## Bucket Elimination: A Unifying Framework for Reasoning

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Citations: | 274 - 60 self |

### BibTeX

@MISC{Dechter_bucketelimination:,

author = {Rina Dechter},

title = {Bucket Elimination: A Unifying Framework for Reasoning},

year = {}

}

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### Abstract

Bucket elimination is an algorithmic framework that generalizes dynamic programming to accommodate many problem-solving and reasoning tasks. Algorithms such as directional-resolution for propositional satisfiability, adaptive-consistency for constraint satisfaction, Fourier and Gaussian elimination for solving linear equalities and inequalities, and dynamic programming for combinatorial optimization, can all be accommodated within the bucket elimination framework. Many probabilistic inference tasks can likewise be expressed as bucket-elimination algorithms. These include: belief updating, finding the most probable explanation, and expected utility maximization. These algorithms share the same performance guarantees; all are time and space exponential in the inducedwidth of the problem's interaction graph. While elimination strategies have extensive demands on memory, a contrasting class of algorithms called "conditioning search" require only linear space. Algorithms in this class split a problem into subproblems by instantiating a subset of variables, called a conditioning set, or a cutset. Typical examples of conditioning search algorithms are: backtracking (in constraint satisfaction), and branch and bound (for combinatorial optimization). The paper presents the bucket-elimination framework as a unifying theme across probabilistic and deterministic reasoning tasks and show how conditioning search can be augmented to systematically trade space for time.

### Citations

11502 |
Computers and Intractability, A Guide to the Theory of NPCompleteness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...ed at length in [19, 20]. 2.2 Bucket elimination for Propositional CNFs Bucket elimination generality can be further illustrated with an algorithm in deterministic reasoning for solving satis ability =-=[26]-=-. Propositional variables take only two valuesftrue� falseg or \1" and \0." We denote propositional variables by uppercase letters P� Q� R� : : :, propositional literals (i.e., P�:P ) stand for P =\tr... |

7493 |
Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference
- Pearl
- 1988
(Show Context)
Citation Context ...are needed, the algorithm can try di erent assignments to the conditioning set. Algorithms such asbacktracking search andbranch and bound may be viewed as conditioning algorithms. Cutset-conditioning =-=[12, 34]-=- applies conditioning to a subset of variables that cut all cycles of the interaction graph and solve the resulting subproblem by bucket-elimination. The complexity of conditioning algorithms is expon... |

1349 |
Local computations with probabilities on graphical structures and their application to expert systems (with discussion
- Lauritzen, Spiegelhalter
- 1988
(Show Context)
Citation Context ...yis dependent on the induced width graph parameter. The algorithms are variations on known algorithms, and, for the most part, are not new in the sense that the basic ideas have existed for some time =-=[8, 34,31,50, 28, 39,32,3,45,46,48,47]-=-. De nition 2 (graph concepts) A directed graph is a pair, G =fV�Eg, where V =fX1�:::�Xng is a set of elements and E =f(Xi�Xj)jXi�Xj2 V�i6= jg is the set of edges. If (Xi�Xj)2E, we say that Xi points ... |

1124 |
A computing procedure for quantification theory
- Davis, Putnam
- 1960
(Show Context)
Citation Context ...ier Science B.V. All rights reserved. PII: S0004-3702(99)00059-442 R. Dechter / Artificial Intelligence 113 (1999) 41–85 activities, including directional resolution for propositional satisfiability =-=[13]-=-, adaptive consistency for constraint satisfaction [22], Fourier and Gaussian elimination for linear equalities and inequalities, and dynamic programming for combinatorial optimization [5]. The bucket... |

1016 | Temporal Constraint Networks
- Dechter, I, et al.
- 1991
(Show Context)
Citation Context ...constraints {green,red} {green,red} Constraint networks have beenshown to be useful in formulating diverse problems such as scene labeling, scheduling, natural language parsing and temporal reasoning =-=[13]-=-. Consider the following graph coloring problem in Figure 1. The task is to assign a color to each node in the graph so that adjacent nodes will have di erent colors. Here is one way tosolve this prob... |

