U. Bertel e and F. Brioschi. Nonserial dynamic programming. Academic Press, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= 3, can be eciently determined. In trees, the width is equal to the induced width ( 1) hence any minimal width ordering, is also an optimal 14 induced width ordering, and it can be found in linear time. A linear time algorithm recognizing problems having w 2 is also available [Arnborg 85, Bertele Brioschi 72] The algorithm selects as last a node having a smallest degree, eliminates it, connects its neighbors in the residual graph (if they were not previously connected) and continues recursively. If the result is an ordering having w 2 it can be concluded that the graph, too, has w 2. ....

.... in the cycle cutset method) the depth of a DFS tree m (in backjumping) and the size of largest non separable component r (appearing in the tree component scheme) It is clear that for any problem structure, the relationships m w ; r w holds, and it can also be shown that w c 1 [Bertele Brioschi 72] m and r are not comparable, sometimes m r (e.g. trees) and sometimes r m (e.g. mesh) Another parameter mentioned in the literature, bandwidth [Zabih 90] is also dominated by w . It can be concluded, therefore, that w provides the most informative graph parameter, and it can be ....

Bertele, U., and Brioschi, F., Nonserial dynamic programming, Academic Press, New York, 1972.

....algorithm. 2. 2 Decomposable Triangulated Graphs and General Graphical Models For general graphical models, the labeling problem is the most probable configuration problem on the graph and can be solved through max propagation on junction trees [18] 21] 28] The dynamic programming algorithm [2] and the max propagation algorithm essentially have the same order of complexity which is determined by the maximum clique size of the graph. Themaximum clique size for a decomposable triangulated graph is three. Since any graph with maximum clique size equal to or less than three can be ....

U. Bertele and F. Brioschi, Nonserial Dynamic Programming. Academic Press, 1971.

....search algorithms need to explore the whole search tree. Nevertheless, in practice they typically do much better. Dynamic programming algorithms solve a problem by a sequence of transformations that reduce the problem size, while preserving the value of the best cost attainable in the problem [2]. Bucket Elimination (BE) 8] is a complete algorithm that relies on the basic step of variable elimination. The algorithm proceeds by selecting one variable at a time and replacing it by a new constraint which summarizes the e#ect of the chosen variable. Once all variables have been eliminated, ....

....of depth first BB is linear. The time complexity is bounded by the product of the search space size d and the complexity per node L, O(d L) 2.3. Bucket Elimination Bucket Elimination (BE ) 8] is an algorithm for COP solving which falls into the category of dynamic programming methods [2]. It is based upon the following two operators over functions: The sum of two functions f and g denoted (f g) is a new function with scope var(f) var(g) which returns for each tuple the sum of costs of f and g, f g) X) f(X) g(X) The elimination of variable x i from f , denoted elim ....

Bertele, U. and F. Brioschi: 1972, Nonserial Dynamic Programming. Academic Press.

....is such that when eliminating the i th vertex, it is in exactly one triangle of the current triangulated polygon. The matching algorithm works by sequentially eliminating the vertices of T , using the nice elimination order. This is an instance of a well known dynamic programming technique (see [3]) After eliminating v 1 , v i 1 , vertex v i is in exactly one triangle, say with nodes v j and v k . The two nodes v j and v k are the parents of v i , which we indicate by letting p[i] a = j and p[i] b = k. We compute the cost of the best placement for v i as a function of the locations ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, 1972.

....j (Z j ) which is a linear combination of m functions f j , each one with domain restricted to Z j . Our goal is to find max x F (x) i.e. to find the state x over which F is maximized. As observed by Koller and Parr [2000] we can maximize such a function F using nonserial dynamic programming [Bertele and Brioschi, 1972] or cost networks [Dechter, 1999] See [Guestrin et al. 2001] for a description of the algorithm. The second key computational step is a projection of a vector into the linear subspace induced by a set of basis functions. The form of the projection depends on our choice of norm. More formally: ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, New York, 1972.

....had to deal with was f 3 , having 3 variables and 12 entries. This is much better than directly using Equation 2.2 which led to a factor with 5 variables and 48 entries. In larger networks the savings are usually even more significant. We are now ready to state the variable elimination algorithm [BB72, ZP94] We first convert all our CPDs into a set of factors F . As long as F contains variables which do not appear in the query we choose such variable Y and eliminate it. The elimination is done by extracting all the factors to which Y belongs, multiplying them together, summing out Y and ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, New York, 1972.

....their actions to optimize some particular Q i . Fig. 1(a) shows the coordination graph for an example where Q = Q 1 (a 1 ; a 2 ) Q 2 (a 2 ; a 4 ) Q 3 (a 1 ; a 3 ) Q 4 (a 3 ; a 4 ) A graph structure suggests the use of a cost network [5] which can be solved using non serial dynamic programming [1] or a variable elimination algorithm which is virtually identical to variable elimination in a Bayesian network. The key idea is that, rather than summing all functions and then doing the maximization, we maximize over variables one at a time. Specifically, when maximizing over a l , only ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, 1972.

