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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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Cited by 38 (3 self)
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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.
Towards a tighter coupling of bottomup and topdown sparse matrix ordering methods
 BIT
, 2001
"... Most stateoftheart ordering schemes for sparse matrices are a hybrid of a bottomup method such as minimum degree and a top down scheme such as George's nested dissection. In this paper we present an ordering algorithm that achieves a tighter coupling of bottomup and topdown methods. In our ..."
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Cited by 36 (0 self)
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Most stateoftheart ordering schemes for sparse matrices are a hybrid of a bottomup method such as minimum degree and a top down scheme such as George's nested dissection. In this paper we present an ordering algorithm that achieves a tighter coupling of bottomup and topdown methods. In our methodology vertex separators are interpreted as the boundaries of the remaining elements in an unfinished bottomup ordering. As a consequence, we are using bottomup techniques such as quotient graphs and special node selection strategies for the construction of vertex separators. Once all separators have been found, we are using them as a skeleton for the computation of several bottomup orderings. Experimental results show that the orderings obtained by our scheme are in general better than those obtained by other popular ordering codes.
Autotuning Performance on Multicore Computers
, 2008
"... personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires pri ..."
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Cited by 32 (10 self)
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personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific
An Efficient Algorithm to Compute Row and Column Counts for Sparse Cholesky Factorization
 SIAM J. MATRIX ANAL. APPL
, 1994
"... Let an undirected graph G be given, along with a specified depthfirst spanning tree T . We give almostlineartime algorithms to solve the following two problems: First, for every vertex v, compute the number of descendants w of v for which some descendant of w is adjacent (in G) to v. Second, ..."
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Cited by 31 (6 self)
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Let an undirected graph G be given, along with a specified depthfirst spanning tree T . We give almostlineartime algorithms to solve the following two problems: First, for every vertex v, compute the number of descendants w of v for which some descendant of w is adjacent (in G) to v. Second, for every vertex v, compute the number of ancestors of v that are adjacent (in G) to at least one descendant of v. These problems arise in Cholesky and QR factorizations of sparse matrices. Our algorithms can be used to determine the number of nonzero entries in each row and column of the triangular factor of a matrix from the zero/nonzero structure of the matrix. Such a prediction makes storage allocation for sparse matrix factorizations more efficient. Our algorithms run in time linear in the size of the input times a slowlygrowing inverse of Ackermann's function. The best previously known algorithms for these problems ran in time linear in the sum of the nonzero counts, which is...
Data Structures and Programming Techniques for the Implementation of Karmarkar's Algorithm
, 1989
"... This paper describes data structures and programming techniques used in an implementation of Karmarkar's algorithm for linear programming. Most of oar discussion focuses on applying Gaussian elimination toward the solution of a sequence of sparse symmetric positive dermite systems of linear equ ..."
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Cited by 31 (5 self)
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This paper describes data structures and programming techniques used in an implementation of Karmarkar's algorithm for linear programming. Most of oar discussion focuses on applying Gaussian elimination toward the solution of a sequence of sparse symmetric positive dermite systems of linear equations, the main requirement in Karmarkar's algorithm. Oar approach relies on a direct factorization scheme, with an extensive symbolic factodzation step performed in a preparatory stage of the linear programming algorithm. An interpretatire version of Gaussian elimination makes use of the symbolic information to perform the actual numerical computations at each iteration of algorithm. We also discuss ordering algorithms that attempt to reduce the mount offilldn in the LU factors, a procedare to build the linear system solved at each iteration, the use of a dense window data structure in the Gaussian elimination method, a preprecesslng procedare designed to increase the sparsity of the linear programming coefficient matrix, and the special treatment of dense columns in the coefficient matrix
A widerange algorithm for minimal triangulation from an arbitrary ordering
 JOURNAL OF ALGORITHMS
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On triangulating dynamic graphical models
 In Proceed. 19th Conf. on Uncertainty in Artificial Intelligence
, 2003
"... This paper introduces improved methodology to triangulate dynamic graphical models and dynamic Bayesian networks (DBNs). In this approach, a standard DBN template can be modified so the repeating and unrolled graph section may dissect the original DBN time slice and may also span (and partially inte ..."