773 |
A machine program for theorem proving
- Davis, Longemann, et al.
- 1962
(Show Context)
Citation Context ...h for solving constraint satisfaction problems. For a recent survey see [18]. The most well known version of backtracking for propositional satis ability istheDavis-Logemann-Loveland (DPLL) algorithm =-=[10]-=-, 13sWorst-case time Average time Space Output Conditioning O( exp( n ) ) better than worst-case O( n ) one solution Elimination O( n exp( w* )) w* n Same as worst-cas O( n exp( w* )) w* n knowledge c... |

402 |
Evaluating influence diagrams
- Shachter
- 1986
(Show Context)
Citation Context ...ingly-connected networks [37]. The two main approaches for extending this propagation algorithm to multiply-connected networks are the cycle-cutset approach, (cutset-conditioning), and treeclustering =-=[34,37,46]-=-. These methods work well for sparse networks with small cyclecutsets or clusters. In subsequent sections bucket-elimination algorithms for each of these tasks will be presented and their relationship... |

401 |
Network-based heuristics for constraint satisfaction problems
- Dechter, Pearl
- 1987
(Show Context)
Citation Context ...commodate algorithms for many complex problem-solving and reasoning activities, including directional resolution for propositional satis ability [11], adaptive consistency for constraint satisfaction =-=[19]-=-, Fourier and 1sGaussian elimination for linear equalities and inequalities, and dynamic programming for combinatorial optimization [5]. The bucket elimination framework will be demonstrated by presen... |

303 | Context-specic independence in Bayesian networks
- Boutilier, Friedman, et al.
- 1996
(Show Context)
Citation Context ... to exploit compilation vs. run-time resources. These issues should be addressed. In particular, improvements exploiting the structure of the conditional probability matrices as presented recently in =-=[7,40,43]-=- can be incorporated on top of bucket-elimination. Acknowledgement A preliminary version of this paper appeared in [17]. An extension restricted to probabilistic reasoning only appears in [20]. I woul... |

295 |
Bayesian updating in causal probabilistic networks by local computations
- Jensen, Lauritzen, et al.
- 1990
(Show Context)
Citation Context ...yis dependent on the induced width graph parameter. The algorithms are variations on known algorithms, and, for the most part, are not new in the sense that the basic ideas have existed for some time =-=[8, 34,31,50, 28, 39,32,3,45,46,48,47]-=-. De nition 2 (graph concepts) A directed graph is a pair, G =fV�Eg, where V =fX1�:::�Xng is a set of elements and E =f(Xi�Xj)jXi�Xj2 V�i6= jg is the set of edges. If (Xi�Xj)2E, we say that Xi points ... |

294 | Bucket elimination: A unifying framework for probabilistic inference
- Dechter
- 1996
(Show Context)
Citation Context ...ure of the conditional probability matrices as presented recently in [42, 7, 37] can be incorporated on top of bucket-elimination. 47s13 Acknowledgment A preliminary version of this paper appeared in =-=[14]-=-. An extension restricted to probabilistic reasoning only appears in [17]. I would like to thank Irina Rish and Nir Freidman for their useful comments on di erent versions of this paper. This work was... |

287 |
A sufficient condition for backtrack-free search
- Freuder
- 1982
(Show Context)
Citation Context ...gorithm generates only unary relationships and is therefore very efficient. It is known that finding w ∗ (and the minimizing ordering) is NP-complete [2]. However greedy heuristic ordering algorithms =-=[5,28]-=- and approximation orderings exist [4,50]. Also, the induced width of a given ordering is easy to compute. Algorithm Adaptive-consistency and its properties are discussed at length in [22,23]. 2.2. Bu... |

284 |
Enhancement schemes for constraint processing: Backjumping, learning, and cutset decomposition
- Dechter
- 1990
(Show Context)
Citation Context ...are needed, the algorithm can try di erent assignments to the conditioning set. Algorithms such asbacktracking search andbranch and bound may be viewed as conditioning algorithms. Cutset-conditioning =-=[12, 34]-=- applies conditioning to a subset of variables that cut all cycles of the interaction graph and solve the resulting subproblem by bucket-elimination. The complexity of conditioning algorithms is expon... |

