Zero-suppressed Binary Decision Diagrams Research Articles

Finding well-optimized special quasirandom structures with decision diagram

The advanced data structure of the zero-suppressed binary decision diagram (ZDD) enables us to efficiently enumerate nonequivalent substitutional structures. Not only can the ZDD store a vast number of structures in a compressed manner, but also a set of structures satisfying given constraints can be extracted from the ZDD efficiently. Here, we present a ZDD-based efficient algorithm for exhaustively searching for special quasirandom structures (SQSs) that mimic the perfectly random substitutional structure. We demonstrate that the current approach can extract only a tiny number of SQSs from a ZDD composed of many substitutional structures ($>{10}^{12}$). As a result, we find SQSs that are optimized better than those proposed in the literature. A series of SQSs should be helpful for estimating the properties of substitutional solid solutions. Furthermore, the present ZDD-based algorithm should be useful for applying the ZDD to the other structure enumeration problems.

Storing Set Families More Compactly with Top ZDDs

Zero-suppressed Binary Decision Diagrams (ZDDs) are data structures for representing set families in a compressed form. With ZDDs, many valuable operations on set families can be done in time polynomial in ZDD size. In some cases, however, the size of ZDDs for representing large set families becomes too huge to store them in the main memory. This paper proposes top ZDD, a novel representation of ZDDs which uses less space than existing ones. The top ZDD is an extension of the top tree, which compresses trees, to compress directed acyclic graphs by sharing identical subgraphs. We prove that navigational operations on ZDDs can be done in time poly-logarithmic in ZDD size, and show that there exist set families for which the size of the top ZDD is exponentially smaller than that of the ZDD. We also show experimentally that our top ZDDs have smaller sizes than ZDDs for real data.

Efficient Formal Verification of Galois-Field Arithmetic Circuits Using ZDD Representation of Boolean Polynomials

In this study, we present a new formal method for verifying the functionality of Galois-field (GF) arithmetic circuits. Assuming that the input–output relation (i.e., the specification of a GF arithmetic circuit) can be represented as polynomials over 2, the proposed method formally checks the equivalence between GF polynomials derived from a netlist and the specification. To efficiently verify the equivalence, we employ a zero-suppressed binary decision diagram (ZDD) to represent polynomials over 2. Even though polynomial reduction is the most time-consuming process of verification (i.e., equivalence checking), our new algorithm can efficiently reduce the GF polynomials in the form of a zero-suppressed binary decision diagram derived from the target netlist. The proposed algorithm derives the polynomials representing all intermediate nodes (i.e., the outputs of all gates) in the order from primary inputs to those primary outputs that are in accordance with the reverse topological traversal order. We demonstrated the efficiency and effectiveness of the proposed method via a set of experimental verifications. In particular, we confirmed that the proposed method can verify practical GF multipliers (including those used in standardized elliptic curve cryptography) approximately 30 times faster on average and at most 170 times faster than the best conventional method.

Applying Zero-suppressed Binary Decision Diagram to Triadic Concept Analysis

Applying Zero-suppressed Binary Decision Diagram to Triadic Concept Analysis

Enumeration of nonequivalent substitutional structures using advanced data structure of binary decision diagram.

A derivative structure is a nonequivalent substitutional atomic configuration derived from a given primitive cell. The enumeration of derivative structures plays an essential role in searching for the ground states in multicomponent systems. However, it is computationally difficult to enumerate derivative structures if the number of derivative structures of a target system becomes huge. In this study, we introduce a novel compact data structure of the zero-suppressed binary decision diagram (ZDD) for enumerating derivative structures much more efficiently. We show its simple applications to the enumeration of structures derived from the face-centered cubic and hexagonal close-packed lattices in binary, ternary, and quaternary systems. The present ZDD-based procedure should contribute to computational approaches based on derivative structures in physics and materials science.

Dynamic game-based approach for optimizing merging vehicle trajectories using time-expanded decision diagram

