## Bucket Elimination: a Unifying Framework for Structure-driven Inference (1998)

Venue: | Artificial Intelligence |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Dechter98bucketelimination:,

author = {Rina Dechter},

title = {Bucket Elimination: a Unifying Framework for Structure-driven Inference},

journal = {Artificial Intelligence},

year = {1998}

}

### OpenURL

### Abstract

Bucket elimination is an algorithmic framework that generalizes dynamic programming to accommodate many complex 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. All these algorithms share the same performance guarantees; all are time and space exponential in the induced-width of the problem's interaction graph. While elimination strategies have extensive demands on memory, pure "conditioning" algorithms require only linear space. Conditioning is a generic name for a...

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Citation Context ... 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 satisfiability =-=[27]. Propositional vari-=-ables takes only two values ftrue; falseg or "0" and "1". We denote propositional variables, by uppercase letters P; Q; R; : : :, propositional literals (i.e., P; :P ) stand for P ... |

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Citation Context ... needed, the algorithm can try different assignments to the conditioning set. Algorithms such as backtrackingssearch and branch and bound may be viewed as conditioning algorithms. Cutset-conditioning =-=[12, 35]-=- 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... |

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Citation Context ...s the relationship between bucket-elimination algorithms and tree-clustering algorithms. Tree clustering algorithms were developed for probabilistic reasoning and constraint processing, independently =-=[20, 33]-=-. The similarity between these two classes of algorithms, observed for constraint satisfaction, can be extended to probabilistic reasoning. Finally, the last two decades have seen prosperity in constr... |

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Citation Context ...rage time Space worst-case better than exp( n ) O( ) Worst-case time knowledge compilation one solution Output Figure 12: Comparing elimination and conditioning basic ideas have existed for some time =-=[8, 35, 33, 49, 30, 39, 34, 3, 45, 46, 48, 47]-=-. What we are presenting here is a syntactic and uniform exposition emphasizing these algorithms' form as a straightforward elimination algorithm. The presentation allows ideas and techniques to flow ... |

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Citation Context ...r 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, 38]-=- can be incorporated on top of bucket-elimination. 13 Acknowledgment A preliminary version of this paper appeared in [14]. An extension restricted to probabilistic reasoning only appears in [17]. I wo... |

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Citation Context ...ucture of the conditional probability matrices as presented recently in [42, 7, 38] can be incorporated on top of bucket-elimination. 13 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 different versions of this paper. This work wa... |

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Citation Context ... needed, the algorithm can try different assignments to the conditioning set. Algorithms such as backtrackingssearch and branch and bound may be viewed as conditioning algorithms. Cutset-conditioning =-=[12, 35]-=- 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... |

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Citation Context ...s the relationship between bucket-elimination algorithms and tree-clustering algorithms. Tree clustering algorithms were developed for probabilistic reasoning and constraint processing, independently =-=[20, 33]-=-. The similarity between these two classes of algorithms, observed for constraint satisfaction, can be extended to probabilistic reasoning. Finally, the last two decades have seen prosperity in constr... |

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Citation Context ...ed by the schematic execution of adaptive-consistency along d, the algorithm generates only unary relationships and is therefore very efficient. It is known that finding the minimum w* is NP-complete =-=[2]-=-. However greedy heuristic ordering algorithms [5, 26] and approximation orderings exist. Also, the induced width of a given ordering is easy to compute. Algorithm Adaptive-consistency and its propert... |

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Citation Context ...sfiability [11], adaptive consistency for constraint satisfaction [19], Fourier and Gaussian 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... |

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Citation Context ...lso related and were applied to Boltzmann trees [43]. In addition, unifying frameworks observing the common features between various algorithms have appeared both previously [47] and more recently in =-=[6]-=-. The work we show here fits also 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 pr... |

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Citation Context ...ithm elim-bel compute the posterior belief P (x 1 je) for any given ordering of the variables which is initiated by X 1 . Both the peeling algorithm for genetic trees [8], Zhang and Poole's algorithm =-=[50]-=-, and the SPI algorithm by D'ambrosio et.al [39] are variations of elim-bel. Decimation algorithms in statistical physics are also related and were applied to Boltzmann trees [43]. 21 Algorithm elim-b... |

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Citation Context ...g the common features between various algorithms have appeared both previously [47] and more recently in [6]. The work we show here fits also 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... |

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Citation Context ... semantics which allows restricting the computation to relevant portions of the belief network. TheseSuch restrictions are already available in the literature in the context ofthe existing algorithms =-=[28, 44]-=-. 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... |

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Citation Context ...mination algorithm for propositional satisfiability which we call directional resolution. Algorithm directional resolution (DR), is the core of the well-known DavisPutnam algorithm for satisfiability =-=[11, 21]-=-. Algorithm DR (see Figure 8) is described using buckets partitioning the set of clauses in the theory '. We call its output theory, E d ('), the directional extension of '. Given an ordering d = Q 1 ... |

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Citation Context ...ates finding an ordering with a smallest induced width. Finding an ordering with the smallest induced width is hard [2], but useful greedy heuristics as well as approximation algorithms are available =-=[13, 4]-=-. In summary, the complexity of algorithm elim-bel is dominated by the time and space needed to process a bucket. Recording a function on all the bucket's variables is time and space exponential in th... |

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Citation Context ...viewed as approximating variable elimination, they recently inspired a general approximation scheme of bucket-elimination. These were recently applied successfully to a variety of probabilistic tasks =-=[22, 29]-=-. 3 Preliminaries for probabilistic reasoning The belief-network algorithms we present next have much in common with directional resolution and adaptive-consistency. They all possess the property of c... |

