Exact algorithm

Abstract: Ready availability has prompted the use of computed tomography (CT) data in various applications in radiation therapy. For example, some radiation treatment planning systems now utilize CT data in heterogeneous dose calculations algorithms. In radiotherapy imaging applications, CT data are projected onto specified planes, thus producing "radiographs," which are compared with simulator radiographs to assist in proper patient positioning and delineation of target volumes. All these applications share the common geometric problem of evaluating the radiological path through the CT array. Due to the complexity of the three-dimensional geometry and the enormous amount of CT data, the exact evaluation of the radiological path has proven to be a time consuming and difficult problem. This paper identifies the inefficient aspect of the traditional exact evaluation of the radiological path as that of treating the CT data as individual voxels. Rather than individual voxels, a new exact algorithm is presented that considers the CT data as consisting of the intersection volumes of three orthogonal sets of equally spaced, parallel planes. For a three-dimensional CT array of N3 voxels, the new exact algorithm scales with 3N, the number of planes, rather than N3, the number of voxels. Coded in FORTRAN-77 on a VAX 11/780 with a floating point option, the algorithm requires approximately 5 ms to calculate an average radiological path in a 100(3) voxel array.

Abstract: Hidden Markov models (HMMs) have proven to be one of the most widely used tools for learning probabilistic models of time series data. In an HMM, information about the past is conveyed through a single discrete variable—the hidden state. We discuss a generalization of HMMs in which this state is factored into multiple state variables and is therefore represented in a distributed manner. We describe an exact algorithm for inferring the posterior probabilities of the hidden state variables given the observations, and relate it to the forward–backward algorithm for HMMs and to algorithms for more general graphical models. Due to the combinatorial nature of the hidden state representation, this exact algorithm is intractable. As in other intractable systems, approximate inference can be carried out using Gibbs sampling or variational methods. Within the variational framework, we present a structured approximation in which the the state variables are decoupled, yielding a tractable algorithm for learning the parameters of the model. Empirical comparisons suggest that these approximations are efficient and provide accurate alternatives to the exact methods. Finally, we use the structured approximation to model Bach‘s chorales and show that factorial HMMs can capture statistical structure in this data set which an unconstrained HMM cannot.

Abstract: The problem addressed in this paper is that of orthogonally packing a given set of rectangular-shaped items into the minimum number of three-dimensional rectangular bins. The problem is strongly NP-hard and extremely difficult to solve in practice. Lower bounds are discussed, and it is proved that the asymptotic worst-case performance ratio of the continuous lower bound is ?. An exact algorithm for filling a single bin is developed, leading to the definition of an exact branch-and-bound algorithm for the three-dimensional bin packing problem, which also incorporates original approximation algorithms. Extensive computational results, involving instances with up to 90 items, are presented: It is shown that many instances can be solved to optimality within a reasonable time limit.