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30
Labeling schemes for small distances in trees
 In Proceedings of the ACMSIAM Symposium on Discrete Algorithms
, 2003
"... Abstract. We consider labeling schemes for trees, supporting various relationships between nodes at small distance. For instance, we show that given a tree T and an integer k we can assign labels to each node of T such that given the label of two nodes we can decide, from these two labels alone, if ..."
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Cited by 27 (2 self)
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Abstract. We consider labeling schemes for trees, supporting various relationships between nodes at small distance. For instance, we show that given a tree T and an integer k we can assign labels to each node of T such that given the label of two nodes we can decide, from these two labels alone, if the distance between v and w is at most k and if so compute it. For trees with n nodes and k ≥ 2, we give a lower bound on the maximum label length of log n + Ω(log log n) bits, and for constant k, we give an upper bound of log n+O(log log n). Bounds for ancestor, sibling, connectivity and bi and triconnectivity labeling schemes are also presented. Key words. Labeling schemes, trees. AMS subject classifications. 68R10, 68W01
Distributed Verification of Minimum Spanning Trees
 Proc. 25th Annual Symposium on Principles of Distributed Computing
, 2006
"... The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in ..."
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Cited by 19 (17 self)
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The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows ” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (except when W ≤ log n). Both our bounds improve previously known bounds for the problem. Our techniques (both for the lower bound and for the upper bound) may indicate a strong relation between the fields of proof labeling schemes and implicit labeling schemes. For the related problem of tree sensitivity also presented by Tarjan, our method yields rather efficient schemes for both the distributed and the sequential settings.
Labeling Schemes for Dynamic Tree Networks
 Theory of Computing Systems
, 2002
"... Distance labeling schemes are composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute the distance between any two vertices directly from their labels (without using any additional information). As applications f ..."
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Cited by 16 (12 self)
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Distance labeling schemes are composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute the distance between any two vertices directly from their labels (without using any additional information). As applications for distance labeling schemes concern mainly large and dynamically changing networks, it is of interest to study distributed dynamic labeling schemes. The current paper considers the problem on dynamic trees, and proposes efficient distributed schemes for it. The paper first presents a labeling scheme for distances in the dynamic tree model, with amortized message complexity O(log 2 n) per operation, where n is the size of the tree at the time the operation takes place. The protocol maintains O(log 2 n) bit labels. This label size is known to be optimal even in the static scenario. A more general labeling scheme is then introduced for the dynamic tree model, based on extending an existing static tree labeling scheme to the dynamic setting. The approach fits a number of natural tree functions, such as distance, separation level and flow. The main resulting scheme incurs an overhead of a O(log n) multiplicative factor in both the label size and amortized message complexity in the case of dynamically growing trees (with no vertex deletions). If an upper bound on n is known in advance, this method yields a different tradeoff, with an O(log 2 n / log log n) multiplicative overhead on the label size but only an O(log n / log log n) overhead on the amortized message complexity. In the fullydynamic model the scheme incurs also an increased additive overhead in amortized communication, of O(log 2 n) messages per operation.
Labeling Schemes for Weighted Dynamic Trees
 In Proc. 30th Int. Colloq. on Automata, Languages & Prog
, 2003
"... A Distance labeling scheme is a type of localized network representation in which short labels are assigned to the vertices, allowing one to infer the distance between any two vertices directly from their labels, without using any additional information sources. As most applications for network repr ..."
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Cited by 16 (11 self)
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A Distance labeling scheme is a type of localized network representation in which short labels are assigned to the vertices, allowing one to infer the distance between any two vertices directly from their labels, without using any additional information sources. As most applications for network representations in general, and distance labeling schemes in particular, concern large and dynamically changing networks, it is of interest to focus on distributed dynamic labeling schemes. The paper considers dynamic weighted trees where the vertices of the trees are fixed but the (positive integral) weights of the edges may change. The two models considered are the edgedynamic model, where from time to time some edge changes its weight by a fixed quanta, and the increasingdynamic model in which edge weights can only grow. The paper presents distributed approximate distance labeling schemes for the two dynamic models, which are efficient in terms of the required label size and communication complexity involved in updating the labels following the weight changes.
A SketchBased Distance Oracle for WebScale Graphs
"... We study the fundamental problem of computing distances between nodes in large graphs such as the web graph and social networks. Our objective is to be able to answer distance queries between pairs of nodes in real time. Since the standard shortest path algorithms are expensive, our approach moves t ..."
