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23
Polylogarithmic deterministic fullydynamic graph algorithms I: connectivity and minimum spanning tree
 JOURNAL OF THE ACM
, 1997
"... Deterministic fully dynamic graph algorithms are presented for connectivity and minimum spanning forest. For connectivity, starting with no edges, the amortized cost for maintaining a spanning forest is O(log² n) per update, i.e. per edge insertion or deletion. Deciding connectivity between any two ..."
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Cited by 125 (6 self)
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Deterministic fully dynamic graph algorithms are presented for connectivity and minimum spanning forest. For connectivity, starting with no edges, the amortized cost for maintaining a spanning forest is O(log² n) per update, i.e. per edge insertion or deletion. Deciding connectivity between any two given vertices is done in O(log n= log log n) time. This matches the previous best randomized bounds. The previous best deterministic bound was O( 3 p n log n) amortized time per update but constant time for connectivity queries. For minimum spanning trees, first a deletionsonly algorithm is presented supporting deletes in amortized time O(log² n). Applying a general reduction from Henzinger and King, we then get a fully dynamic algorithm such that starting with no edges, the amortized cost for maintaining a minimum spanning forest is O(log^4 n) per update. The previous best deterministic bound was O( 3 p n log n) amortized time per update, and no better randomized bounds were ...
SingleStrip Triangulation of Manifolds with Arbitrary Topology
, 2004
"... Triangle strips have been widely used for efficient rendering. It is NPcomplete to test whether a given triangulated model can be represented as a single triangle strip, so many heuristics have been proposed to partition models into few long strips. In this paper, we present a new algorithm for cre ..."
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Cited by 16 (5 self)
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Triangle strips have been widely used for efficient rendering. It is NPcomplete to test whether a given triangulated model can be represented as a single triangle strip, so many heuristics have been proposed to partition models into few long strips. In this paper, we present a new algorithm for creating a single triangle loop or strip from a triangulated model. Our method applies a dual graph matching algorithm to partition the mesh into cycles, and then merges pairs of cycles by splitting adjacent triangles when necessary. New vertices are introduced at midpoints of edges and the new triangles thus formed are coplanar with their parent triangles, hence the visual fidelity of the geometry is not changed. We prove that the increase in the number of triangles due to this splitting is 50 % in the worst case, however for all models we tested the increase was less than 2%. We also prove tight bounds on the number of triangles needed for a singlestrip representation of a model with holes on its boundary. Our strips can be used not only for efficient rendering, but also for other applications including the generation of space filling curves on a manifold of any arbitrary topology.
Lombardi Drawings of Graphs
"... We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each ..."
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Cited by 9 (6 self)
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We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.
SQuad: Compact representation for triangle meshes
 Computer Graphics Forum
, 2011
"... The SQuad data structure represents the connectivity of a triangle mesh by its “S table ” of about 2 rpt (integer references per triangle). Yet it allows for a simple implementation of expected constanttime, randomaccess operators for traversing the mesh, including inorder traversal of the triang ..."
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Cited by 8 (3 self)
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The SQuad data structure represents the connectivity of a triangle mesh by its “S table ” of about 2 rpt (integer references per triangle). Yet it allows for a simple implementation of expected constanttime, randomaccess operators for traversing the mesh, including inorder traversal of the triangles incident upon a vertex. SQuad is more compact than the Corner Table (CT), which stores 6 rpt, and than the recently proposed SOT, which stores 3 rpt. However, incore access is generally faster in CT than in SQuad, and SQuad requires rebuilding the S table if the connectivity is altered. The storage reduction and memory coherence opportunities it offers may help to reduce the frequency of page faults and cache misses when accessing elements of a mesh that does not fit in memory. We provide the details of a simple algorithm that builds the S table and of an optimized implementation of the SQuad operators.
Simultaneous diagonal flips in plane triangulations
 In Proc. 17th Annual ACMSIAM Symp. on Discrete Algorithms (SODA ’06
, 2006
"... Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every nvertex triangulation with at least six vertices has a simultaneous flip into a 4connected triangulation, and that it can be computed in O(n) time. It follows that every triangulation has a simultaneous f ..."
