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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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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
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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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
personal communication
, 2002
"... 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 equiangularly spaced around ..."
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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 equiangularly spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs. Submitted:
Short bisection implementation in MATLAB
, 2006
"... ABSTRACT. This is the documentation of the local mesh refinement using newest bisection or longest bisection in MATLAB. The new feature of our implementation is the edge marking strategy to ensure the conformity. The short implementation is helpful for the teaching of adaptive finite element methods ..."
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ABSTRACT. This is the documentation of the local mesh refinement using newest bisection or longest bisection in MATLAB. The new feature of our implementation is the edge marking strategy to ensure the conformity. The short implementation is helpful for the teaching of adaptive finite element methods and programming in more advanced languages. 1.
An Algorithm for Computing Simple kFactors
"... A kfactor of graph G is defined as a kregular spanning subgraph of G. For instance, a 2factor of G is a set of cycles that span G. 2factors have multiple applications in Graph Theory, Computer Graphics, and Computational Geometry [5, 4, 6, 11]. We define a simple 2factor as a 2factor without d ..."
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A kfactor of graph G is defined as a kregular spanning subgraph of G. For instance, a 2factor of G is a set of cycles that span G. 2factors have multiple applications in Graph Theory, Computer Graphics, and Computational Geometry [5, 4, 6, 11]. We define a simple 2factor as a 2factor without degenerate cycles. In general, simple kfactors are defined as kregular spanning subgraphs where no edge is used more than once. We propose a new algorithm for computing simple kfactors for all values of k ≥ 2. 1
OPTIMAL MULTILEVEL METHODS FOR GRADED BISECTION GRIDS
"... Abstract. We design and analyze optimal additive and multiplicative multilevel methods for solving H 1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs ..."
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Abstract. We design and analyze optimal additive and multiplicative multilevel methods for solving H 1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be performed only for three vertices the new vertex and its two parent vertices. We show that our methods lead to optimal complexity for any dimensions and polynomial degree. The theory hinges on a new decomposition of bisection grids in any dimension, which is of independent interest and yields a corresponding decomposition of spaces. We use the latter to bridge the gap between graded and quasiuniform grids, for which the multilevel theory is wellestablished.
Lineartime algorithms to color topological graphs
, 2005
"... We describe a lineartime algorithm for 4coloring planar graphs. We indeed give an O(V + E + χ  + 1)time algorithm to Ccolor Vvertex Eedge graphs embeddable on a 2manifold M of Euler characteristic χ where C(M) is given by Heawood’s (minimax optimal) formula. Also we show how, in O(V + E) ..."
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We describe a lineartime algorithm for 4coloring planar graphs. We indeed give an O(V + E + χ  + 1)time algorithm to Ccolor Vvertex Eedge graphs embeddable on a 2manifold M of Euler characteristic χ where C(M) is given by Heawood’s (minimax optimal) formula. Also we show how, in O(V + E) time, to find the exact chromatic number of a maximal planar graph (one with E = 3V − 6) and a coloring achieving it. Finally, there is a lineartime algorithm to 5color a graph embedded on any fixed surface M except that an Mdependent constant number of vertices are left uncolored. All the algorithms are simple and practical and run on a deterministic pointer machine, except for planar graph 4coloring which involves enormous constant factors and requires an integer RAM with a random number generator. All of the algorithms mentioned so far are in the ultraparallelizable deterministic computational complexity class “NC. ” We also have more practical planar 4coloring algorithms that can run on pointer machines in O(V log V) randomized time and O(V) space, and a very simple deterministic O(V)time coloring algorithm for planar graphs which conjecturally uses 4 colors.
GraphTheoretic Solutions to Computational Geometry Problems
, 908
"... Abstract. Many problems in computational geometry are not stated in graphtheoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graphtheoretic algorithm on it. Often, the efficiency of the algorithm depends on the special properties of the graph construct ..."
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Abstract. Many problems in computational geometry are not stated in graphtheoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graphtheoretic algorithm on it. Often, the efficiency of the algorithm depends on the special properties of the graph constructed in this way. We survey the art gallery problem, partition into rectangles, minimumdiameter clustering, rectilinear cartogram construction, mesh stripification, angle optimization in tilings, and metric embedding from this perspective. 1
RIMS1731 Covering Cuts in Bridgeless Cubic Graphs By
, 2011
"... In this paper we are interested in algorithms for finding 2factors that cover certain prescribed edgecuts in bridgeless cubic graphs. We present an algorithm for finding a minimumweight 2factor covering all the 3edge cuts in weighted bridgeless cubic graphs, together with a polyhedral descripti ..."
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In this paper we are interested in algorithms for finding 2factors that cover certain prescribed edgecuts in bridgeless cubic graphs. We present an algorithm for finding a minimumweight 2factor covering all the 3edge cuts in weighted bridgeless cubic graphs, together with a polyhedral description of such 2factors and that of perfect matchings intersecting all the 3edge cuts in exactly one edge. We further give an algorithm for finding a 2factor covering all the 3 and 4edge cuts in bridgeless cubic graphs. Both of these algorithms run in O(n 3) time, where n is the number of vertices. As an application of the latter algorithm, we design a 6/5approximation algorithm for finding a minimum 2edgeconnected subgraph in 3edgeconnected cubic graphs, which improves upon the previous best ratio of 5/4. The algorithm begins with finding a 2factor covering all 3 and 4edge cuts, which is the bottleneck in terms of complexity, and thus it has running time O(n 3). We then improve this time complexity to O(n 2 log 4 n) by relaxing the condition of the initial 2factor and elaborating the subsequent processes.