded depth. 1