Connected and automated technologies for vehicles pave the way for major changes in traffic control methodology. A decentralized control system based on a communication network using vehicle-to-vehicle communications and each vehicle’s on-board controller will be one of the predominant approaches to vehicle trajectory optimization when the deployment of connected and automated vehicles (CAVs) has advanced, because of its scalability and fault resistance compared to the centralized system. In this study, we propose a dynamic game-based vehicle trajectory optimization approach that can be utilized in a decentralized control system based on CAV technologies for merging segments, which is one of the bottlenecks of highways. Since the decentralized system is constructed by equal individual vehicles that decide their own optimal control in conflicting situations, we introduce a game-theory-based approach for obtaining a satisfactory result for those vehicles. Further, for achieving the optimal result at the end of the target segment, we employ a dynamic game to design the whole interrelated time-series trajectories of competing pairs of vehicles. To tackle the curse of dimensionality, we use an efficient method to enumerate the sets of combinations – a zero-suppressed binary decision diagram (ZDD) – and propose an algorithm that uses this ZDD to solve the dynamic game. The performance of the proposed approach is demonstrated via numerical experiments, and is compared to a static game-based model and an individual dynamic decision model. The results prove the effectiveness of the proposed approach—it induces efficiently adjusted behaviors of competing vehicles and enables merging success, especially in a situation where the initial gaps between vehicles are not enough for immediate lane change, which is achieved by the combined effect of the game-theoretic approach and the dynamic decision approach.

Chain Reduction for Binary and Zero-Suppressed Decision Diagrams

Chain reduction enables reduced ordered binary decision diagrams (BDDs) and zero-suppressed binary decision diagrams (ZDDs) to each take advantage of the others' ability to symbolically represent Boolean functions in compact form. For any Boolean function, its chain-reduced ZDD (CZDD) representation will be no larger than its ZDD representation, and at most twice the size of its BDD representation. The chain-reduced BDD (CBDD) of a function will be no larger than its BDD representation, and at most three times the size of its CZDD representation. Extensions to the standard algorithms for operating on BDDs and ZDDs enable them to operate on the chain-reduced versions. Experimental evaluations on representative benchmarks for encoding word lists, solving combinatorial problems, and operating on digital circuits indicate that chain reduction can provide significant benefits in terms of both memory and execution time.

Practical Frank–Wolfe Method with Decision Diagrams for Computing Wardrop Equilibrium of Combinatorial Congestion Games

Computation of equilibria for congestion games has been an important research subject. In many realistic scenarios, each strategy of congestion games is given by a combination of elements that satisfies certain constraints; such games are called combinatorial congestion games. For example, given a road network with some toll roads, each strategy of routing games is a path (a combination of edges) whose total toll satisfies a certain budget constraint. Generally, given a ground set of n elements, the set of all such strategies, called the strategy set, can be large exponentially in n, and it often has complicated structures; these issues make equilibrium computation very hard. In this paper, we propose a practical algorithm for such hard equilibrium computation problems. We use data structures, called zero-suppressed binary decision diagrams (ZDDs), to compactly represent strategy sets, and we develop a Frank–Wolfe-style iterative equilibrium computation algorithm whose per-iteration complexity is linear in the size of the ZDD representation. We prove that an ϵ-approximate Wardrop equilibrium can be computed in O(poly(n)/ϵ) iterations, and we improve the result to O(poly(n) log ϵ−1) for some special cases. Experiments confirm the practical utility of our method.

Analysis of two-terminal network reliability based on efficient data structure

The present work is depend upon a different data structure to perform the network reliability estimation efficiently. Zero suppressed binary decision diagram (ZBDD) is an well ordered and effective method of representing not only the boolean functions but also the sets of combinations than the conventional binary decision diagrams (BDD). This paper proposes new algorithm to manipulate the ZBDDs the variant of BDD on some benchmark networks. The 2-terminal network reliability problem has been studied extensively and effective results have been obtained.

Bicluster Method to Predict Gene Patterns to Classify Differential Gene Expressions in Non-Small Cell Lung Cancer

In recent years, there are numerous efforts to overcome the constraints of data mining approaches to classify "BIG DATA". There are several types of data which has identical and similar expressions but there are dependent classification algorithms to predict these classes of expressions. Totally Different algorithms have been developed and enforced to research and differentiate the categories of data groups based on their functions. Zero-suppressed Binary Decision Diagram (ZBDD) algorithms help to classify the data with several categories. In the present study, lung cancer gene expression datasets 25 samples contain 10 mouth buccal cavity epithelial tissue samples and 15 nasal epithelial tissue samples from never smokers and current smokers were used to classify the genes and their expressions with various conditions. Using R and BioConductor software to normalize and predict differential expressed genes by Affy, Affycore tools and Limma packages to predict the gene expression with various functional properties. ZBDD algorithm and parallel coordination helps to predict the functional genes and the results shows 345 nasal epithelial genes were predict of which 54 genes were present in bicluster and 35 genes from mouth epithelial tissues show 14 were present in ZBDD bicluster. The results conclude that ZBDD algorithm has great advantage to classify big data and this algorithm can introduce in any large datasets for the accurate predict of large datasets.

