On the use of OBDDs in model-based diagnosis: An approach based on the partition of the model

In this paper, we discuss how Ordered Binary Decision Diagrams (OBDDs) can be exploited for the computation of consistency-based diagnoses in model-based diagnosis. Since it is not always possible to efficiently encode the whole system model within a single OBDD, we propose to build a set of OBDDs, each one encoding a portion of the original model. For each portion of the model, we compute an OBDD encoding the set of local diagnoses; the OBDD encoding global diagnoses is then obtained by merging all the local-diagnoses OBDDs. Finally, minimal-cardinality diagnoses can be efficiently computed and extracted.The paper reports formal results about soundness, completeness and computational complexity of the proposed algorithm. Thanks to the fact that encoding diagnoses is in general much simpler than encoding the whole system model, this approach allows for the successful computation of global diagnoses even if the system model could not be compiled into a single OBDD. This is exemplified referring to a challenging combinatorial digital circuit taken from the ISCAS85 benchmark.     

On the Minimization of (Complete) Ordered Binary Decision Diagrams

Ordered binary decision diagrams (OBDDs) are a popular data structure for Boolean functions. Some applications work with a restricted variant called complete OBDDs which is strongly related to nonuniform deterministic finite automata. One of its complexity measures is the width which has been investigated in several areas in computer science like machine learning, property testing, and the design and analysis of implicit graph algorithms. For a given function the size and the width of a (complete) OBDD is very sensitive to the choice of the variable ordering but the computation of an optimal variable ordering for the OBDD size is known to be NP-hard. Since optimal variable orderings with respect to the OBDD size are not necessarily optimal for the complete model or the OBDD width and hardly anything about the relation between optimal variable orderings with respect to the size and the width is known, this relationship is investigated. Here, using a new reduction idea it is shown that the size minimization problem for complete OBDDs and the width minimization problem are NP-hard.     

A reducibility concept for problems defined in terms of ordered binary decision diagrams

Reducibility concepts are fundamental in complexity theory. Usually, they are defined as follows: A problem Π is reducible to a problems Σ if Π can be computed using a program or device for Σ as a subroutine. However, this approach has its limitations if restricted computational models are considered. In the case of ordered binary decision diagrams (OBDDs), it allows the use of merely the almost unmodified original program for the subroutine. Here we propose a new reducibility for OBDDs: We say that Π is reducible to Σ if and OBDD for Π can be constructed by applying a sequence of elementary operations to an OBDD for Σ. In contrast to traditional reducibility notions, the newly introduced reduction is able to reflect the real needs of a reducibility concept in the context of OBDD-based complexity classes: it allows the reduction of problems to others which are computable with the same amount of OBDD-resources and it gives a tool to carry over lower and upper bounds. The authors are grateful to DAAD Acciones Integradas, Grant No. 322-ai-e-dr.     

The Theory of Zero-Suppressed BDDs and the Number of Knight's Tours

Zero-suppressed binary decision diagrams (ZBDDs) have been introduced by Minato [14–17] who presents applications for cube set representations, fault simulation, timing analysis and the n-queens problem. Here the structural properties of ZBDDs are worked out and a generic synthesis algorithm is presented and analyzed. It is proved that ZBDDs can be at most by a factor n + 1 smaller or larger than ordered BDDs (OBDDs) for the same function on n variables. Using ZBDDs the best known upper bound on the number of knight's tours on an 8 × 8 chessboard is improved significantly.     

Lower bounds on the OBDD size of two fundamental functions' graphs

Ordered Binary Decision Diagrams (OBDDs) are a data structure for Boolean functions which supports many useful operations. Among others it finds applications in CAD, model checking, and symbolic graph algorithms. Nevertheless, many simple functions are known to have exponential OBDD size with respect to their number of variables. In order to investigate the limits of symbolic graph algorithms which work on OBDD-represented graph instances, it is useful to have simply-structured graphs whose OBDD representation has exponential size. Therefore, we consider two fundamental functions with exponential lower bounds on their OBDD size and transfer these results to their corresponding graphs. Concretely, we consider the Indirect Storage Access function and the Hidden Weighted Bit function.     

