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Parallel Tree Contraction Part 2: Further Applications
 SIAM JOURNAL ON COMPUTING
, 1991
"... This paper applies the parallel tree contraction techniques developed in Miller and paper [Randomness and Computation, 5, S. Micali, ed., JAI Press, 1989, pp. 4772] to a number of fundamental graph problems. The paper presents an time and processor, a 0sided randomized algorithm for testing the i ..."
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Cited by 29 (3 self)
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This paper applies the parallel tree contraction techniques developed in Miller and paper [Randomness and Computation, 5, S. Micali, ed., JAI Press, 1989, pp. 4772] to a number of fundamental graph problems. The paper presents an time and processor, a 0sided randomized algorithm for testing the isomorphism of trees, and an n) time, nprocessor algorithm for maximal isomorphism and for common subexpression elimination. An time, nprocessor algorithm for computing the canonical forms of trees and subtrees is given. An Ologn time algorithm for computing the tree of 3connected components of a graph, an n)time algorithm for computing an explicit planar embedding of a planar graph, and an n)time algorithm for computing a canonical form for a planar graph are also given. All these latter algorithms use only processors on a Parallel Random Access Machine (PRAM) model with concurrent writes and concurrent reads.
An O(n log n) algorithm for maximum stflow in a directed planar graph
"... We give the first correct O(n log n) algorithm for finding a maximum stflow in a directed planar graph. After a preprocessing step that consists in finding singlesource shortestpath distances in the dual, the algorithm consists of repeatedly saturating the leftmost residual stot path. ..."
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Cited by 28 (0 self)
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We give the first correct O(n log n) algorithm for finding a maximum stflow in a directed planar graph. After a preprocessing step that consists in finding singlesource shortestpath distances in the dual, the algorithm consists of repeatedly saturating the leftmost residual stot path.
Parallel Open Ear Decomposition with Applications to Graph Biconnectivity and Triconnectivity
 Synthesis of Parallel Algorithms
, 1992
"... This report deals with a parallel algorithmic technique that has proved to be very useful in the design of efficient parallel algorithms for several problems on undirected graphs. We describe this method for searching undirected graphs, called "open ear decomposition", and we relate this decompos ..."
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Cited by 25 (9 self)
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This report deals with a parallel algorithmic technique that has proved to be very useful in the design of efficient parallel algorithms for several problems on undirected graphs. We describe this method for searching undirected graphs, called "open ear decomposition", and we relate this decomposition to graph biconnectivity. We present an efficient parallel algorithm for finding this decomposition and we relate it to a sequential algorithm based on depthfirst search. We then apply open ear decomposition to obtain an efficient parallel algorithm for testing graph triconnectivity and for finding the triconnnected components of a graph.
Smarandache MultiSpace Theory
, 2011
"... Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countr ..."
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Cited by 13 (5 self)
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Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countries appear and each of them has its own system. All of these show that our WORLD is not in homogenous but in multiple. Besides, all things that one can acknowledge is determined by his eyes, or ears, or nose, or tongue, or body or passions, i.e., these six organs, which means the WORLD consists of have and not have parts for human beings. For thousands years, human being has never stopped his steps for exploring its behaviors of all kinds. We are used to the idea that our space has three dimensions: length, breadth and height with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. Today, we have known two heartening notions for sciences. One is the Smarandache multispace came into being by purely logic.
Graphs as rotations
 KAM Series, 96327, Prague,1996
"... Using a notation of corner between edges when graph has a fixed rotation, i.e. cyclical order of edges around vertices, we define combinatorial objects combinatorial maps as pairs of permutations, one for vertices and one for faces. Further, we define multiplication of these objects, that coincides ..."
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Cited by 12 (9 self)
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Using a notation of corner between edges when graph has a fixed rotation, i.e. cyclical order of edges around vertices, we define combinatorial objects combinatorial maps as pairs of permutations, one for vertices and one for faces. Further, we define multiplication of these objects, that coincides with the multiplication of permutations. We consider closed under multiplication classes of combinatorial maps that consist of closed classes of combinatorial maps with fixed edges where each such class is defined by a knot. One class among them is special, containing selfconjugate maps. 1
A new paradigm for changing topology during subdivision modeling
 In Proceedings of Pacific Graphics
, 2000
"... In this paper, we present a new paradigm that allows dynamically changing the topology of 2manifold polygonal meshes. Our new paradigm always guarantees topological consistency of polygonal meshes. Based on our paradigm, by simply adding and deleting edges, handles can be created and deleted, holes ..."
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Cited by 11 (4 self)
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In this paper, we present a new paradigm that allows dynamically changing the topology of 2manifold polygonal meshes. Our new paradigm always guarantees topological consistency of polygonal meshes. Based on our paradigm, by simply adding and deleting edges, handles can be created and deleted, holes can be opened or closed, polygonal meshes can be connected or disconnected. These edge insertion and edge deletion operations are highly consistent with subdivision algorithms. In particular, these operations can be easily included into a subdivision modeling system such that the topological changes and subdivision operations can be performed alternatively during model construction. We demonstrate practical examples of topology changes based on this new paradigm and show that the new paradigm is convenient, effective, efficient, and friendly to subdivision surfaces. 1
Local specification of surface subdivision algorithms
 AGTIVE 2003
, 2004
"... Many polygon mesh algorithms operate in a local manner, yet are formally specified using global indexing schemes. This obscures the essence of these algorithms and makes their specification unnecessarily complex, especially if the mesh topology is modified dynamically. We address these problems by d ..."
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Cited by 8 (2 self)
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Many polygon mesh algorithms operate in a local manner, yet are formally specified using global indexing schemes. This obscures the essence of these algorithms and makes their specification unnecessarily complex, especially if the mesh topology is modified dynamically. We address these problems by defining a set of local operations on polygon meshes represented by graph rotation systems. We also introduce the vv programming language, which makes it possible to express these operations in a machineâˆ’readable form. The usefulness of the vv language is illustrated by the application examples, in which we concentrate on subdivision algorithms for the geometric modeling of surfaces. The algorithms are specified as short, intuitive vv programs, directly executable by the corresponding modeling software.
Fully Dynamic Planarity Testing with Applications
"... The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without ..."
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Cited by 6 (0 self)
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The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O(n2=3) time, where n is the number of vertices in the graph. The bound for tests and deletions is worstcase, while the bound for insertions is amortized. This is the first algorithm for this problem with sublinear running time, and it affirmatively answers a question posed in [11]. The same data structure has further applications in maintaining the biconnected and triconnected components of a dynamic planar graph. The time bounds are the same: O(n2=3) worstcase time per edge deletion, O(n2=3) amortized time per edge insertion, and O(n2=3) worstcase time to check whether two vertices are either biconnected or triconnected.
On usefulness of combinatorial maps in graphtopological computations, prepared for some conference
, 1996
"... Having through the use of combinatorial maps a oneone correspondence between, say, permutations and graphs on surfaces, we try to find out simple formulas with permutations for non trivial calculations in the graphs on surfaces. 1 ..."
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Cited by 6 (5 self)
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Having through the use of combinatorial maps a oneone correspondence between, say, permutations and graphs on surfaces, we try to find out simple formulas with permutations for non trivial calculations in the graphs on surfaces. 1