Maximum Cardinality Search for Computing Minimal Triangulations of Graphs

Berry, Alain; Blair, Jean R. S.; Heggernes, Pinar; Peyton, Barry W.

doi:10.1007/s00453-004-1084-3

Cited by 103 publications

(97 citation statements)

References 18 publications

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“…Despite its complexity merits, LEX M yields only a restricted family of minimal triangulations, and the size of the resulting fill is not small in general. Recently a new algorithm for computing minimal triangulations, which can be regarded as a simplification of LEX M, has been introduced [5]. This algorithm, called MCS-M, has the same asymptotic time complexity and the same kind of properties regarding fill as LEX M.…”

Section: Background and Motivationmentioning

confidence: 99%

A wide-range algorithm for minimal triangulation from an arbitrary ordering

Berry

Bordat

Heggernes

et al. 2006

Journal of Algorithms

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We present a new algorithm, called LB-Triang, which computes minimal triangulations. We give both a straightforward O(nm ) time implementation and a more involved O(nm) time implementation, thus matching the best known algorithms for this problem.Our algorithm is based on a process by Lekkerkerker and Boland for recognizing chordal graphs which checks in an arbitrary order whether the minimal separators contained in each vertex neighborhood are cliques. LB-Triang checks each vertex for this property and adds edges whenever necessary to make each vertex obey this property. As the vertices can be processed in any order, LB-Triang is able to compute any minimal triangulation of a given graph, which makes it significantly different from other existing triangulation techniques.We examine several interesting and useful properties of this algorithm, and give some experimental results. Background and motivationComputing a triangulation consists in embedding a given graph into a triangulated, or chordal, graph by adding a set of edges called a fill. If no proper subset of the fill can generate a chordal graph when added to the given graph, then this fill is said to be minimal, and the resulting chordal graph is called a minimal triangulation. The fill is said to be minimum if its cardinality is the smallest over all possible minimal fills, and the corresponding triangulation is called a minimum *

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Section: Background and Motivationmentioning

confidence: 99%

A wide-range algorithm for minimal triangulation from an arbitrary ordering

Berry

Bordat

Heggernes

et al. 2006

Journal of Algorithms

Self Cite

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“…But like Lex-M, MCS-M does not always perform well. For example, for the graph representing an N × N square grid, MCS-M produces exactly the same fill as Lex-M does, and as pointed out in [43], the minimum fill for such graphs is O(n 2 log n) whereas the fill produced by Lex-M and MCS-M is O(n 3 ) [4].…”

Section: Related Researchmentioning

confidence: 75%

“…Recently Berry et al [4] have introduced a new algorithm called MCS-M. This is a simplification of Lex-M so that cardinality weights are used instead of lexicographic labels.…”

Section: Related Researchmentioning

confidence: 99%

Exploiting the Probability of Observation for Efficient Bayesian Network Inference

Mousumi

Grant

2012

Advances in Artificial Intelligence

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It is well-known that the observation of a variable in a Bayesian network can affect the effective connectivity of the network, which in turn affects the efficiency of inference.Unfortunately, the observed variables may not be known until runtime, which limits the amount of compile-time optimization that can be done in this regard. This thesis considers how to improve inference when users know the likelihood of a variable being observed. It demonstrates how these probabilities of observation can be exploited to improve existing heuristics for choosing elimination orderings for inference. Empirical tests over a set of benchmark networks using the Variable Elimination algorithm show reductions of up to 50% and 70% in multiplications and summations, as well as runtime reductions of up to 55%. Similarly, tests using the Elimination Tree algorithm show reductions by as much as 64%, 55%, and 50% in recursive calls, total cache size, and runtime, respectively. iv Acknowledgments I take much pleasure to express my profound gratitude to my supervisor Dr. Kevin Grant for his persistent and inspiring supervision and helping me learn the ABC of Bayesian Networks. Without his endless help and continuous reassurance at the most difficult times, this thesis would not have been a reality. I am indebted to him for his support, encouragement, suggestions, generosity, and the invaluable knowledge he shared with me.

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“…The complexity of learning the B i in [5] is polynomial, as long as the maximal degree of a vertex is bounded by a polynomial. One can find a minimal triangulation in polynomial time [6]. Maximum cardinality search can be completed in linear time [6, §3].…”

Section: Introductionmentioning

confidence: 99%

Objective Bayesian nets from consistent datasets

Landes¹,

Williamson²

2016

AIP Conference Proceedings

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Abstract. This paper addresses the problem of finding a Bayesian net representation of the probability function that agrees with the distributions of multiple consistent datasets and otherwise has maximum entropy. We give a general algorithm which is significantly more efficient than the standard brute-force approach. Furthermore, we show that in a wide range of cases such a Bayesian net can be obtained without solving any optimisation problem.

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Maximum Cardinality Search for Computing Minimal Triangulations of Graphs

Cited by 103 publications

References 18 publications

A wide-range algorithm for minimal triangulation from an arbitrary ordering

A wide-range algorithm for minimal triangulation from an arbitrary ordering

Exploiting the Probability of Observation for Efficient Bayesian Network Inference

Objective Bayesian nets from consistent datasets

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