Translation among CNFs, characteristic models and ordered binary decision diagrams

We consider translation among conjunctive normal forms (CNFs), characteristic models, and ordered binary decision diagrams (OBDDs) of Boolean functions. It is shown in this paper that Horn OBDDs can be translated into their Horn CNFs in polynomial time. As for the opposite direction, the problem can be solved in polynomial time if the ordering of variables in the resulting OBDD is specified as an input. In case that such ordering is not specified and the resulting OBDD must be of minimum size, its decision version becomes NP-complete. Similar results are also obtained for the translation in both directions between characteristic models and OBDDs. We emphasize here that the above results hold on any class of functions having a basis of polynomial size.     

基于OBDD的编码算法

论文提出一个新的无损图像压缩算法,主要是通过有序二叉决策图(OBDD)的方法,寻找图像中重复的模式来减少其存储空间的一种变换编码,因而成为表示图像的另一种数据结构。我们通过该算法来寻找OBDD以精确地表示图像,并给出了其OBDD的有效编码,所获得的结果表明,所提出的算法及其编码是实现无损图像压缩的一种有效的方法。     

Attributed point matching for automatic groundtruth generation

Geometric groundtruth at the character, word, and line levels is crucial for designing and evaluating optical character recognition (OCR) algorithms. Kanungo and Haralick proposed a closed-loop methodology for generating geometric groundtruth for rescanned document images. The procedure assumed that the original image and the corresponding groundtruth were available. It automatically registered the original image to the rescanned one using four corner points and then transformed the original groundtruth using the estimated registration transformation. In this paper, we present an attributed branch-and-bound algorithm for establishing the point correspondence that uses all the data points. We group the original feature points into blobs and use corners of blobs for matching. The Euclidean distance between character centroids is used as the error metric. We conducted experiments on synthetic point sets with varying layout complexity to characterize the performance of two matching algorithms. We also report results on experiments conducted using the University of Washington dataset. Finally, we show examples of application of this methodology for generating groundtruth for microfilmed and FAXed versions of the University of Washington dataset documents. Received: July 24, 2001 / Accepted: May 20, 2002     

OBDDs of a Monotone Function and Its Prime Implicants

Coudert made a breakthrough in the two-level logic minimization problem with Ordered Binary Decision Diagrams (OBDDs for short) recently [3]. This paper discusses the relationship between the two OBDDs of a monotone function and its prime implicant set to clarify the complexity of this practically efficient method. We show that there exists a monotone function which has an O(n) size sum-of-products but cannot be represented by a polynomial size OBDD. In other words, we cannot obtain the OBDD of the prime implicant set of a monotone function in an output-size sensitive manner once we have constructed the OBDD of that function as in [3], in the worst case. A positive result is also given for a meaningful class of matroid functions. Received April 1997, and in revised form December 1997, and in final form February 1998.     

Complexity Theoretical Results on Partitioned (Nondeterministic) Binary Decision Diagrams

Ordered binary decision diagrams (OBDDs) have found many applications in the verification of combinational and sequential circuits, protocols, and the synthesis and analysis of systems. The applications are limited, since the expressive power of polynomial-size OBDDs is too restricted. Therefore, several more general BDD models are used. Partitioned OBDDs are OBDD models allowing restricted use of nondeterminism and different variable orderings. They are restricted enough such that the essential operations can be performed efficiently and they allow polynomial-size representations for many more functions than OBDDs. Here the expressive power of polynomial-size partitioned OBDDs is investigated. A tight hierarchy with respect to the number of parts in the partition is proved and partitioned OBDDs are compared with other BDD models. Received April 1998, and in final form September 1998.     