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Cited by 7 (3 self)
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Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every nvertex triangulation with at least six vertices has a simultaneous flip into a 4connected triangulation, and that it can be computed in O(n) time. It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. This result is used to prove that for any two nvertex triangulations, there exists a sequence of O(log n) simultaneous flips to transform one into the other. The total number of edges flipped in this sequence is O(n). The maximum size of a simultaneous flip is then studied. It is proved that every triangulation has a simultaneous flip of at least 1 (n − 2) edges. On the other hand, every simultaneous flip has at most n − 2 edges, 3 and there exist triangulations with a maximum simultaneous flip of 6 (n − 2) edges. 7
Networks of Relations
, 2005
"... Project, and my advisor Shuki Bruck for supporting me during my studies. I would also like to thank Shuki for being a good advisor and collaborator. I am grateful not only to Shuki but to all the people I have worked with, including Erik Winfree and David Soloveichik, in collaboration with whom the ..."
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Cited by 5 (2 self)
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Project, and my advisor Shuki Bruck for supporting me during my studies. I would also like to thank Shuki for being a good advisor and collaborator. I am grateful not only to Shuki but to all the people I have worked with, including Erik Winfree and David Soloveichik, in collaboration with whom the material in section 3.4.2 was produced. My family has supported my adventure of being a student, especially my wife Éva, my children András, Adam, and Emma, my mother Sarah, and my grandfather Howard, and to them I am very grateful. iv Relations are everywhere. In particular, we think and reason in terms of mathematical and English sentences that state relations. However, we teach our students much more about how to manipulate functions than about how to manipulate relations. Consider functions. We know how to combine functions to make new functions, how to evaluate functions efficiently, and how to think about compositions of functions. Especially in the area of boolean functions, we have become experts in the theory and art of designing combinations of functions to yield what we want, and this expertise has led to techniques that enable
S.: Euclidean Representation of 3D electronic institutions: Automatic Generation
 In: Proceedings of the 8th International Working Conference on Advanced Visual Interfaces (AVI
, 2006
"... In this paper we present the 3D Electronic Institutions metaphor and show how it can be used for the specification of highly secure Virtual Worlds and how 3D Virtual Worlds can be automatically generated from this specification. To achieve the generation task we propose an algorithm for automatic tr ..."
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Cited by 5 (3 self)
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In this paper we present the 3D Electronic Institutions metaphor and show how it can be used for the specification of highly secure Virtual Worlds and how 3D Virtual Worlds can be automatically generated from this specification. To achieve the generation task we propose an algorithm for automatic transformation of the Performative Structure graph into a 3D Virtual World, using the rectangular dualization technique. The nodes of the initial graph are transformed into rooms, the connecting arcs between nodes determine which rooms have to be placed next to each other and define the positions of the doors connecting those rooms. The proposed algorithm is sufficiently general to be used for transforming any planar graph into a 3D Virtual World.
From Graphs to Euclidean Virtual Worlds: Visualization of 3D Electronic Institutions
"... In this paper we propose an algorithm for automatic transformation of a graph into a 3D Virtual World and its Euclidean map, using the rectangular dualization technique. The nodes of the initial graph are transformed into rooms, the connecting arcs between nodes determine which rooms have to be plac ..."
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Cited by 4 (4 self)
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In this paper we propose an algorithm for automatic transformation of a graph into a 3D Virtual World and its Euclidean map, using the rectangular dualization technique. The nodes of the initial graph are transformed into rooms, the connecting arcs between nodes determine which rooms have to be placed next to each other and define the positions of the doors connecting those rooms. The proposed algorithm is general enough to be used for automatic generation of 3D Virtual Worlds representation of any planar graph, however, our research is particulary focused on the automatic generation of 3D Electronic Institutions from the Performative Structure graph.
WorstCaseOptimal Algorithms for Guarding Planar Graphs and Polyhedral Surfaces
, 2003
"... We present an optimal \Theta (n)time algorithm for the selection of a subset of the vertices of an nvertex plane graph G so that each of the faces of G is covered by (i.e. incident with) one or more of the selected vertices. At most bn=2c vertices are selected, matching the worstcase requiremen ..."
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Cited by 4 (0 self)
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We present an optimal \Theta (n)time algorithm for the selection of a subset of the vertices of an nvertex plane graph G so that each of the faces of G is covered by (i.e. incident with) one or more of the selected vertices. At most bn=2c vertices are selected, matching the worstcase requirement. Analogous results for edgecovers are developed for two different notions of "coverage". In particular,our lineartime algorithm selects at most n \Gamma 2 edges to strongly cover G, at most bn=3c diagonals to cover G, and in the case where G has no quadrilateral faces, at most bn=3c edges to cover G. All these bounds are optimal in the worstcase. Most of our results flow from the study of a relaxation of thefamiliar notion of a 2coloring of a plane graph which we call a facerespecting 2coloring that permits