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30
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 157 (7 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 ...
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 20 (11 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.
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 19 (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.
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 13 (8 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
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 9 (4 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.
Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids
 In Multiscale, Nonlinear and Adaptive Approximation
, 2009
"... Abstract We give an overview of multilevel methods, such as Vcycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasiuniform meshes and extend them to graded meshes and completely unstructured grids. We firs ..."
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Cited by 9 (2 self)
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Abstract We give an overview of multilevel methods, such as Vcycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasiuniform meshes and extend them to graded meshes and completely unstructured grids. We first discuss the classical multigrid theory on the basis of the method of subspace correction of Xu and a key identity of Xu and Zikatanov. We next extend the classical multilevel methods in H(grad) to graded bisection grids upon employing the decomposition of bisection grids of Chen, Nochetto, and Xu. We finally discuss a class of multilevel preconditioners developed by Hiptmair and Xu for problems discretized on unstructured grids and extend them to H(curl) and H(div) systems over graded bisection grids. 1
Networks of Relations
, 2005
"... 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 ..."
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Cited by 8 (2 self)
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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
Hierarchyless simplification, stripification and compression of triangulated twomanifolds
 COMPUT. GRAPH. FORUM
, 2005
"... In this paper we explore the algorithmic space in which stripification, simplification and geometric compression of triangulated 2manifolds overlap. Edgecollapse/uncollapse based geometric simplification algorithms develop a hierarchy of collapses such that during uncollapse the reverse order has ..."
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Cited by 6 (3 self)
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In this paper we explore the algorithmic space in which stripification, simplification and geometric compression of triangulated 2manifolds overlap. Edgecollapse/uncollapse based geometric simplification algorithms develop a hierarchy of collapses such that during uncollapse the reverse order has to be maintained. We show that restricting the simplification and refinement operations only to, what we call, the collapsible edges creates hierarchyless simplification in which the operations on one edge can be performed independent of those on another. Although only a restricted set of edges is used for simplification operations, we prove topological results to show that, with minor retriangulation, any triangulated 2manifold can be reduced to either a single vertex or a single edge using the hierarchyless simplification, resulting in extreme simplification. The set of collapsible edges helps us analyze and relate the similarities between simplification, stripification and geometric compression algorithms. We show that the maximal set of collapsible edges implicitly describes a triangle strip representation of the original model. Further, these strips can be effortlessly maintained on multiresolution models obtained through any sequence of hierarchyless simplifications on these collapsible edges. Due to natural relationship between stripification and geometric compression, these multiresolution models can also be efficiently compressed using traditional compression algorithms. We present algorithms to find the maximal set of collapsible edges and to reorganize these edges to get the minimum
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 6 (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 &quot;coverage&quot;. 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