Group Decision Diagram (GDD): A Compact Representation for Permutations

Permutation is a fundamental combinatorial object appeared in various areas in mathematics, computer science, and artificial intelligence. In some applications, a subset of a permutation group must be maintained efficiently. In this study, we develop a new data structure, called group decision diagram (GDD), to maintain a set of permutations. This data structure combines the zero-suppressed binary decision diagram with the computable subgroup chain of the permutation group. The data structure enables efficient operations, such as membership testing, set operations (e.g., union, intersection, and difference), and Cartesian product. Our experiments demonstrate that the data structure is efficient (i.e., 20–300 times faster) than the existing methods when the permutation group is considerably smaller than the symmetric group, or only subsets constructed by a few operations over generators are maintained.

Boolean Gröbner Basis Reductions on Finite Field Datapath Circuits Using the Unate Cube Set Algebra

Recent developments in formal verification of arithmetic datapaths make efficient use of symbolic computer algebra algorithms. The circuit is modeled as an ideal in polynomial rings, and Grobner basis (GB) reductions are performed over these polynomials to derive a canonical representation. As they model logic gates of the circuit, the ideals comprise largely of Boolean (or pseudo-Boolean) polynomials. This paper considers a logic synthesis analogue of GB reductions over Boolean polynomials, by interpreting symbolic algebra as the unate cube set algebra over characteristic sets. By representing Boolean polynomials as characteristic sets using zero-suppressed binary decision diagrams (ZDDs), implicit algorithms are efficiently designed for GB-reduction for datapath circuits. We show that the imposition of circuit-topology-based monomial orders exposes a special structure on the ZDD representation of the polynomials. The subexpressions employed in the GB-reduction are readily visible as subgraphs on the ZDDs, which are directly used to compose the result. Our division algorithms effectively cancel multiple monomials implicitly in one-step, simplify the search for divisors, and avoid intermediate size explosion. Experiments performed over various finite field arithmetic architectures demonstrate the efficiency of our algorithms and implementations; our approach is orders of magnitude faster as compared to conventional methods.

Evaluation of hotspot cluster detection using spatial scan statistic based on exact counting

In this paper, we propose a novel approach to the detection of spatial clusters based on linkage information of a map dataset. Spatial scan statistic has been widely used for detecting a hotspot cluster (or a coldspot cluster) in various fields, such as astronomy, biosurveillance, natural disasters, and forestry. This approach is based on the idea of finding a connected regional subset that maximizes likelihood in the whole study area. To detect a hotspot cluster, which aggregates high-risk regions so as to be maximum likelihood, we only just search such a cluster from all patterns of connected regional subsets. However, except when there are extremely few regions of the study area, since the total number of connected regional patterns usually becomes enormous, we cannot investigate all of them. This means that we have not been able to know whether a detected hotspot which is obtained under certain rules, such as using the previous studies, has the truly maximum likelihood within a given study area. A zero-suppressed binary decision diagram (ZDD), one approach to frequent item set mining, enables us to extract all of the potential cluster regions at a realistic computational load. In this study, we propose a hotspot detection method using ZDD-based enumeration, and apply it to sudden infant death syndrome in North Carolina. This completely new method enables us to detect a true hotspot cluster that has the truly maximum likelihood. To evaluate our proposed method, we compare the properties of that with existing methods such as flexible scan and echelon scan, and discuss their suitability for different purposes of detecting hotspot.

Flexible job shop scheduling using zero-suppressed binary decision diagrams

Flexible job shop scheduling using zero-suppressed binary decision diagrams

Boosting over non-deterministic ZDDs

We propose a new approach to large-scale machine learning, learning over compressed data: First compress the training data somehow and then employ various machine learning algorithms on the compressed data, with the hope that the computation time is significantly reduced when the training data is well compressed. As a first step toward this approach, we consider a variant of the Zero-Suppressed Binary Decision Diagram (ZDD) as the data structure for representing the training data, which is a generalization of the ZDD by incorporating non-determinism. For the learning algorithm to be employed, we consider a boosting algorithm called AdaBoost⁎ and its precursor AdaBoost. In this paper, we give efficient implementations of the boosting algorithms whose running times (per iteration) are linear in the size of the given ZDD.