一种基于熵的OBDD变量排序算法

有序二叉决策图（OBDD）是一种有效表示布尔函数的数据结构，其大小依赖于所采用的变量序。熵是定量描述布尔函数中变量重要性的一种方法。基于变量的熵值分析了高质量变量序的特征，给出了一种基于熵的OBDD变量排序算法。实验结果表明：该算法与模拟退火算法和遗传算法结果相当。时间仅为相应算法的80．84％和29．79％。     

Intelligent multimedia data: data + indices + inference

In this paper, we present a novel approach for multimedia data indexing and retrieval that is machine independent and highly flexible for sharing multimedia data across applications. Traditional multimedia data indexing and retrieval problems have been attacked using the central data server as the main focus, and most of the indexing and query-processing for retrieval are highly application dependent. This precludes the use of created indices and query processing mechanisms for multimedia data which, in general, have a wide variety of uses across applications. The approach proposed in this paper addresses three issues: 1. multimedia data indexing; 2. inference or query processing; and 3. combining indices and inference or query mechanism with the data to facilitate machine independence in retrieval and query processing. We emphasize the third issue, as typically multimedia data are huge in size and requires intra-data indexing. We describe how the proposed approach addresses various problems faced by the application developers in indexing and retrieval of multimedia data. Finally, we present two applications developed based on the proposed approach: video indexing; and video content authorization for presentation.     

Approximate query processing using wavelets

Approximate query processing has emerged as a cost-effective approach for dealing with the huge data volumes and stringent response-time requirements of today's decision support systems (DSS). Most work in this area, however, has so far been limited in its query processing scope, typically focusing on specific forms of aggregate queries. Furthermore, conventional approaches based on sampling or histograms appear to be inherently limited when it comes to approximating the results of complex queries over high-dimensional DSS data sets. In this paper, we propose the use of multi-dimensional wavelets as an effective tool for general-purpose approximate query processing in modern, high-dimensional applications. Our approach is based on building wavelet-coefficient synopses of the data and using these synopses to provide approximate answers to queries. We develop novel query processing algorithms that operate directly on the wavelet-coefficient synopses of relational tables, allowing us to process arbitrarily complex queries entirely in the wavelet-coefficient domain. This guarantees extremely fast response times since our approximate query execution engine can do the bulk of its processing over compact sets of wavelet coefficients, essentially postponing the expansion into relational tuples until the end-result of the query. We also propose a novel wavelet decomposition algorithm that can build these synopses in an I/O-efficient manner. Finally, we conduct an extensive experimental study with synthetic as well as real-life data sets to determine the effectiveness of our wavelet-based approach compared to sampling and histograms. Our results demonstrate that our techniques: (1) provide approximate answers of better quality than either sampling or histograms; (2) offer query execution-time speedups of more than two orders of magnitude; and (3) guarantee extremely fast synopsis construction times that scale linearly with the size of the data. Received: 7 August 2000 / Accepted: 1 April 2001 Published online: 7 June 2001     

Synchronization support and group-membership services for reliable distributed multimedia applications

We describe a collection of algorithms designed to support reliable synchronization and group membership services for distributed multimedia applications. In particular, we consider those applications that require interactivity, isochronous rendering of multimedia data, and high reliability. We show that the algorithms we propose (i) provide reliable support for the synchronization of multimedia data streams, despite the occurrence of possible communication failures, (ii) maintain a consistent view of the relative group membership of all the nonfaulty application components, (iii) guarantee time-bounded delay of component failure detection and join, and (iv) meet effectively possible scalability requirements of the applications.     