263 |
Tree clustering for constraint networks
- Dechter, Pearl
- 1989
(Show Context)
Citation Context ...rithms (i.e., bucket-elimination and join-tree clustering) are closely related, and their worst-case complexity (time and space) is essentially the same (as already observed for constraint processing =-=[20]-=-). Join-tree clustering is initiated by triangulating the moral graph along a given variable ordering. The maximal cliques (i.e., maximally fully connected subgraphs) of the triangulated graph are use... |

207 |
Efficient algorithms for combinatorial problems on graphs with bounded decomposability - a survey
- Arnborg
- 1985
(Show Context)
Citation Context ...matic execution of adaptive-consistency along d, the algorithm generates only unary relationships and is therefore very e cient. It is known that nding w* (and the minimizing ordering) is NP-complete =-=[2]-=-. However greedy heuristic ordering algorithms [5, 25] and approximation 7sW*(D)= 3 {1,2} D E {1,2} C A B {1,2} {1,2} {1,2,3} Figure 4: A modi ed graph coloring problem E D C B A W*(d) = 3 A D C B E W... |

183 |
Nonserial Dynamic Programming
- Bertele, Brioschi
- 1972
(Show Context)
Citation Context ... ability [11], adaptive consistency for constraint satisfaction [19], Fourier and 1sGaussian elimination for linear equalities and inequalities, and dynamic programming for combinatorial optimization =-=[5]-=-. The bucket elimination framework will be demonstrated by presenting reasoning algorithms for processing both deterministic knowledge-bases such as constraint networks and cost networks as well as pr... |

172 | Semiring-based Constraint Satisfaction and Optimization
- Bistarelli, Montanari, et al.
- 1997
(Show Context)
Citation Context ...veral works in the last decade. In [38] the connection between optimization and constraint satisfaction and its relationship to dynamic programming is explicated. In the work of [33, 47] and later in =-=[6]-=- an axiomatic framework that characterize tasks that can be solved polynomially over hyper-trees, is introduced. Such tasks can be described using combination and projection operators over real-valued... |

160 |
Linear time algorithms for NP-hard problems restricted to partial k-trees
- Arnborg, Proskurowski
- 1989
(Show Context)
Citation Context ...s are also made explicit. Task speci c properties are also studied (e.g, irrelevant buckets in belief updating). The work we showhere also ts into the framework developed by Arnborg and Proskourowski =-=[2, 1]-=-. They present table-based reductions for various NPhard graph problems such as the independent-set problem, network reliability, vertex cover, graph k-colorability, and Hamilton circuits. Here and el... |

160 | Exploiting causal independence in Bayesian network inference
- Zhang, Poole
- 1996
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Citation Context ...3 Algorithm elim-bel computes the posterior belief P (x1je) for any given ordering of the variables which is initiated byX1. 2 The peeling algorithm for genetic trees [8], Zhang and Poole's algorithm =-=[51]-=-, as well as the SPI algorithm by D'Ambrosio et. al., [39] are all variations of elim-bel. Decimation algorithms in statistical physics are also related and were applied to Boltzmann trees [43]. 21sG ... |

147 | Probabilistic inference and influence diagrams - Shachter - 1988 |

115 |
Valuation-based systems for Bayesian Decision Analysis
- Shenoy
- 1992
(Show Context)
Citation Context ...yis dependent on the induced width graph parameter. The algorithms are variations on known algorithms, and, for the most part, are not new in the sense that the basic ideas have existed for some time =-=[8, 34,31,50, 28, 39,32,3,45,46,48,47]-=-. De nition 2 (graph concepts) A directed graph is a pair, G =fV�Eg, where V =fX1�:::�Xng is a set of elements and E =f(Xi�Xj)jXi�Xj2 V�i6= jg is the set of edges. If (Xi�Xj)2E, we say that Xi points ... |