....previously, such a model can be completely specified in terms of the distribution p(x(0) at the root node and the parent child transition distributions p(x(s) x(s#) for every node s #=0. Such models have a long history, extending back to studies in statistical physics [26] dynamic programming [32], artificial intelligence and other investigations of graphical models [267, 294, 295, 89, 128, 7, 169] and signal and image processing [42, 80, 261, 199, 58, 59, 283, 175, 281, 213] Later in this paper we will illustrate examples of such models for two di#erent purposes. One is a class of image ....

.... structure and spirit, something that has been emphasized in several investigations [294, 295, 7, 169] Computing the MAP estimate involves a generalization of the well known Viterbi algorithm [118] one that can be traced at least back to the study of so called nonserial dynamic programming [32] and to the work of others in artificial intelligence and graphical models [267, 294, 295, 89, 7, 169] A description of the algorithm that mirrors very closely the two pass structure of the estimation algorithms we have described so far (and that also makes clear how this algorithm generalizes ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, New York, 1972.

....based on ordinary multiplication primarily because we are dealing with probabilities. By defining Omega differently (e.g. based on addition) our relational data model can be easily extended to solve a number of apparently different but closely related problems such as dynamic programming [2], solving sparse linear equations [11] and constraint propagation [3] ....

U. Bertel`e and F. Brioschi, Nonserial Dynamic Programming. Academic Press, 1972.

.... problem solving and reasoning activities, including directional resolution for propositional satisfiability [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 probabilistic databases such as belief networks and influence diagrams. The main virtues of the ....

....1. Indeed as is demonstrated 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 w (and the minimizing ordering) is NP complete [2] However greedy heuristic ordering algorithms [5, 25] and approximation 7 A B D 1,2 1,2,3 1,2 1,2 1,2 Figure 4: A modified graph coloring problem E D A D C B A C B W (D) 2 W (d) 3 W (d) 2 Figure 5: The induced width along the orderings: d 1 = A; B; C; D;E and d 2 = E; B; C; D;A 8 B C D E F G bucket(G) ....

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U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, 1972.

....a problem in a nonserial formula tion into a serial one. 1. Introduction Dynamic programming (DP) is a powerful optimization methodology that is widely applicable to a large number of areas including optimal control, industrial engineering, economics, and artificial intelligence [3] [5], 9] 23] 29] Many practical problems involving a sequence of interrelated decisions can be solved efficiently by DP. Bellman has characterized DP through the Principle of Optimality, which states that an optimal sequence of decisions has the property that whatever the initial state and ....

.... monotone cost function [10] DP can also be formulated as a special case of the branch and bound algorithm, which is a general top down OR tree search procedure with dominance tests [22] 13] 18] Lastly, nonserial DP has been shown to be optimal among all nonoverlapping comparison algorithms [5], 25] Although DP has long been recognized as a powerful approach to solving a wide spectrum of optimization problems, its applicability has been somewhat limited due to the large computational requirements. Recent advances in very large scale integration (VLSI) and multiprocessor technologies ....

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U. Bertele and F. Brioschi, Nonserial Dynamic Programming, Academic Press, New York, 1972.

....k(n) log log n, then we can get an O(n log n) algorithm with 0(1 x log log n ) relative error. 2. Nonserial dynamic programming. We can define many NP complete problems, such as the maximum independent set problem, the graph coloring problem, and so on, as nonserial dynamic programming. [2] An additional concept need to descripted, the restriction of an objective functionf tn = k= fk to a set of variables xi, x2 is the objective functionj = fk Ifk depends only upon x, x2 . We will use algorithm to solve the problem: maximizef subject to the constraints on the ....

U. BERTELE AND F. BRIOSCHI, Nonserial Dynamic Programming, Academic Press, New York, 1972

....their actions to optimize some particular Q j . Fig. 1 shows the coordination graph for an example where Q = Q 1 (a 1 , a 2 ) Q 2 (a 2 , a 4 ) Q 3 (a 1 , a 3 ) Q 4 (a 3 , a 4 ) A graph structure suggests the use of a cost network [6] which can be solved using non serial dynamic programming [3] or a variable elimination algorithm which is virtually identical to variable elimination in a Bayesian network. We review this construction here, as it is a key component. The idea is that, rather than summing all functions and then maximizing, we maximize over variables one at a time. When ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, New York, 1972.