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Cited by 30 (16 self)
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This paper introduces improved methodology to triangulate dynamic graphical models and dynamic Bayesian networks (DBNs). In this approach, a standard DBN template can be modified so the repeating and unrolled graph section may dissect the original DBN time slice and may also span (and partially intersect) many such slices. We introduce the notion of a “boundary ” which divides a graph into multislice partitions each of which has an interface, and define the “boundary algorithm”, a method to find the best boundary (and corresponding interface) between partitions in such models. We prove that, after using this algorithm, the sizes of the best forward and backward interface (and also the corresponding fillins) are identical. The boundary algorithm allows for constrained elimination orders (and therefore graph triangulations) that are impossible using standard slicebyslice constrained elimination. We describe the above using the Graphical Model ToolKit (GMTK)’s notion of dynamic graphical model, slightly generalizing standard DBN templates. We report triangulation results on handconcocted graphs, novel speech recognition DBNs, and random graphs, and find that the boundary algorithm can significantly improve both tree width and graph weight. 1
The Hierarchical Basis Multigrid Method And Incomplete LU Decomposition
 In Seventh International Symposium on Domain Decomposition Methods for Partial Differential Equations
, 1994
"... . A new multigrid or incomplete LU technique is developed in this paper for solving large sparse algebraic systems from discretizing partial differential equations. By exploring some deep connection between the hierarchical basis method and incomplete LU decomposition, the resulting algorithm can be ..."
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Cited by 27 (7 self)
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. A new multigrid or incomplete LU technique is developed in this paper for solving large sparse algebraic systems from discretizing partial differential equations. By exploring some deep connection between the hierarchical basis method and incomplete LU decomposition, the resulting algorithm can be effectively applied to problems discretized on completelyunstructured grids. Numerical experiments demonstrating the efficiency of the method are also reported. Key words. Finite element, hierarchical basis, multigrid, incomplete LU . AMS(MOS) subject classifications. 65F10, 65N20 1. Introduction. In this work, we explore the connection between the methods of sparse Gaussian elimination [8][13], incomplete LU (ILU) decomposition [9][10] and the hierarchical basis multigrid (HBMG) [16][4]. Hierarchical basis methods have proved to be one of the more robust classes of methods for solving broad classes of elliptic partial differential equations, especially the large systems arising in conju...
A Polynomial Approximation Algorithm for the Minimum FillIn Problem
, 1998
"... In the minimum fillin problem, one wishes to find a set of edges of smallest size, whose addition to a given graph will make it chordal. The problem has important applications in numerical algebra and has been studied intensively since the 1970s. We give the first polynomial approximation algorithm ..."
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Cited by 27 (2 self)
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In the minimum fillin problem, one wishes to find a set of edges of smallest size, whose addition to a given graph will make it chordal. The problem has important applications in numerical algebra and has been studied intensively since the 1970s. We give the first polynomial approximation algorithm for the problem. Our algorithm constructs a triangulation whose size is at most eight times the optimum size squared. The algorithm builds on the recent parameterized algorithm of Kaplan, Shamir and Tarjan for the same problem. For bounded degree graphs we give a polynomial approximation algorithm with a polylogarithmic approximation ratio. We also improve the parameterized algorithm.
A Practical Algorithm for Making Filled Graphs Minimal
 THEOR. COMP. SC
, 2001
"... For an arbitrary filled graph G + of a given original graph G, we consider the problem of removing fill edges from G + in order to obtain a graph M that is both a minimal filled graph of G and a subgraph of G + . For G + with f fill edges and e original edges, we give a simple O(f(e+f)) a ..."
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Cited by 27 (13 self)
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For an arbitrary filled graph G + of a given original graph G, we consider the problem of removing fill edges from G + in order to obtain a graph M that is both a minimal filled graph of G and a subgraph of G + . For G + with f fill edges and e original edges, we give a simple O(f(e+f)) algorithm which solves the problem and computes a corresponding minimal elimination ordering of G. We report on experiments with an implementation of our algorithm, where we test graphs G corresponding to some real sparse matrix applications and apply wellknown and widely used ordering heuristics to find G + . Our findings show the amount of fill that is commonly removed by a minimalization for each of these heuristics, and also indicate that the runtime of our algorithm on these practical graphs is better than the presented worstcase bound.