113 |
Identifying independence in Bayesian networks
- Geiger, Verma, et al.
- 1990
(Show Context)
Citation Context ...ial semantics which allows restricting the computation to relevant portions of the belief network. These restrictions are already available in the literature in the context of the existing algorithms =-=[27, 44]-=-. Since summation over all values of a probability function is 1, the recorded functions of some buckets will degenerate to the constant 1. Ifwe can predict these cases in advance, we can avoid needle... |

112 | Dynamic programming and influence diagrams
- Tatman, Shachter
- 1990
(Show Context)
Citation Context ...d with utility components (i.e., an influence diagram) in O(n exp(w ∗ (d, e)) time and space, where w ∗ (d, e) is the adjusted induced width along d of the augmented moral graph. Tatman and Schachter =-=[51]-=- have published an algorithm for the general influence diagram that is a variation of elim-meu. Kjaerulff’s algorithm [32] can be viewed as a variation of elim-meu tailored to dynamic probabilistic ne... |

101 | Directional resolution: The Davis-Putnam procedure, revisited
- Dechter, Rish
- 1994
(Show Context)
Citation Context ... E� D�C�B�A algorithm for propositional satis ability whichwecall directional resolution. Algorithm directional resolution, (DR), is the core of the well-known DavisPutnam algorithm for satis ability =-=[11,21]-=-. Algorithm DR (see Figure 8) is described using buckets partitioning the set of clauses in the theory '. We call its output theory Ed('), the directional extension of '. Given an ordering d = Q1� :::... |

71 | A sufficiently fast algorithm for finding close to optimal junction trees
- Becker, Geiger
- 1996
(Show Context)
Citation Context ...s and is therefore very efficient. It is known that finding w ∗ (and the minimizing ordering) is NP-complete [2]. However greedy heuristic ordering algorithms [5,28] and approximation orderings exist =-=[4,50]-=-. Also, the induced width of a given ordering is easy to compute. Algorithm Adaptive-consistency and its properties are discussed at length in [22,23]. 2.2. Bucket elimination for propositional CNFs B... |

68 | Symbolic probabilistic inference in belief networks
- Shachter, D’Ambrosio, et al.
- 1990
(Show Context)
Citation Context |

65 | Local and global relational consistency
- Dechter, Beek
- 1997
(Show Context)
Citation Context ...resent table-based reductions for various NPhard graph problems such as the independent-set problem, network reliability, vertex cover, graph k-colorability, and Hamilton circuits. Here and elsewhere =-=[22, 16]-=-we extend the approach to a di erent set of problems. The following paragraphs summarize and generalizes the bucket elimination algorithm using two operators of combination and marginalization. The ta... |

60 | Dynamic Variable Ordering in CSPs
- Bacchus, Run
- 1995
(Show Context)
Citation Context |

59 |
Probability functions on complex pedigrees
- Cannings, Thompson, et al.
- 1978
(Show Context)
Citation Context |

57 | Topological parameters for time-space tradeoff
- Dechter, Fattah
- 2001
(Show Context)
Citation Context ...ning search. This complementary behavior calls for algorithms that combine the two approaches. Indeed, such algorithms are being developed for constraint-satisfaction and propositional satisfiability =-=[10,15,18,41]-=-. In the following sections we will focus in more detail on deriving bucket elimination algorithms for processing probabilistic networks. We are presenting a syntactic and uniform exposition emphasizi... |

56 | A practical algorithm for finding optimal triangulations
- Shoikhet, Geiger
- 1997
(Show Context)
Citation Context ...s and is therefore very efficient. It is known that finding w ∗ (and the minimizing ordering) is NP-complete [2]. However greedy heuristic ordering algorithms [5,28] and approximation orderings exist =-=[4,50]-=-. Also, the induced width of a given ordering is easy to compute. Algorithm Adaptive-consistency and its properties are discussed at length in [22,23]. 2.2. Bucket elimination for propositional CNFs B... |