....actions to optimize some particular Q i . Fig. 1 shows the coordination graph for an example where V = Q 1 (a 1 , a 2 ) Q 2 (a 2 , a 4 ) Q 3 (a 1 , a 3 ) Q 4 (a 3 , a 4 ) A graph structure suggests the use of a cost network [6] which can be solved using non serial dynamic programming [3] or a variable elimination algorithm which is virtually identical to variable elimination in a Bayesian network. We review this construction here, as it is a key component in the rest the paper. The key idea is that, rather than summing all functions and then doing the maximization, we maximize ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, New York, 1972.

....action choices A. Our task now is to select a joint action a that maximizes P j Q j (a) The structure of the coordination graph will allow us to design an efficient coordination strategy for the agents. Maximization in a graph structure suggests the use of non serial dynamic programming [1] , or variable elimination. To exploit structure in rules, we use a variable elimination algorithm similar to variable elimination in a Bayesian network with context specific independence [13] Intuitively, the algorithm operates by having an individual agent collect value rules relevant to ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, New York, 1972.

....actions to optimize some particular Q i . Fig. 1 shows the coordination graph for an example where Q = Q 1 (a 1 ; a 2 ) Q 2 (a 2 ; a 4 ) Q 3 (a 1 ; a 3 ) Q 4 (a 3 ; a 4 ) A graph structure suggests the use of a cost network [6] which can be solved using non serial dynamic programming [3] or a variable elimination algorithm which is virtually identical to variable elimination in a Bayesian network. We review this construction here, as it is a key component in the rest the paper. The key idea is that, rather than summing all functions and then doing the maximization, we maximize ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, New York, 1972.

....G is associated with a deterministic function: G = DF . The rest of the CPTs are positive. The moral graph is given in Figure 1b. Bucket elimination. Bucket elimination is a unifying algorithmic framework for variable elimination algorithms applicable to probabilistic and deterministic reasoning [Bertele and Brioschi, 1972, N. L. Zhang and Poole, 1994, Dechter, 1996] The input to a bucket elimination algorithm is a set of functions or relations. Given a variable ordering, the algorithm partitions the functions (e.g. CPTs) into buckets, where a function is placed in the bucket of its latest argument in the ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, 1972.

....their actions to optimize some particular Q i . Fig. 1(a) shows the coordination graph for an example where V = Q 1 (a 1 ; a 2 ) Q 2 (a 2 ; a 4 ) Q 3 (a 1 ; a 3 ) Q 4 (a 3 ; a 4 ) A graph structure suggests the use of a cost network [6] which can be solved using non serial dynamic programming [2] or a variable elimination algorithm which is virtually identical to variable elimination in a Bayesian network. The key idea is that, rather than summing all functions and then doing the maximization, we maximize over variables one at a time. Specifically, when maximizing over a l , only ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, 1972.

....that BTE can be viewed as an instance of treedecomposition algorithms appearing in a wide range of automated reasoning tasks. Bucket elimination (BE) is a unifying algorithmic framework for dynamic programming algorithms applicable to a wide variety of probabilistic and deterministic reasoning [Bertele and Brioschi, 1972, Dechter, 1999] such as belief updating, finding the most probable explanation (MPE) finding the maximum aposteriori hypothesis (MAP) and finding a collection of decisions that maximize the expected utility in influence diagrams. As well, the scheme is applicable to constraint satisfaction and ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, 1972.

....elimination. Bucket elimination is a unifying algorithmic framework for variable elim D G A B C F A B C F G D Figure 1: Belief network P (g; f; d; c; b; a) P (gjf; d)P (f jc; b)P (djb; a)P (bja)P (cja)P (a) ination algorithms applicable to probabilistic and deterministic reasoning [Bertele and Brioschi, 1972, N. L. Zhang and Poole, 1994, Dechter, 1996] The input to a bucket elimination algorithm is a set of functions or relations. Given a variable ordering, the algorithm partitions the functions (e.g. CPTs) into buckets, where a function is placed in the bucket of its latest argument in the ....

U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, 1972.

.... However all these task are NP complete and to solve them effectively we translate the original Bayesian network into a secondary structure, a join tree called also a Markov tree (consult [4] or [12] Within such a structure all the queries can be solved by using the idea of dynamic programming [1]. There are three main architectures implementing such a recipe: the Lauritzen Spiegelhalter architecture [6] the Shenoy Shafer architecture [9] and the HUGIN architecture [4] In all these architectures the nodes of a Markov tree act as independent processors communicating with their ....

Bertele, U., and Brioschi, F. Nonserial Dynamic Programming, Academic Press, 1972,

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U. Bertel e and F. Brioschi. Nonserial dynamic programming. Academic Press, 1972.

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U.BerteleandF.Brioschi.Nonserial Dynamic Programming. Academic Press, 1972.

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U. Bertele and F. Brioschi. Nonserial Dynamic Programming. Academic Press, New York, 1972.

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U. Bertel`e and F. Brioschi (1972), Nonserial dynamic programming, Mathematics in Science and Engineering, Vol. 91, Academic Press.

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