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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 =-=[23, 16]-=- we extend the approach to a different set of problems. 12 Summary and Conclusion Using the bucket-elimination framework which generalizes dynamic programming, constraint processing and resolution pro... |

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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 =-=[12, 40, 15]-=-. In the following sections we will focus in more detail on deriving bucket elimination algorithms for processing probabilistic networks. This will allow borrowing ideas and methods already developed ... |

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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 =-=[23, 16]-=- we extend the approach to a different set of problems. 12 Summary and Conclusion Using the bucket-elimination framework which generalizes dynamic programming, constraint processing and resolution pro... |

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Citation Context ...stigated various approaches to finding the mpe in a belief network. (See, e.g., [35, 9, 36, 37]). Recent proposals include best first-search algorithms [48] and algorithms based on linear programming =-=[41]-=-. The problem is to find x 0 such that P (x 0 ) = max x P (x; e) = max x \Pi i P (x i ; ejx pa i ) where x = (x 1 ; :::; xn ) and e is a set of observations, on subsets of the variables. Computing for... |

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Citation Context ...r 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, 38]-=- can be incorporated on top of bucket-elimination. 13 Acknowledgment A preliminary version of this paper appeared in [14]. An extension restricted to probabilistic reasoning only appears in [17]. I wo... |

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Citation Context ...e most likely input message which was transmitted over a noisy channel, given the observed output. Researchers have investigated various approaches to finding the mpe in a belief network. (See, e.g., =-=[35, 9, 36, 37]-=-). Recent proposals include best first-search algorithms [48] and algorithms based on linear programming [41]. The problem is to find x 0 such that P (x 0 ) = max x P (x; e) = max x \Pi i P (x i ; ejx... |

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Citation Context ...ment needs to be maintained. Intensive research in the last two decades, had 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 satisfiability is the Davis-Logemann-Loveland algorithm [10]. 13 2.5 Summary We have given an overview of three well known procedures in... |

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Citation Context ...nd Poole's algorithm [50], and the SPI algorithm by D'ambrosio et.al [39] are variations of elim-bel. Decimation algorithms in statistical physics are also related and were applied to Boltzmann trees =-=[43]-=-. 21 Algorithm elim-bel Input: A belief network BN = fP 1 ; :::; Png; an ordering of the variables, d = X 1 ; :::; Xn ; evidence e. Output: The belief in X 1 = x 1 . 1. Initialize: Generate an ordered... |

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Citation Context ... semantics which allows restricting the computation to relevant portions of the belief network. TheseSuch restrictions are already available in the literature in the context ofthe existing algorithms =-=[28, 44]-=-. 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... |

13 | An evaluation of structural parameters for probabilistic reasoning: results on benchmark circuits
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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, 25]-=-. The details of these approaches are universal across tasks. 45 11 Additional related work We have mentioned throughout this paper algorithms in probabilistic and deterministic reasoning that can be ... |

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Citation Context ...r 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, 38]-=- can be incorporated on top of bucket-elimination. 13 Acknowledgment A preliminary version of this paper appeared in [14]. An extension restricted to probabilistic reasoning only appears in [17]. I wo... |

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Citation Context ...e most likely input message which was transmitted over a noisy channel, given the observed output. Researchers have investigated various approaches to finding the mpe in a belief network. (See, e.g., =-=[35, 9, 36, 37]-=-). Recent proposals include best first-search algorithms [48] and algorithms based on linear programming [41]. The problem is to find x 0 such that P (x 0 ) = max x P (x; e) = max x \Pi i P (x i ; ejx... |

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Citation Context ...nstraints and processing each pair of relevant inequalities in a bucket by linear elimination, yields a bucket elimination algorithm which coincides with well known Fourier elimination algorithm (see =-=[32]-=-). From the general principle of variable elimination and as is already known the algorithm decides the solvability of any set of linear inequalities over the rationals, and generates a problem repres... |

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Citation Context ...viewed as approximating variable elimination, they recently inspired a general approximation scheme of bucket-elimination. These were recently applied successfully to a variety of probabilistic tasks =-=[22, 29]-=-. 3 Preliminaries for probabilistic reasoning The belief-network algorithms we present next have much in common with directional resolution and adaptive-consistency. They all possess the property of c... |

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A ComputerBased Medical Diagnosis Aid that Integrates Causal and Probabilistic Knowledge
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Citation Context ...e most likely input message which was transmitted over a noisy channel, given the observed output. Researchers have investigated various approaches to finding the mpe in a belief network. (See, e.g., =-=[35, 9, 36, 37]-=-). Recent proposals include best first-search algorithms [48] and algorithms based on linear programming [41]. The problem is to find x 0 such that P (x 0 ) = max x P (x; e) = max x \Pi i P (x i ; ejx... |

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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 =-=[12, 40, 15]-=-. In the following sections we will focus in more detail on deriving bucket elimination algorithms for processing probabilistic networks. This will allow borrowing ideas and methods already developed ... |

2 |
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Citation Context ... induced width along d of the augmented moral graph. Tatman and Schachter [49] have published an algorithm for the general influence diagram, that is a variation of elim-meu and Kjaerulff's algorithm =-=[31]-=- can be viewed as a variation of elim-meu tailored to dynamic probabilistic networks. 35 A B C F D G (a) D D 2 1 u(f,g) u(d) u(b,c) Figure 24: An influence diagram bucket G : P (f jg; D 1 ), g = 1, u(... |