54 |
A su cient condition for backtrack-free search
- Freuder
(Show Context)
Citation Context ...he algorithm generates only unary relationships and is therefore very e cient. It is known that nding w* (and the minimizing ordering) is NP-complete [2]. However greedy heuristic ordering algorithms =-=[5, 25]-=- and approximation 7sW*(D)= 3 {1,2} D E {1,2} C A B {1,2} {1,2} {1,2,3} Figure 4: A modi ed graph coloring problem E D C B A W*(d) = 3 A D C B E W*(d) = 2 W*(D)= 2 Figure 5: The induced-width along th... |

43 | Mini-buckets: A general scheme for generating approximations in automated reasoning
- Dechter
- 1997
(Show Context)
Citation Context ...resent table-based reductions for various NPhard graph problems such as the independent-set problem, network reliability, vertex cover, graph k-colorability, and Hamilton circuits. Here and elsewhere =-=[22, 16]-=-we extend the approach to a di erent set of problems. The following paragraphs summarize and generalizes the bucket elimination algorithm using two operators of combination and marginalization. The ta... |

42 | On the Generation of Alternative Explanations with Implications for Belief Revision
- Santos
- 1991
(Show Context)
Citation Context ...earchers have investigated various approaches to nding the mpe in a belief network [34, ?, 35, 36]. Recent proposals include best rst-search algorithms [48] and algorithms based on linear programming =-=[41]-=-. The problem is to nd x 0 such that P (x 0 ) = maxx P (x� e) = maxx iP (xi�ejxpai) where x =(x1� :::� xn) ande is a set of observations, on subsets of the variables. Computing for a given ordering X1... |

41 |
A Computing Procedure for Quanti cation Theory
- Davis, Putnam
- 1960
(Show Context)
Citation Context ...ic framework that generalizes dynamic programming to accommodate algorithms for many complex problem-solving and reasoning activities, including directional resolution for propositional satis ability =-=[11]-=-, adaptive consistency for constraint satisfaction [19], Fourier and 1sGaussian elimination for linear equalities and inequalities, and dynamic programming for combinatorial optimization [5]. The buck... |

41 |
Efficient inference in Bayes networks as a combinatorial optimization problem
- Li, D’Ambrosio
- 1994
(Show Context)
Citation Context |

37 | Probabilistic partial evaluation: Exploiting rule structure in probabilistic inference
- Poole
- 1997
(Show Context)
Citation Context ... to exploit compilation vs. run-time resources. These issues should be addressed. In particular, improvements exploiting the structure of the conditional probability matrices as presented recently in =-=[42, 7, 37]-=- can be incorporated on top of bucket-elimination. 47s13 Acknowledgment A preliminary version of this paper appeared in [14]. An extension restricted to probabilistic reasoning only appears in [17]. I... |

37 |
A new algorithm for finding MAP assignments to belief networks
- Shimony, Charniak
(Show Context)
Citation Context ...lose enough and is often used in applications. Researchers have investigated various approaches to finding the mpe in a belief network [9,37–39]. Recent proposals include best first-search algorithms =-=[49]-=- and algorithms based on linear programming [42]. The problem is to find x0 such that P(x 0 ) = max x ∏ P(x,e)= max P(xi,e|xpa ) i x i where x = (x1,...,xn) and e is a set of observations, on subsets ... |

33 | A connectionist model for diagnostic problem solving - Peng, Reggia - 1989 |

31 | Backtracking algorithms for constraint satisfaction problems
- Dechter, Frost
- 1999
(Show Context)
Citation Context ...nment needs to be maintained. Intensive research in the last two decades has been done on improving the basic backtracking search for solving constraint satisfaction problems. For a recent survey see =-=[18]-=-. The most well known version of backtracking for propositional satis ability istheDavis-Logemann-Loveland (DPLL) algorithm [10], 13sWorst-case time Average time Space Output Conditioning O( exp( n ) ... |

28 |
Gaussian in uence diagrams
- Shachter, Kenley
- 1989
(Show Context)
Citation Context |