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Cited by 16 (1 self)
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We study the fundamental problem of computing distances between nodes in large graphs such as the web graph and social networks. Our objective is to be able to answer distance queries between pairs of nodes in real time. Since the standard shortest path algorithms are expensive, our approach moves the timeconsuming shortestpath computation offline, and at query time only looks up precomputed values and performs simple and fast computations on these precomputed values. More specifically, during the offline phase we compute and store a small “sketch ” for each node in the graph, and at querytime we look up the sketches of the source and destination nodes and perform a simple computation using these two sketches to estimate the distance. Categories and Subject Descriptors G.2.2 [Graph Theory]: Graph algorithms, path and circuit problems
General Compact Labeling Schemes for Dynamic Trees
 In Proc. 19th Int. Symp. on Distributed Computing
, 2005
"... Let F be a function on pairs of vertices. An F labeling scheme is composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute F (u, v) of any two vertices u and v directly from their labels. As applications for labe ..."
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Cited by 13 (9 self)
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Let F be a function on pairs of vertices. An F labeling scheme is composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute F (u, v) of any two vertices u and v directly from their labels. As applications for labeling schemes concern mainly large and dynamically changing networks, it is of interest to study distributed dynamic labeling schemes. This paper investigates labeling schemes for dynamic trees. We consider two dynamic tree models, namely, the leafdynamic tree model in which at each step a leaf can be added to or removed from the tree and the leafincreasing tree model in which the only topological event that may occur is that a leaf joins the tree. A general method for constructing labeling schemes for dynamic trees (under the above mentioned dynamic tree models) was previously developed in [29]. This method is based on extending an existing static tree labeling scheme to the dynamic setting. This approach fits many natural functions on trees, such as distance, separation level, ancestry relation, routing (in both the adversary and the designer port models), nearest common ancestor etc.. Their
Small InducedUniversal Graphs and Compact Implicit Graph Representations
 In Proc. 43’rd annual IEEE Symp. on Foundations of Computer Science
, 2002
"... We show that there exists a graph G with n 2 nodes, where any forest with n nodes is a nodeinduced subgraph of G. Furthermore, the result implies existence of a graph with n nodes that contains all nnode graphs of fixed arboricity k as nodeinduced subgraphs. We provide a lower bound ..."
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Cited by 12 (0 self)
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We show that there exists a graph G with n 2 nodes, where any forest with n nodes is a nodeinduced subgraph of G. Furthermore, the result implies existence of a graph with n nodes that contains all nnode graphs of fixed arboricity k as nodeinduced subgraphs. We provide a lower bound of the size of such a graph. The upper bound is obtained through a simple labeling scheme for parent queries in rooted trees.
Onthefly maintenance of seriesparallel relationships in forkjoin multithreaded programs
 IN PROCEEDINGS OF THEACM SYMPOSIUM ON PARALLEL ALGORITHMS AND ARCHITECTURES (SPAA
, 2004
"... A key capability of datarace detectors is to determine whether one thread executes logically in parallel with another or whether the threads must operate in series. This paper provides two algorithms, one serial and one parallel, to maintain seriesparallel (SP) relationships “on the fly” for fork ..."
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Cited by 8 (2 self)
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A key capability of datarace detectors is to determine whether one thread executes logically in parallel with another or whether the threads must operate in series. This paper provides two algorithms, one serial and one parallel, to maintain seriesparallel (SP) relationships “on the fly” for forkjoin multithreaded programs. The serial SPorder algorithm runs in O(1) amortized time per operation. In contrast, the previously best algorithm requires a time per operation that is proportional to Tarjan’s functional inverse of Ackermann’s function. SPorder employs an ordermaintenance data structure that allows us to implement a more efficient “EnglishHebrew ” labeling scheme than was used in earlier race detectors, which immediately yields an improved determinacyrace detector. In particular, any forkjoin program running in T1 time on a single processor can be checked on the fly for determinacy races in
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.
Compact reachability labeling for graphstructured data
 In Proceedings of the 2005 ACM International Conference on Information and Knowledge Management (CIKM
, 2004
"... Testing reachability between nodes in a graph is a wellknown problem with many important applications, including knowledge representation, program analysis, and more recently, biological and ontology databases inferencing as well as XML query processing. Various approaches have been proposed to enc ..."
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Cited by 8 (2 self)
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Testing reachability between nodes in a graph is a wellknown problem with many important applications, including knowledge representation, program analysis, and more recently, biological and ontology databases inferencing as well as XML query processing. Various approaches have been proposed to encode graph reachability information using node labeling schemes, but most existing schemes only work well for specific types of graphs. In this paper, we propose a novel approach, HLSS(Hybrid Labeling of SubStructures), which identifies different types of substructures within a graph and encodes them using techniques suitable to the characteristics of each of them. We implement HLSS with an efficient twophase algorithm, where the first phase identifies and encodes strongly connected components as well as tree substructures, and the second phase encodes the remaining reachability relationships by compressing dense rectangular submatrices in the transitive closure matrix. For the important subproblem of finding densest submatrices, we demonstrate the hardness of the problem and propose several practical algorithms. Experiments show that HLSS handles different types of graphs well, while existing approaches fall prey to graphs with substructures they are not designed to handle. Finally, we also discuss how to update reachability labels when the graph is updated, and qualitatively show that HLSS supports more efficient updates than existing approaches. 1