The symbolic OBDD scheme for generating mechanical assembly sequences

Assembly sequence planning is one of typical combinatorial optimization problems, where the size of parts involved is a significant and often prohibitive difficulty. The compact storage and efficient evaluation of feasible assembly sequences is one crucial concern. Ordered binary decision diagram (OBDD) is a canonical form to represent and manipulate the Boolean functions efficiently, and appears to give improved results for large-scale combinatorial optimization problems. In this paper, assembly knowledge models of liaison graph and translation function are formulated by OBDDs, and OBDD-based representation of assembly sequences is proposed. A novel OBDD-based procedure was presented to generate all geometrically feasible assembly sequences from the OBDDs of liaison graph and translation relation. This procedure can be used conveniently on the computer and all the feasible sequences can be derived. The great advantage of OBDD-based scheme is that the storage space of OBDD-based representation of feasible assembly sequences does not increase with the part count of assembly dramatically so quickly as that of AND/OR graph does. We developed the prototype tool for generating assembly sequence using Visual C++ and CUDD package, and undertake some experimental tests. It was shown that the OBDD scheme generated feasible assembly sequences correctly and completely.     

LADAR target detection using morphological shared-weight neural networks

Morphological shared-weight neural networks (MSNN) combine the feature extraction capability of mathematical morphology with the function-mapping capability of neural networks in a single trainable architecture. The MSNN method has been previously demonstrated using a variety of imaging sensors, including TV, forward-looking infrared (FLIR) and synthetic aperture radar (SAR). In this paper, we provide experimental results with laser radar (LADAR). We present three sets of experiments. In the first set of experiments, we use the MSNN to detect different types of targets simultaneously. In the second set, we use the MSNN to detect only a particular type of target. In the third set, we test a novel scenario, referred to as the Sims scenario: we train the MSNN to recognize a particular type of target using very few examples. A detection rate of 86% with a reasonable number of false alarms was achieved in the first set of experiments and a detection rate of close to 100% with very few false alarms was achieved in the second and third sets of experiments. In all the experiments, a novel pre-processing method is used to create a pseudo-intensity images from the original LADAR range images.     

Video OCR: indexing digital news libraries by recognition of superimposed captions

The automatic extraction and recognition of news captions and annotations can be of great help locating topics of interest in digital news video libraries. To achieve this goal, we present a technique, called Video OCR (Optical Character Reader), which detects, extracts, and reads text areas in digital video data. In this paper, we address problems, describe the method by which Video OCR operates, and suggest applications for its use in digital news archives. To solve two problems of character recognition for videos, low-resolution characters and extremely complex backgrounds, we apply an interpolation filter, multi-frame integration and character extraction filters. Character segmentation is performed by a recognition-based segmentation method, and intermediate character recognition results are used to improve the segmentation. We also include a method for locating text areas using text-like properties and the use of a language-based postprocessing technique to increase word recognition rates. The overall recognition results are satisfactory for use in news indexing. Performing Video OCR on news video and combining its results with other video understanding techniques will improve the overall understanding of the news video content.     

Minimal area for surface reconstruction from cross sections

Surface reconstruction from cross-sectional data is important in a variety of applications. It is usually possible to generate a surface in many ways, but only reasonable ones are acceptable. A surface of minimal area has been considered as one of the most natural optimal criteria for the original tiling method of surface reconstruction from cross sections. In the paper, we consider minimal surfaces for continuous generalization of the tiling approach and in the general situation of reconstruction from cross sections. We show that in these cases the minimal area criterion leads to defective surfaces and is thus unacceptable. Published online: 23 July 2002 Correspondence to: D. Berzin     

Larger lower bounds on the OBDD complexity of integer multiplication

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Recently, the question whether the OBDD complexity of the most significant bit of integer multiplication is exponential has been answered affirmatively. In this paper a larger general lower bound is presented using a simpler proof. Furthermore, we prove a larger lower bound for the variable order assumed to be one of the best ones for the most significant bit. Moreover, the best known lower bound on the OBDD complexity for the so-called graph of integer multiplication is improved.     

On the size of (generalized) OBDDs for threshold functions

Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Among the many areas of application are hardware verification, model checking, and symbolic graph algorithms. Threshold functions are the basic functions for discrete neural networks and are used as building blocks in the design of some symbolic graph algorithms. In this paper the first exponential lower bound on the size of a more general model than OBDDs and the first nontrivial asymptotically optimal bound on the OBDD size for a threshold function are presented.