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A Fully Dynamic Algorithm for Maintaining the Transitive Closure
 In Proc. 31st ACM Symposium on Theory of Computing (STOC'99
, 1999
"... This paper presents an efficient fully dynamic graph algorithm for maintaining the transitive closure of a directed graph. The algorithm updates the adjacency matrix of the transitive closure with each update to the graph. Hence, each reachability query of the form "Is there a directed path from i t ..."
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Cited by 43 (1 self)
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This paper presents an efficient fully dynamic graph algorithm for maintaining the transitive closure of a directed graph. The algorithm updates the adjacency matrix of the transitive closure with each update to the graph. Hence, each reachability query of the form "Is there a directed path from i to j?" can be answered in O(1) time. The algorithm is randomized; it is correct when answering yes, but has O(1/n^c) probability of error when answering no, for any constant c. In acyclic graphs, worst case update time is O(n^2). In general graphs, update time is O(n^(2+alpha)), where alpha = min {.26, maximum size of a strongly connected component}. The space complexity of the algorithm is O(n^2).
Fully dynamic algorithms for chordal graphs
 In Proceedings of the 10th Annual ACMSIAM Symposium on Discrete Algorithms (SODA'99
, 1999
"... We present the rst dynamic algorithm that maintains a clique tree representation of a chordal graph and supports the following operations: (1) query whether deleting or inserting an arbitrary edge preserves chordality, (2) delete or insert an arbitrary edge, provided it preserves chordality. Wegivet ..."
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Cited by 30 (1 self)
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We present the rst dynamic algorithm that maintains a clique tree representation of a chordal graph and supports the following operations: (1) query whether deleting or inserting an arbitrary edge preserves chordality, (2) delete or insert an arbitrary edge, provided it preserves chordality. Wegivetwo implementations. In the rst, each operation runs in O(n) time, where n is the numberofvertices. In the second, an insertion query runs in O(log 2 n) time, an insertion in O(n) time, a deletion query in O(n) time, and a deletion in O(n log n) time. We also present a data structure that allows a deletion query to run in O ( p m) time in either implementation, where m is the current number of edges. Updating this data structure after a deletion or insertion requires O(m) time. We also present avery simple dynamic algorithm that supports each of the following operations in O(1) time on a general graph: (1) query whether the graph is split, (2) delete or insert an arbitrary edge. 1
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
The hindex of a graph and its application to dynamic subgraph statistics
 in Algorithms and Data Structures, ser. Lecture Notes in Computer Science
"... Abstract. We describe a data structure that maintains the number of triangles in a dynamic undirected graph, subject to insertions and deletions of edges and of degreezero vertices. More generally it can be used to maintain the number of copies of each possible threevertex subgraph in time O(h) pe ..."
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Cited by 4 (0 self)
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Abstract. We describe a data structure that maintains the number of triangles in a dynamic undirected graph, subject to insertions and deletions of edges and of degreezero vertices. More generally it can be used to maintain the number of copies of each possible threevertex subgraph in time O(h) per update, where h is the hindex of the graph, the maximum number such that the graph contains h vertices of degree at least h. We also show how to maintain the hindex itself, and a collection of h highdegree vertices in the graph, in constant time per update. Our data structure has applications in social network analysis using the exponential random graph model (ERGM); its bound of O(h) time per edge is never worse than the Θ ( √ m) time per edge necessary to list all triangles in a static graph, and is strictly better for graphs obeying a power law degree distribution. In order to better understand the behavior of the hindex statistic and its implications for the performance of our algorithms, we also study the behavior of the hindex on a set of 136 realworld networks. 1
Efficient Data Structures for Maintaining Set Partitions (Extended Abstract)
 Proceedings of Seventh Scandinavian Workshop on Algorithm Theory
, 1999
"... ) Michael Bender Saurabh Sethia Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 117944400 fbendersaurabhskienag@cs.sunysb.edu April 22, 1999 1 Introduction Each test or feature in a classification system defines a set partition on a class of object ..."