24 | Learning in Boltzmann trees
- Saul, Jordan
- 1994
(Show Context)
Citation Context ...orithm [51], as well as the SPI algorithm by D'Ambrosio et. al., [39] are all variations of elim-bel. Decimation algorithms in statistical physics are also related and were applied to Boltzmann trees =-=[43]-=-. 21sG B C D F A G B C F A (a) (b) (c) D Figure 18: Two orderings of the moral graph of our example problem 4.2 Complexity We see that although elim-bel can be applied using any ordering, its complexi... |

24 | A computational scheme for reasoning in dynamic probabilistic networks
- Kjærulff
- 1992
(Show Context)
Citation Context ...sted induced width along d of the augmented moral graph. Tatman and Schachter [51] have published an algorithm for the general influence diagram that is a variation of elim-meu. Kjaerulff’s algorithm =-=[32]-=- can be viewed as a variation of elim-meu tailored to dynamic probabilistic networks. 8. Cost networks and dynamic programming As we have mentioned at the outset, bucket-elimination algorithms are var... |

17 |
Composition principles for synthesis of optimal multistage processes
- Mitten
- 1964
(Show Context)
Citation Context ...h was recognized by several works in the last decade. In [38] the connection between optimization and constraint satisfaction and its relationship to dynamic programming is explicated. In the work of =-=[33, 47]-=- and later in [6] an axiomatic framework that characterize tasks that can be solved polynomially over hyper-trees, is introduced. Such tasks can be described using combination and projection operators... |

14 | An evaluation of structural parameters for probabilistic reasoning: Results on benchmark circuits
- Fattah, Y, et al.
- 1996
(Show Context)
Citation Context ...). Another method which uses the super-bucket approach collects a set of consecutive buckets into one super-bucket that it processes by conditioning, thus avoiding recording some intermediate results =-=[15, 24]-=-. See also [9]. 11 Additional related work Wehave mentioned throughout this paper algorithms in probabilistic and deterministic reasoning that can be viewed as bucket-elimination algorithms. Among tho... |

13 | Hybrid algorithms for approximate belief updating in bayes nets
- Santos, Shimony, et al.
(Show Context)
Citation Context ... to exploit compilation vs. run-time resources. These issues should be addressed. In particular, improvements exploiting the structure of the conditional probability matrices as presented recently in =-=[42, 7, 37]-=- can be incorporated on top of bucket-elimination. 47s13 Acknowledgment A preliminary version of this paper appeared in [14]. An extension restricted to probabilistic reasoning only appears in [17]. I... |

12 |
Contextspeci c independence in bayesian networks
- Boutilier, Friedman, et al.
- 1996
(Show Context)
Citation Context ... to exploit compilation vs. run-time resources. These issues should be addressed. In particular, improvements exploiting the structure of the conditional probability matrices as presented recently in =-=[42, 7, 37]-=- can be incorporated on top of bucket-elimination. 47s13 Acknowledgment A preliminary version of this paper appeared in [14]. An extension restricted to probabilistic reasoning only appears in [17]. I... |

12 |
An ordered examination of influence diagrams, Networks 20
- Shachter
- 1990
(Show Context)
Citation Context ...ial semantics which allows restricting the computation to relevant portions of the belief network. These restrictions are already available in the literature in the context of the existing algorithms =-=[30,45]-=-. Since summation over all values of a probability function is 1, the recorded functions of some buckets will degenerate to the constant 1. If we can predict these cases in advance, we can avoid needl... |

11 |
Topological parameters for time-space trade-o
- Dechter
- 1996
(Show Context)
Citation Context ...ioning search. This complementary behavior calls for algorithms that combine the two approaches. Indeed, such algorithms are being developed for constraint-satisfaction and propositional satis ability=-=[12,40,9,15]-=-. In the following sections we will focus in more detail on deriving bucket elimination algorithms for processing probabilistic networks. We are presenting a syntactic and uniform exposition emphasizi... |

11 | Plausability of diagnostic hypothesis - Peng, Reggia - 1986 |