Path Representation in Circuit Netlists Using Linear-Sized ZDDs with Optimal Variable Ordering

The efficient representation and manipulation of a large number of paths in a Directed Acyclic Graph (DAG) requires the usage of special data structures that may become of exponential size with respect to the size of the graph. Several methodologies targeting Electronic Design Automation problems such as timing analysis, physical design, verification and testing involve path representation and necessary manipulation. Previous works proposed an encoding using Zero-suppressed Binary Decision Diagrams (ZDDs), which has been shown experimentally to cope well when representing structural or logical paths in VLSI circuits. However, it is well known that the ordering of the variables in a ZDD highly affects its size and, therefore, the efficiency of the methodologies utilizing these data structures. In this work, we show that using a reverse topological order for the ZDD variables bounds the number of nodes in the ZDD representing structural paths to the number of edges in the DAG considered, hence, making the ZDD size linear to the DAG’s size. This result, supported here both theoretically and experimentally, is very important as it can render methodologies with questionable scalability applicable to larger industrial designs. We demonstrate the applicability of the proposed variable ordering in one such methodology which utilizes ZDDs to grade the Path Delay Fault coverage of a given test set.

A Branch-and-Price Algorithm for Parallel Machine Scheduling Using ZDDs and Generic Branching

We study the parallel machine scheduling problem to minimize the sum of the weighted completion times of the jobs to be scheduled (problem Pm∥∑wj Cj in the standard three-field notation). We use the set covering formulation that was introduced by van den Akker et al. [van den Akker J, Hoogeveen J, van de Velde S (1999) Parallel machine scheduling by column generation. Oper. Res. 47(6):862–872.] for this problem, and we improve the computational performance of their branch-and-price (B&P) algorithm by a number of techniques, including a different generic branching scheme, zero-suppressed binary decision diagrams (ZDDs) to solve the pricing problem, dual-price smoothing as a stabilization method, and Farkas pricing to handle infeasibilities. We report computational results that show the effectiveness of the algorithmic enhancements, which depends on the characteristics of the instances. To the best of our knowledge, we are also the first to use ZDDs to solve the pricing problem in a B&P algorithm for a scheduling problem. The online supplement is available at https://doi.org/10.1287/ijoc.2018.0809 .

The robust machine availability problem – bin packing under uncertainty

We define and solve the robust machine availability problem in a parallel machine environment, which aims to minimize the number of identical machines required while completing all the jobs before a given deadline. The deterministic version of this problem essentially coincides with the bin packing problem. Our formulation preserves a user-defined robustness level regarding possible deviations in the job durations. For better computational performance, a branch-and-price procedure is proposed based on a set covering reformulation. We use zero-suppressed binary decision diagrams for solving the pricing problem, which enable us to manage the difficulty entailed by the robustness considerations as well as by extra constraints imposed by branching decisions. Computational results are reported that show the effectiveness of a pricing solver with zero-suppressed binary decision diagrams compared with a mixed integer programming solver.

Diagnozowanie błędów w systemach złożonych na podstawie analizy niezawodności oraz danych z czujników z uwzględnieniem niepewności epistemicznej

This paper presents an information fusion method to diagnose system fault based on dynamic fault tree (DFT) analysis and dynamic evidential network (DEN). In the proposed method, firstly, it uses a DFT to describe the dynamic fault characteristics and evaluates the failure rate of components using interval numbers to deal with the epistemic uncertainty. Secondly, qualitative analysis of a DFT is to generate the characteristic function via a traditional zero-suppressed binary decision diagram, while quantitative analysis is to calculate some importance measures by mapping a DFT into a DEN. Thirdly, these reliability results are updated according to sensors data and used to design a novel diagnostic algorithm to optimize system diagnosis. Furthermore, a diagnostic decision tree (DDT) is obtained to guide the maintenance workers to recover the system. Finally, the performance of the proposed method is evaluated by applying it to a train-ground wireless communication system. The results of simulation analysis show the feasibility and effectiveness of this methodology

DenseZDD: A Compact and Fast Index for Families of Sets †

In this article, we propose a succinct data structure of zero-suppressed binary decision diagrams (ZDDs). A ZDD represents sets of combinations efficiently and we can perform various set operations on the ZDD without explicitly extracting combinations. Thanks to these features, ZDDs have been applied to web information retrieval, information integration, and data mining. However, to support rich manipulation of sets of combinations and update ZDDs in the future, ZDDs need too much space, which means that there is still room to be compressed. The paper introduces a new succinct data structure, called DenseZDD, for further compressing a ZDD when we do not need to conduct set operations on the ZDD but want to examine whether a given set is included in the family represented by the ZDD, and count the number of elements in the family. We also propose a hybrid method, which combines DenseZDDs with ordinary ZDDs. By numerical experiments, we show that the sizes of our data structures are three times smaller than those of ordinary ZDDs, and membership operations and random sampling on DenseZDDs are about ten times and three times faster than those on ordinary ZDDs for some datasets, respectively.