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Cited by 3 (1 self)
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) Michael Bender Saurabh Sethia Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 117944400 fbendersaurabhskienag@cs.sunysb.edu April 22, 1999 1 Introduction Each test or feature in a classification system defines a set partition on a class of objects. Adding new features refines the classification, whereas deleting features may result in merging previously distinguished classes. As an illustration, consider the set of automobile types f VW Beetle, Toyota, Lexus, Cadillac g. The feature size partitions the cars into sets of small and large cars, ff VW Beetle, Toyotag, f Lexus, Cadillac gg. The feature domesticorigin partitions the cars into ff VW Beetle, Toyota, Lexus g, f Cadillac gg. The feature uglyshape distinguishes f VW Beetle, Cadillac g from f Toyota, Lexus g. Incorporating both size and origin induces the refined partition ff VW Beetle, Toyotag, f Lexus g, f Cadillac gg, whereas the union of all three features completely di...
Dynamic Routing Schemes for Graphs with Low Local Density
, 2007
"... This paper studies approximate distributed routing schemes on dynamic communication networks. The paper focuses on dynamic weighted general graphs where the vertices of the graph are fixed but the weights of the edges may change. Our main contribution concerns bounding the cost of adapting to dynam ..."
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Cited by 2 (0 self)
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This paper studies approximate distributed routing schemes on dynamic communication networks. The paper focuses on dynamic weighted general graphs where the vertices of the graph are fixed but the weights of the edges may change. Our main contribution concerns bounding the cost of adapting to dynamic changes. The update efficiency of a routing scheme is measured by the time needed in order to update the routing scheme following a weight change. A naive dynamic routing scheme, which updates all vertices following a weight change, requires Ω(Diam) time in order to perform the updates after every weight change, where Diam is the diameter of the underlying graph. In contrast, this paper presents approximate dynamic routing schemes with average time complexity ˜Θ(D), per topological change, where D is the local density parameter of the underlying graph. Following a weight change, our scheme never incurs more than Diam time, thus, our scheme is particularly efficient on graphs which have low local density and large diameter. The paper also establishes upper and lower bounds on the size of the databases required by the scheme at each site.
Compact Separator Decompositions in Dynamic Trees and Applications to Labeling Schemes
"... In this paper we construct an efficient scheme maintaining a separator decomposition representation in dynamic trees. Our dynamic scheme uses asymptotically optimal labels. In order to maintain the short label, the scheme uses O(log 3 n) amortized message complexity, per topology change, where n is ..."
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Cited by 1 (0 self)
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In this paper we construct an efficient scheme maintaining a separator decomposition representation in dynamic trees. Our dynamic scheme uses asymptotically optimal labels. In order to maintain the short label, the scheme uses O(log 3 n) amortized message complexity, per topology change, where n is the current number of nodes in the tree. We note that a separator decomposition is a fundamental decomposition of trees used extensively as a component in many static graph algorithms. Therefore, our dynamic separator decomposition may be used for constructing dynamic versions to these algorithms. Going along these lines, we then show how to use our dynamic separator decomposition to construct rather efficient routing schemes on dynamic trees, for both the designer and the adversary port models. Since passing messages from one place to another is usually the main purpose of a network, constructing efficient dynamic routing schemes is an important task in the fields of distributed computing and communication networks. Our dynamic routing schemes use O(log 3 n) amortized message complexity and the labels they maintains are optimal up to a multiplicative factor of O(log log n). In addition, we show how to use our dynamic separator decomposition to construct dynamic labeling schemes supporting the ancestry and NCA relations using asymptotically optimal labels and O(log 3 n) amortized message complexity. Finally, we show how to use our dynamic separator decomposition to extend a known result on dynamic distance labeling schemes.
Communicated by:
, 2012
"... We describe a data structure that maintains the number of triangles in a dynamic undirected graph, subject to insertions and deletions of edges and of degreezero vertices. More generally it can be used to maintain the number of copies of each possible threevertex subgraph in time O(h) per update, ..."
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We describe a data structure that maintains the number of triangles in a dynamic undirected graph, subject to insertions and deletions of edges and of degreezero vertices. More generally it can be used to maintain the number of copies of each possible threevertex subgraph in time O(h) per update, where h is the hindex of the graph, the maximum number such that the graph contains h vertices of degree at least h. We also show how to maintain the hindex itself, and a collection of h highdegree vertices in the graph, in constant time per update. Our data structure has applications in social network analysis using the exponential random graph model (ERGM); its bound of O(h) time per edge is never worse than the Θ ( √ m) time per edge necessary to list all triangles in a static graph, and is strictly better for graphs obeying a power law degree distribution. In order to better understand the behavior of the hindex statistic and its implications for the performance of our algorithms, we also study the behavior of the hindex on a set of 136 realworld networks. Submitted: