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Minimal triangulations of graphs: A survey
 Discrete Mathematics
"... Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was ..."
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Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms. 1 Introduction and
Combinatorial and algebraic tools for optimal multilevel algorithms
, 2007
"... This dissertation presents combinatorial and algebraic tools that enable the design of the first linear work parallel iterative algorithm for solving linear systems involving Laplacian matrices of planar graphs. The major departure of this work from prior suboptimal and inherently sequential approac ..."
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Cited by 3 (0 self)
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This dissertation presents combinatorial and algebraic tools that enable the design of the first linear work parallel iterative algorithm for solving linear systems involving Laplacian matrices of planar graphs. The major departure of this work from prior suboptimal and inherently sequential approaches is centered around: (i) the partitioning of planar graphs into fixed size pieces that share small boundaries, by means of a local ”bottomup ” approach that improves the customary ”topdown ” approach of recursive bisection, (ii) the replacement of monolithic global preconditioners by graph approximations that are built as aggregates of miniature preconditioners. In addition, we present extensions to the theory and analysis of Steiner tree preconditioners. We construct more general Steiner graphs that lead to natural linear time solvers for classes of graphs that are known a priori to have certain structural properties. We also present a graphtheoretic approach to classical algebraic multigrid algorithms. We show that their design can be
Improving the performance of the vertex elimination algorithm for derivative calculation
 in AD2004: Proceedings of the 4th International Conference on Automatic Differentiation
, 2005
"... heuristics aiming to find elimination sequences that minimise the number of floatingpoint operations (flops) for vertex elimination Jacobian code. We also used the depthfirst traversal algorithm to reorder the statements of the Jacobian code with the aim of reducing the number of memory accesses. ..."
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Cited by 2 (1 self)
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heuristics aiming to find elimination sequences that minimise the number of floatingpoint operations (flops) for vertex elimination Jacobian code. We also used the depthfirst traversal algorithm to reorder the statements of the Jacobian code with the aim of reducing the number of memory accesses. In this work, we study the effects of reducing flops or memory accesses within the vertex elimination algorithm for Jacobian calculation. On RISC processors, we observed that for data residing in registers, the number of flops gives a good estimate of the execution time, while for outofregister data, the execution time is dominated by the time for memory access operations. We also present a statement reordering scheme based on a greedylist scheduling algorithm using ranking functions. This statement reordering will enable us to tradeoff the exploitation of the instruction level parallelism of such processors with the reduction in memory accesses. Key words: vertex elimination, Jacobian accumulation, performance analysis, statement reordering, greedylist scheduling algorithms